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Modeling, Simulation and Control with Modelica 3.0 and Dymola 7

1. Modelica Electrical Analog Sources SineVoltage S V 220 freqHz 50 Modelica Electrical Analog Basic Resistor R1 R 100 Modelica Electrical Analog Basic Resistor R2 R 100 Modelica Electrical Analog Basic Capacitor C C 0 001 Modelica Electrical Analog Basic Inductor L L 0 001 Modelica Electrical Analog Basic Ground G equation connect S p R1l p connect S p R2 p connect L n G p connect G p C n connect G p S n connect R1 n C p connect R2 n L p end SimpleCircuit When selecting Simulation Simulate in the toolbar of Dymola the above circuit is translated into C code the C code is compiled into object code the object code is loaded into the simulation environment and then the circuit is simulated and the result can be visualized in the plotting environment of Dymola It is now ana lyzed how the translation process is performed The first step is to map the Modelica model above into a set of equations by a replacing component declarations with the component equations from the small library above and b replacing all connect statements with corresponding equations see section 2 2 The result is shown in the next table Table 12 2 Basic equations of SimpleCircuit model 0 Rli tRli 0 R2i R2 i RI kia RIv RI v R2 R2 u R2 v R2 v Rlu RIR RLi R2 u R2 R R2 i Draft Jan 21 2009 12 1
2. type potential flow connector name Modelica library icon variables _ variables magnetic magnetic magnetic Modelica_Magnetic Interfaces PositiveMagneticPort al pot flux translational distance force Modelica Mechanics Translational Interfaces Flange_a et rotational angle torque Modelica Mechanics Rotational Interfaces Flange_a 3 dim mechanics 1 3 T 3 3 f 3 t 3 Modelica Mechanics MultiBody Interfaces Frame_a thermal temperature heat flow Modelica Thermal HeatTransfer Interfaces HeatPort_a E E na rate ldi h pressure mass flow aim thermo rate Modelica_Fluid Interfaces FluidPort_a fluid flow stream variables h Xi C signal connectors Real none Modelica Blocks Interfaces RealInput Output be signal Integer Modelica Blocks Interfaces IntegerInput Output gt gt Boolean Modelica Blocks Interfaces BooleanInput Output gt gt electric digital aa RoRS Modelica Electrical Digital Interfaces DigitalSignal E state machine Booleans none Modelica StateGraph Interfaces Step_in Transition_in p signal bus configurable none Icon Modelica Icons SignalBus SS Column type is the physical domain column potential variables defines the variables in the connector that have no flow prefix column flow variables defines the variables with a flow prefix column connector name defines the connector class name where t
3. Si Sj Equality of positions Qi Qj Equality of angles T T Equality of temperatures Di DPj Equality of pressures In order that the above boundary conditions are generated from connect equations the corresponding variables have to be defined as potential variables in the connector Examples connector XXX import SI Modelica SIunits SI Voltage ME SI MagneticPotential V_mag SI Position S SI Angle phi SI Temperature Ty SI Pressure p end XXX Examples for balance equations at an infinitesimal small connection point are given in the next table Table 3 2 Examples of connector balance equations at an infinitesimal small connection point 0 i Kirchhoff s current law sum of the currents is zero 0 x Sum of the magnetic fluxes is zero 0 gt f Force balance sum of the cut forces is zero 0 b2 T Torque balance sum of the cut torques is zero 0 ba OF acs Energy balance of heat transfer sum of the heat flow rates is zero 0 y M pow Mass balance of a fluid sum of the mass flow rates is zero 0 gt Gane Momentum balance of a fluid sum of the momentum flow rates Cina M oy V is zero The above equations can easily be derived by formulating the corresponding balance equation for a finite volume and then making the limit for an infinitesimal small volume For example Newton s momentum bal ance in one dimension states that ms gt fi gt
4. SI Angle phi annotation Dialog group Initialization __Dymola_initialDialog true SI AngularVelocity w annotation Dialog group Initialization __Dymola_initialDialog true SI AngularAcceleration a annotation Dialog group Initialization __Dymola_initialDialog true end Inertia This definition leads to the following parameter menu Draft Jan 21 2009 4 5 Initialization fails 71 Figure 4 11 Parameter menu of model Inertia using the __Dymola_initialDialog annotation Parameters J kg m2 Moment of inertia Initialization phi start M deg Absolute rotation angle of component w start rad s Absolute angular velocity of component der phi a start T rad s2 Absolute angular acceleration of component der w The Modelica standard annotation Dialog group Initialization defines that an input field of the corresponding variable is placed in a group with the name Initialization in the parameter menu The Dymola specific annotation __ Dymola_initialDialog true defines that this input field is for the start attribute of the variable and that additionally a check box for the f ixed attribute shall be added Note the start attribute is still undefined with these definitions and therefore if either phi w or a is se lected as iteration variable and no start value is provided when using model Inertia then Dymola prints a warning message
5. A end 1 A end end product e i J Returns the product of the scalar expression e i j evaluated for for 1 in u all combinations ofi in u j in v j in v e u l v 1 e u 2 v 1 e ufend v 1 e u end v end The type of product e i j for i in u j in v is the same as the type ofe i j Example product A i j B j i for i in 1 n1l j in 1 n2 236 Chapter 22 Literature Draft Jan 21 2009 Chapter 22 Literature Adams R J and S Banda 1993 Robust Flight Control Design Using Dynamic Inversion and Structured Singular Value Synthesis EEE Transactions on Control Systems Technology 1 2 80 92 June 1993 Arzen K E R Olsson and J Akesson 2002 Grafchart for Procedural Operator Support Tasks Proceedings of the 15th IFAC World Congress Barcelona Spain Astr m K J and B Wittenmark 1997 Computer Controlled Systems Theory and Design Prentice Hall 3rd edition Barton P 1992 The Modelling and Simulation of Combined Discrete Continuous Processes PhD thesis University of London Imperial College Bauschat M W Mo nnich D Willemsen and G Looye 2001 Flight testing Robust Autoland Control Laws In Proceedings of the AIAA Guidance Navigation and Control Conference Montreal CA Benveniste A P Caspi S Edwards N Halbwachs P le Guernic and R de Simone 2003 The Synchronous Languages 12 Years L
6. and its derivative v are measured The goal is to move the load to a specified horizontal position s2 For a non linear disturbance observer the inverse model from s to fis needed since s is measured The system is first linearized around the stationary position where the rope hangs vertically down g 0 The transfer function from f to s has 2 conjugate complex zeros on the imaginary axis signaling an undamped oscillation of the inverse model which is not desired This can be improved by including linear damping d in the revolute joint for the inverse plant used in the controller If the damping constant d is large enough the two zeros on the imaginary axis are moved to the negative real axis The disturbance observer is able to force the plant that does have low damping moving in such a way as if there would be high damp ing Draft Jan 21 2009 19 8 Difficulties with Inverse Models 221 The major goal is to position the load i e to control the horizontal position s2 of the load Therefore the feedforward control should use the inverse model from sz to f The transfer function from f to sz of the lin earized model has no zeros and a relative degree of 4 Constructing the inverse model from the non linear plant model requires however a filter of order 2 instead of 4 as suggested by the linearized model Simulat ing the inverse model results in a division by zero if 0 or g 180 To sum
7. fr osci f ns 12 13 Do Ea Ma f 0 u u4 uo fs 0 u u Uu U fs 15For example it is not possible to solve f4 for z2 since z is already computed in the previous step from f and because fz depends only on zs this is the only possible way to compute z2 158 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 The variables can be categorized as states x state derivatives dx dt and algebraic variables y du uy t x d y 12 14 l2 di U4 dt i Variables that appear differentiated x are assumed to be known and the unknowns are dx dt and y The functional dependency from the unknowns is therefore given by the left column below 0 f u i 0 f u u Rri w fin 0 fs uz u4 u uu 0 f3 du dt ii sorting BLT 0 f6 u uz u4 solving for the u u u i 0 f di dt u results in 0 filu i P i am R 0 f u u4 0 fli i dudt ilC 0 felu ur ua 0 f d izl dt us di ldt u lL Note that fz depends only on w during sorting because i is a state and therefore assumed to be known Sort ing or BLT transformation results in the middle column Using the actual equations it is then possible to solve for the unknowns in a forward sequence without algebraic loops as shown in the right column above Several algorithms are known to transform to BLT form see e g Duff et al 1986 The mostly used approach splits the pr
8. A B Comment cD gt C D Model Path Modelica_LinearSystems Sampled StateSpace Comment Continuous or discrete state space system block Parameters system 3 Continuous linear time invariant system OK Info Cancel Here either the name of a StateSpace record can be given or when clicking on the table symbol at the right end of the input field another menu pops up in which the 4 matrices can be defined as shown in Figure 17 12 Figure 17 12 Parameter menu of StateSpace record General Add modifiers Component r icon Name system StateSpace es om Comment Model Path Modelica_LinearSystems StateSpace Record containing the data of a Comment linear time invariant differential equation system in state space form der x A x B u y C x D u By clicking again on the table symbol on the right side of one of the input fields a matrix editor appears that allows to conveniently define the corresponding matrix Some of the blocks of the Sampled library have only a pure discrete representation such as the Fil terFIR UnitDelay Sampler ADconverter DAconverter and Noise blocks The continuous rep resentation of these blocks is defined to be y u i e the output is identical to the input which means that these components are effectively removed in the continuous mode 190 Chapter 18 Hierarchical State Machines StateGraph Library D
9. Conceptually the PositiveImpulse operator is mapped to the following equation this map ping and the following example is from document hybrid2 pdf of H Elmqvist and M Otter from the min utes of the 21st Modelica design meeting in Detroit The two operators are available in Dymola but require an activation flag in order that they can be used Contact Dynasim info dynasim se for more information Draft Jan 21 2009 11 2 Experimental operators to describe impulsive behavior 147 0 if edge condition then v expr else y This equation states that y has an infinite large value at the time instant when the condition becomes true such that the unknown time integral of y is determined so that v expr after the impulse At other time instants y 0 The PositiveImpulse operator is demonstrated at the following ex ample of two colliding mass points Figure 11 3 Two colliding mass points der x1 vl der x2 v2 ml der v1 f m2 der v2 f v rel v2 vl f PositiveImpulse x2 lt x1 v rel e pre v_rel The solution is performed in the following way In principal this is just standard impact mechanics see e g Pfeiffer Glocker 1996 However formulated in a more general framework based on Dirac function algebra and the Pantelides algorithm Case 1 edge x2 lt x1 false der x1 vl input xl x2 vl v2 der x2 v2 ml der vl f der x1 vl
10. Real Izz final unit m4 Geometrical moment of inertia Izz Real It final unit m4 Geometrical torsional moment It end BeamCrossSection record BeamData parameter BeamCrossSection crossSection end BeamData Basically the BeamData record consists of sub record crossSection to define all characteristic numbers of the cross section area that are needed to describe the flexibility of the beam To require this data directly from the user as stated above would be very inconvenient For this reason the actual implementation shown in Figure 5 12 adds an annotation to define a pop down menu record BeamData parameter BeamCrossSection crossSection annotation choices choice BeamChoices rectangle choice BeamChoices squarepipe choice BeamChoices circle JJ end BeamData The choices annotation is a standard Modelica annotation For every choice xxx definition an entry in the pop down menu is defined If an entry is selected interactively the corresponding choice definition is used as modifier For example if the third entry is selected above interactively then the input ar guments of function BeamChoices circle are shown in a menu and after clicking OK a modifier is gen erated with the given inputs For example ifa diameter of 0 04 is defined then the following code is gen Draft Jan 21 2009 5 5 Records 103 erated
11. Real v quantity velocity unit m s start 3 fixed true If this would be the only way to declare attributes such as units this feature would be not often used since it is too much work to build such a declaration manually For this reason Modelica provides type classes in order that a modeler can define own types that are derived from the 4 basic types Examples type Angle Real final quantity Angle final unit rad displayUnit deg type Torque Real final quantity Torque final unit N m type Mass Real final quantity Mass final unit kg min 0 type Pressure Real final quantity Pressure final unit Pa displayUnit bar nominal 1e5 In parenthesis every attribute can be defined If the attribute is prefixed with final it cannot be modified anymore Without the final attribute an attribute can be modified when using this type in a declaration A user defined type class is utilized in the following form parameter Mass m 2 parameter Angle phi 0 1 Pressure p start le6 fixed true displayUnit mPa Note that the displayUnit attribute of Pressure is modified from bar to mPa This is possible since displayUnit in the type definition of Pressure was not prefixed with final A Modelica simulation environment may or may not support all of the attributes In Dymola the unit attributes are uti lized in
12. b as parameter Integer nb max na size b 1 in order that vector b cannot be defined larger as a Unfortunately by default Dymola only prints a warning message if the min and or max attributes are violated Only if the non default option Simula tion Setup Debug Min Max assertions All variables is selected then an error occurs if the bounds de fined by the min and or max attributes are not fulfilled Additionally the error or warning message in the above context is not so easy to interpret because nb is not a variable in the public interface For these reasons in the model above another approach is used Vector bb is defined so that the decla ration is correct even if b has more elements as a Furthermore an assert statement is added that triggers an error during translation because the assert condition can be evaluated during translation since na and nb are dimensions that are fixed during translation and provides a meaningful error mes sage in its second argument The essential part of the controller canonical form is implemented with an if clause since the standard case here the else branch is not correct if the block has no states i e if the dimension of vector x is zero Dymola evaluates an if clause during translation provided the if condition is a parameter expression Only in this special case the number
13. becomes true When the measurement signal y m becomes larger as y ref b it switches back to inactive and the out Tee put u becomes false Again the method of figure Figure 9 1 leads to the following implementation 134 Chapter 9 Finite State Machines Draft Jan 21 2009 active pre inactive and y m lt y ref b or pre active and not y m gt y reftb inactive not active u active active and therefore u not pre u and y m lt y reftb or pre u and not y m gt y ref tb 9 2 Event iteration Both for the StateGraph library and the manual implementation discussed in the previous section event itera tion takes place at every event instant Basically all equations of the respective state machine are evaluated to perform usually one transition Since at least for one state pre state state the equations are again evaluated after setting pre state state The event iteration terminates until no transition fires any more This approach has several drawbacks 1 It is not guaranteed that the event iteration terminates This is critical and not acceptable for a real time control system 2 Even if the event iteration terminates the number of iterations until it terminates is usually not known in advance This is also critical for a real time control system 3 The complete model i e also the continuous parts is evaluated in every iteration which means that after
14. peR f eR 12 3 where is time p are known constants or parameters u are known input signals x are variables that ap pear differentiated in the model equations i e the der operator is applied on them and y are all other unknown variables Variables x are called potential states because the states in the ODE form are ei ther the full vector x of the DAE form above or a subset of it Variables y are also called algebraic variables because no derivatives of these variables appear in the model equations In order to simplify notation the explicit dependency of the model equations from constants parameters and input signals is usually not shown in the DAE form because these are just known variables that do not pose any problems Formally these variables are treated as a special case of the explicit dependency of time t of the DAE form As a result the following shortened DAE form will be utilized in the sequel 0 f x t x t y t t tEIR xER yelR fEIR 12 4 The primary goal is to transform this DAE into the ODE form X t _ elx i e computing X and y from the potential states x and the actual time instant t For an ODE integration algo rithm it is then sufficient to just pass x to the integrator and to hide the algebraic variables y from the integra tor The transformation from DAE to ODE form therefore requires solving a non linear system of equations 0 f x y 12 6 fo
15. ramp2 duration 2 T 0 1 The input output prefixes do not define computational causality i e that an input variable needs to be provided from the outside and that the output variables are computed The reason is that the most basic prin Draft Jan 21 2009 3 4 Connectors for Block Diagrams Blocks library 51 ciple of Modelica is that only equations are defined and that the Modelica tool is responsible to provide an efficient and correct evaluation order of the equations Therefore input output prefixes in connectors only restrict how connectors can be connected together This is e g utilized for inverse models see Chapter 19 where the outputs of blocks are treated as known and the inputs are computed An exception are connectors on the top level of a model Here these prefixes define the interaction with the environment where the model is used Especially the input prefix defines that values for such a vari able have to be provided from the simulation environment and the output prefix defines that the values of the corresponding variable can be directly utilized in the environment e g when exporting a model to Simulink the top level prefixes define the inputs and outputs of the corresponding Simulink model If these prefixes are not used in a connector but still in a model the interpretation is different The output prefix does not have a particular effect and is ignored The input prefix defines t
16. 12 Chapter 1 Introduction Draft Jan 21 2009 Chapter 1 Introduction 1 1 Composition Diagrams Modelica supports both high level modeling by composition of components and low level modeling by im plementing basic components with equations Models of standard components are typically available in mod el libraries Using a graphical model editor from a Modelica environment a model can be defined by draw ing a composition diagram also called schematics by positioning icons that represent the models of the components drawing connections and giving parameter values in dialog boxes Constructs for including graphical annotations in the Modelica language make icons and composition diagrams portable between dif ferent tools An example of a composition diagram of a simple drive train is shown in Figure 1 1 the screen shots are from Dymola Dymola 2008 Figure 1 1 Modelica schematic of a simple drive train consisting of a controlled motor a gear box and a load inertia controller inertia 1 pn i i J 10 resistor inductor hoer R 0 L 0 inertia flange i p e emf J 0 001 x 9f T ground A composition diagram consists of the following elements e An iconic graphical representation of a physical component such as an electrical motor a gear box a pump or a resistor e Interfaces that allow the coupling between c
17. algorithm if u gt uMax then y uMax elseif u lt uMin then y uMin else y u end if end Limiter A function is defined in a similar way as a block i e it has public and or protected sections to declare all used variables All variables in the public sections must have the prefixes input output parame ter or constant A function has exactly one algorithm section In this algorithm section nearly all operations are allowed that are also allowed in an algorithm section of a model or block e g it is pos sible to use array operations if clauses for and while loops It is however not allowed to use the der operator and operators associated with event handling such as the pre operator It is not allowed to modify the input arguments inside a function In the Limiter example above this means that variables u uMax uMin are not allowed to be used as left hand side variables of an assign ment operator As a result of this restriction a Modelica tool is free to decide whether the actual calling mechanism of a function is performed with call by value i e actual values are copied or call by refer ence i e the addresses of the actual values are copied For efficiency reasons it is important that arrays are passed with call by reference which is the case for Dymola in order that only the addresses of arrays are copied and not all array elements A function is executed by calling the
18. can be eas ily performed manually resulting in Zi Zi dig Z Za 0 f a z2 10000 The solution can be directly derived 0 fale i 0 f z i r se 1100 0 1 Solve f for z2 es ees 2 1011 0 2 Solve f4 for z 0 fs 21 Za Zs 09111 0 3 Solve f and f for z and zs 0 f 23 24 00101 4 Solve fi for zs From the BLT form the solution can be directly derived by solving the functions in a meaningful order so that one variable at a time can be computed In the left column above the variables that are computed from the corresponding functions are marked in bold font and are underlined For example z can be computed from 4 since z is already computed in the previous step from f In some cases e g fz and f above two or more variables can only be computed together In such a case a non trivial algebraic loop is present As a result in the example above an algebraic loop of 5 equations is transformed in to a sequence of operations Draft Jan 21 2009 12 2 Sorting BLT 157 where 3 scalar equations are solved for one variable per equation and an algebraic loop with 2 equations has to be solved Obviously the solution problem was simplified by this transformation Since a Modelica translator knows the details of every equation further symbolic processing is possible In many cases there is a linear relationship of the unknown variable in the corresponding function For ex ample assume that f is linear in z which is m
19. flying When flying true the ball is flying and is described by the equation der v g When the ball remains on the ground flying false and the model equations are changed to h 0 However this would require to dynamically change the number of state variables during simulation since flying false would have no states any more The current Model ica tools cannot handle such a situation and therefore the approximation is used that der v 0 instead Since h is close to zero when the model structure is changed it will remain close to zero when the second derivative der v is zero and therefore this solution is a reasonable approximation It remains to discuss when to switch between the defined model structures In the general case when im pact can occur between more than two parts the switching can be described by a finite state machine but the number of states of the finite state machine grows quickly with the number of potential impacts so this is not a good solution but the best what can be currently implemented in Modelica In the simple case of the bouncing ball we have only two discrete states flying false and flying true and one transi tion between these two states when the height is negative and the ball velocity is negative then the ball would fly through the ground and therefore at this point the model structure is changed to flying false Therefore it is easy to
20. function since transformed with f2c to C the function needs to be defined as FORTRAN 77 and not as C in Modelica If the copy operation of arrays shall be avoided to enhance efficiency the FORTRAN function needs to be rewritten before transformed to C by exchanging indices of the arrays in the FORTRAN code Usually this will not be an option since too much work and the risk of errors is high 5 7 Modelica as Scripting Language not yet available 5 8 Component Arrays not yet available 5 9 Component Coupling by Fields not yet available 5 10 Replaceable Models not yet available 107 Part B Modeling of Discontinuous and Variable Structure Systems All numerical integration algorithms require that the functions to be integrated are continuous and differen tiable up to a certain order In this part it is discussed how systems are handled with the Modelica language and in a Modelica simulation environment where this fundamental prerequisite of an integrator does not hold If a physical system is modeled macroscopically detailed enough all variables are continuous and the following considerations are not necessary Discontinuities are introduced due to simplified assumptions e To reduce the simulation time e g because the integrator does not have to follow a very rapid change of a variable if this rapid change is approximated by a discontinuity e To reduce the identification effort e g because the parameters
21. ol 11 12 It is not possible to inquire in a function which of the input and or output arguments are not provided from the outside Therefore always all output arguments must be computed inside the function assuming that all input arguments are provided If it is computationally expensive to calculate one or more of the output ar guments a function could have an additional option argument that allows the caller of a function to decide whether output arguments shall actually be computed in the function or whether only a default value shall be returned For example the function f above could be called as 01 02 11 12 computeOutput3 false and the function could be implemented as function f input Real il i2 input Boolean computeOutput3 true output Real ol 02 03 0 algorithm ol O2 f weF if computeOutput3 then Oo3 t 7 end if end f As a demonstration for a function implementation in the following example the function returns the product of two polynomials A polynomial is described by a vector of its coefficients e g polynomial p 2 x 2 3 x 4 is defined by its coefficient vector a 2 3 4 function polynomialMultiply input Real a b vectors of polynomial coefficients output Real c size a 1 size b 1 1 protected Integer na size a 1 Integer nb size b 1 Integer ne nl n2 1 algorithm for i in 1l nc lo
22. x 3 0 x 1 TF Diy x 1 Ol Array reduction functions a nd operators reduces an array or several scalars to one value min A Returns the smallest element of array expression A min x y Returns the smallest element of the scalars x and y min e i Jj Returns the smallest value of the scalar expression e i j evaluat for 1 an u y ed for all combinations of i in u j in v Example j in v min A i j B j i for i in 1l nl j in 1 n2 max A Returns the largest element of array expression A max x y Returns the largest element of the scalars x and y max e i J Returns the largest value of the scalar expression e i j evaluated for 1 an u forall combinations of i in u j in v Example j in v max A i j B j i for i in 1 nl j in 1 n2 sum A Returns the scalar sum of all the elements of array expression A A 1 1 A 2 1 A end 1 A end end sum e i J Returns the sum of the expression e i j evaluated for all combi for i in U nationsofi in u j in v j in v aliens pelL tetta oe AULI Pct e u end v 1l e ufend v end The type of sum e i j for i in u j in v isthe same as the type ofe i 3 Example sum A i j B j i for i in l nl j in 1 n2 product A Returns the scalar product of all the elements of array expression A A 1l l1 A 2 1
23. yd This means that the continuous equations of the plant are evaluated again but this time using the just com puted controller output as actuator input Since the sample operator is only active once at a time in stant c edge c false and therefore the controller output just keeps its value i e yd pre yd The event iteration is finished since yd pre yd after the model evaluation and the continu ous integration is restarted Note that the above procedure can be interpreted as solving the original non linear system of equations by a fixed point iteration scheme and terminate after one iteration It is clear that in general this will give a different result as when using say a global convergent Newton method and iterate until convergence Since the fix point iteration describes the desired solution of an infinitesimal small computational time delay it is the correct model and the original model with the algebraic loop has to be viewed as erroneous Some types of controllers especially in a car are most of the time not acting A typical example is an ABS anti blocking brake system algorithm that is only acting when the brake is pressed and the car starts to slide In reality the controller is implemented as a sampled data system Since an integrator has to be restart ed after every sample instant a simulation would be very inefficient if such a controller would be modeled as sampled data system an exception is
24. 11 12 21 223 TA Skar 3214 Real M2 3 4 M1 v1 3 v2 31 32 3 6 The operator is a bit more general as presented here In the Modelica specification it is precisely 76 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 defined to concatenate arrays along the first and second dimension The general case is not often needed and therefore we restrict our description to the straightforward applications shown above 66 99 The colon operator is provided to construct regular vectors Integer vl 1 4 same as 1 2 3 4 Integer v2 1 2 7 same as 1 3 5 7 If the operands of the colon operator are all of type Integer the result is an Integer vector If one of the operands is of type Real the result is a Real vector The colon operator is often used to define an index range e g to extract an array out of another array Real M3 2 2 M2 2 3 3 4 M2 from above i e M3 2 5 3 6 Real v3 3 M2 2 1 4 M2 from above i e v3 21 22 2 5 As demonstrated a matrix is generated if the colon operator is applied to 2 index ranges here to construct M3 and a vector is generated if the colon operator is only applied to one index range here to construct v3 The i11 operator is provided to construct arrays of the desired type and dimensions and to fill the ar ray with one value for each element Real M4 111 2 0 2 3 M4 2
25. 21 2009 Operator Description ndims A Returns the number of dimensions k of array expression A with k gt 0 size A i Returns the size of dimension i of array expression A where i shall be gt 0 and lt ndims A size A Returns a vector of length ndims A containing the dimension sizes of A Conversion functions between dimensions scalar A Returns the single element of array A size A i 1 is required for 1 lt i lt ndims A vector A Returns a 1 vector if A is a scalar and otherwise returns a vector containing all the elements of the array provided there is at most one dimension size gt 1 matrix A Returns promote A 2 if A is a scalar or vector and otherwise returns the ele ments of the first two dimensions as a matrix size A 1 1 is required for 2 lt i lt ndims A Specialized array constructor functions identity n Returns the n x n Integer identity matrix with ones on the diagonal and ze ros at the other places diagonal v Returns a square matrix with the elements of vector v on the diagonal and all other elements zero zeros ny Ny we Returns then x n x Integer array with all elements equal to zero 1 2 n gt 0 ones n n Return then x n x Integer array with all elements equal to one n gt 0 fill s nj n Returnsthen x n x array with all elements equal to scalar or ar ray expression s
26. Gaussian elemination with partial pivoting Inputs A Matrix A of A x b b Vector b of A x b Info Copy Call Execute Close It is more convenient to use instead the vector matrix editor by clicking on the table symbol to the right of the input field marked with a red ellipse in Figure 5 9 This opens the menu in Figure 5 10 Draft Jan 21 2009 5 4 Functions 93 Figure 5 10 Array editor of input field of a function or parameter menu in Dymola First the number of rows and number of columns are defined and then all elements of a vector or matrix can be easily provided After clicking OK the vector or matrix is copied in Modelica notation in to the corre sponding input field The result is shown in Figure 5 11 after both matrix A and vector b are defined in this way Figure 5 11 Resulting menu after defining the input arguments in the array editor Modelica Math Matrices solve 2 xi f solve Description Solve real system of linear equations A x b with a b vector Gaussian elemination with partial pivoting Inputs A 1 2 3 4 BS gt MatixAofA x b b 5 11 ES gt Vector b of A x b Info Copy Call Execute Close After clicking OK the function call is copied in to the command window of Dymola and is executed Re member that the construction of the menu for the input arguments of a function call is performed automati cally by Dymola based on
27. However during simulation z will become zero and then a division by zero will take place in the tearing transformed equation When the original equations 2 and 3 are simultaneously solved together this problem does not occur and the equations have a unique solution In other words this application of the tearing trans formation is erroneous Due to this problem nearly all known tearing algorithms introduce some kind of pivoting in order to guarantee that the rank of the system is not changed by the tearing transformation However it is then no longer possible to perform the tearing transformation symbolically once during trans lation Instead the numerical values of the pivots have to be watched during simulation and when the tearing transformation is no longer appropriate in the example above when z comes too close to zero a new selec tion of tearing variables and residue equations has to be determined dynamically see e g Duff et al 1986 Mah 1990 In that case the tearing approach is no longer attractive and other sparse matrix methods are most likely to be more effective Fortunately for Modelica models more information is available as just the information whether a vari able is present in an equation or not It is possible to construct tearing algorithms where the details of the equations are taken into account in order to select tearing variables and residue equations so that it is guaran teed that the transform
28. Matlab A record consists of a set of declarations without equation or algorithm sections For example every medium in the Modelica Media library is defined with the Modelica Media Interfaces PartialMedi um FluidConstants record to identify the medium record FluidConstants Standard data of fluid String iupacName Complete IUPAC name String casRegistryNumber Chemical abstracts sequencing number String chemicalFormula Chemical formula according to Hill String structureFormula Chemical structure formula Modelica SIunits MolarMass molarMass Molar mass end FluidConstants Records can be derived by inheritance from one or more other records They can be instantiated in a similar way as a model to provide actual values for their elements via a modifier If an instantiated record is pre fixed with parameter it appears in the parameter menu of the corresponding class Examples record TwoPhaseFluidConstants Standard data for two phase media import SI Modelica SIunits extends FluidConstants SI Temperature criticalTemperature SI AbsolutePressure criticalPressure SI MolarVolume criticalMolarVolume SI Temperature triplePointTemperature end TwoPhaseFluidConstants package Water constant TwoPhaseFluidConstants fluidConstants ilupacName oxidane casRegistryNumber 7732 18 5 chemicalFormula H20 structureFormula H20 molarMass 0 018015268
29. Mod elica Media and Modelica_ Fluid libraries and for many inverse systems see Chapter 19 constructed with any library Since the Pantelides algorithm together with the Dummy Derivative method are the only known methods to transform generic singular DAEs into a regular form every Modelica simulation environ ment that shall be able to simulate models constructed with the Modelica Standard Library has to support these two algorithms or a variant of them Let us analyze how other simulation environments treat singular DAEs e Modelica simulation environments such as Dymola www Dynasim se MathModelica www mathcore com OpenModelica www ida liu se pelab modelica OpenModelica html support the Pantelides algorithm and the dummy derivative method i e they can handle singular DAEs with discontinuous variable changes e SIMPACK www simpack com is a software system to simulate complex multi body systems with rigid and flexible bodies The mechanical equations can be integrated numerically as a singular DAE index 2 formulation At an event instant singular DAEs are always transformed to regular DAEs using the well known transformation techniques of mechanical systems Therefore SIMPACK can handle certain classes of singular DAEs with discontinuous variable changes e The MathWorks www mathworks com provides the block diagram simulator Simulink and blocksets such as SimMechanics and SimPowerSystems Simulink supports a subse
30. SI units and recommendations for the use of their multiples and of certain other units are de fined see Figure 2 3 Figure 2 3 Predefined unit definitions in package Modelica SIunits gt Stunts Modelica STunits Modelica Text lolx G Eile Edit Simulation Plot Animation Commands Window Help la SUQGNM movan Hh E gt eSB B Z 100 J Space and Time ct er 1 of ISO 31 1992 a Packages type Angle R 5 Math final quan amp Mechanics j OMedia final quantity olidAngle final unit sr OStunits l1 quantity Length final unit m Users Guide s Conversions Angle SolidAngle Length PathLength Position Distance Breadth Height Thickness Radius Diameter z R 2 el gt Ler 1 min 0 ngth min 0 1al quantity Area final unit m2 l final quantity Volume final unit m3 final quantity Time final unit s Wye l final qu final unit Line 1 Modeling There are several possibilities to use the types from the Modelica Standard Library With full name parameter Modelica SIunits Mass m 2 Modelica SIunits Velocity v start ll w LS With shortened name import Modelica SIunits parameter SIunits Mass m 2 reuse last name of import statement SIunits Velocity v start 3 With user defined name import SI Modelica SIunits parameter SI Mass m 2 SI Velocity v start 3 With t
31. The key to the solution is the observation that v 0 in the stuck mode and when forward sliding starts but der v gt 0 when sliding starts and der v 0 in the stuck mode see above figure Since the friction charac teristic in the above figure at zero velocity is no functional relationship again a parameterized curve descrip tion with a new curve parameter s_a has to be used leading to the following equations note at zero veloci ty startFor sa gt 1 startBack sa lt l a der v a if startFor then sa 1 else if startBack then sat l else 0 f if startFor then fO else if startBack then f0 else f0 sa At zero velocity these equations and the equation of the block form again a mixed continuous discrete set of equations which has to be solved at event instants similarly as in the simple rectifier circuit discussed above When switching from sliding to stuck mode the velocity is small or zero Since the derivative of the con straint equation der v 0 is fulfilled in the stuck mode the velocity remains small even if v 0 is not ex plicitly taken into account By this well know procedure the velocity v remains a state in all switching configurations Consequently v is small but may have any sign when switching from stuck to sliding mode if the friction element starts to slide say in the forward direction one has to wait until the velocity is really posi tive before
32. The result is shown in Figure 19 16 In order to not have a jump in the cooling temperature a filter order of 3 instead of 2 is actu ally used The cut off frequency of the filter is set to 1 300 Hz It turns out that a simple P controller is suffi cient to stabilize the system A controller gain of 20 is selected Draft Jan 21 2009 19 7 Example Application 217 Figure 19 16 Control system with nonlinear feedforward controller for mixing unit step fitter startTime 0 Simulation results are shown in Figure 19 17 for a jump of c des 0 492 to 0 237 The straight lines correspond to the nominal case where the plant and the inverse plant model have the same parameters The result is a good control behavior The dashed lines correspond to the case where the parameters of the plant are 50 higher as the parameters of the inverse plant model to check the robustness of the design only pa rameter was not changed because the result is very sensitive to it As can be seen the result is still satisfac torily For an actual design it is useful to perform Monte Carlo simulations by varying all model parameters and initial conditions of the plant systematically in order to determine the robustness of the control system Figure 19 17 Simulation results of mixing unit of Figure 19 16 Blue lines same model parameters for plant and inverse plant model Red lines model parameters of plant are enlarged by 50 PlantvvithSOPercent
33. c3 Complex cl c2 cl c2 Streams print c3 Complex String c3 Streams print c3 Complex String c3 i In the declaration the two record instances c1 and c2 are defined and are added in the algorithm section to construct c3 The value of record c3 is printed with the Complex String functions in different forms resulting in the following print out c3 5 7j c3 5 7i Once operator overloading is available the above statements can be simply written as C3 tanel AZ Streams print c3 String c3 Streams print c3 String c3 i It is then also possible to write Complex j Complex j Complex c4 2 5 j Complex c5 cl c2 c3 c4 In record Math Complex the basic operations for complex numbers are provided In a similar way all other data structures of the Modelica_LinearSystems library are defined They can all be at once used in current Modelica environments and they will be conveniently usable once operator overloading is available in Modelica and in Modelica tools Record Math Polvnomial defines a data structure for polynomials with real valued coefficients and pro vides the most important operations on polynomials Its content is displayed on the right side A Polynomial is constructed by the command Polynomial coeff Vector where the input argument provides the polynomial coefficients in descending order For example the polyno mi
34. every transition the inputs are newly measured and the outputs are newly applied on a continuous plant In reality this is not the case because new inputs are only measured and outputs are newly applied only if the iteration in the state machine is finished The first of these issues could be fixed by evaluating the state machine equations only once at every sample instant For example the basic equation for one state could be changed to when sampling then state inStatel and inCondi or inState2 and inCond2 or inStateN and inCondN or pre state and not outCondl or outCond2 or or outCondM end when and sampling is defined at the highest hierarchical level as sample 0 T i e generating time events every T seconds This means that at every sample instant the equations of a state machine are evaluated just once which in turn implies that usually at most one transition fires The severe drawback of this approach is that the sample period will significantly restrict the step size of the continuous integration although most of the time the output signals applied to the continuous plant will not change The efficiency of the event iteration can be improved with the following approach For the sorting algo rithm BLT partitioning the pre values are assumed to be known The idea is to perform the event iteration not over all model equations but only over the minimal part This part can be determined by assum in
35. gt 1 changes its value However assume that u gt 1 and u is becoming smaller as 1 then the relation u gt 1 changes its value but it will not influence the result since y remains on the upper branch and therefore no event should be generated In principal it is possible to detect such a situation since e g not pre high and u gt 1 is defined and therefore this expression is false whenever pre high is true independently of the value of the relation u gt 1 In the Modelica simulation environment Dymo la Dymola 2008 unnecessary events of this type are avoided The equation for high could also be de fined with the following equation high u gt 1 or pre high and u gt 1 This formulation has the slight drawback that an event will always be triggered when u gt 1 changes its values Therefore unnecessary events will occur since a Modelica simulation environment cannot detect during translation that some events will not have an effect on the result The equation for the Boolean variable high is not obvious How to derive such equations systematically is explained in section XXX To check whether the equation for high is correct the different possibilities are analyzed e pre high false and u lt 1 high true and false or false and false high false i e y remains on the lower branch e pre high false and 1 lt u lt 1 high true and false or fa
36. i e to potentially use either p or T or both as iteration variables or to compute the two variables in a previous step by other equations It is not possible to use p as an iteration variable because e g 9 would then be computed as p pl R T 12 20 Therefore potentially a division by zero can occur In reality this is not possible because a suitable media model will never be evaluated for T 0 K However it is impossible to deduce this property from the model equations and therefore every reliable symbolic tearing algorithm has to assume that T may become zero and that therefore p can not be selected as iteration variable There are more suitable options for a tool if the gas 164 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 law is directly written in the form 12 20 The reason is that the modeler divides by T It is therefore the modeler s responsibility that this variable never becomes zero and for the gas law this is actually the case A tool can now either use p or p as iteration variable because in both cases the rank of the Jacobian of this equation is not reduced when solving for the remaining unknown If the modeler uses p and T as state vari ables which is a natural choice for gases then this is a good choice because potentially p is computed di rectly from this equation The choice of 12 19 is not as good because a tool would have to divide by the state T to compute p and a reliable tool
37. if condition then exprla exprlb expr2a expr2b else expr3a expr3b else expr3a expr3b 0 if condition then expr2a expr2b expr4a expr4b else expr4a expr4b end if It is mapped to two equations having if expressions After this mapping the two equations might be sorted independently to each other i e in the generated code the two equations need not be present one after the other A more general if clause is the following parameter Integer levelOfDetail min 1 max 3 Modeling level equation if levelOfDetail 1 then 0 u2 ul else L der i u2 ul end if Here different equations are selected depending on parameter levelOfDetail Modelica has the special semantics that the branch of an if clause must be selected during translation if all conditions of an if clause depend on parameter expressions i e expressions with parameters constants and or literals In such a case it is no longer possible to set new values to the parameters used in the if conditions after transla tion The reason for this behavior is that Modelica simulation environments are currently not able to handle situations as above where either an algebraic or a differential equation shall be selected A Modelica simula tion environment is typically not able to handle this selection a switch between an algebraic and a differen tial equation during simulation since the symbolic pre processing is different
38. locally balanced Rule 2 Restrictions on Number of Equations locally balanced The number of equations in a non partial model or block must be number of unknowns number of inputs number of flow variables Note Variables with the constant and parameter prefix are treated as known variables model and block instances are not taken into account for the equation count due to rule 3 below but variables inside a connector or record instance are taken into account The equation count is demonstrated on the basis of a few examples Draft Jan 21 2009 2 7 Balanced Models 39 connector Pin Modelica SIunits Voltage Vv flow Modelica SIunits Current i end Pin model Capacitor parameter Modelica SlIunits Capacitance C Modelica SIunits Voltage u Pine py we equation OS Pek ob naky u p v n v C der u p i end Capacitor The Capacitor model has 5 unknowns u p v p i n v n i and 2 flow variables p i n i It is therefore required that this model has 5 2 3 equations and the model fulfills this requirement model VoltageSource input Modelica SIunits Voltage u Pin p n equation u p v n v OOS o Hb Inky end VoltageSource The VoltageSource model has 5 unknowns u p v p i n v n i 2 flow variables p i n i and J input variable u It is therefore required that this model has 5 2 1 2 equations and the model ful fills this requirement With the above two
39. m2 der v2 f gt der x2 v2 v_rel v2 vl der v1 0 f 0 der v2 0 v_rel v2 vil f 0 Case 2 edge x2 lt x1 true der x1 vl der x2 v2 ml der vl f m2 der v2 f v_rel v2 vl v_rel e pre v_rel These are 6 equations in 6 unknowns Since the last two equations are two equations for one unknown the Pantelides algorithm and the dummy derivative method have to be applied which requires differentiating the last two equations However v_re1 is a discontinuous function Differentiating a discontinuous function re sults in a Dirac impulse 5 t t0 at time t0 We use the well known formula that differentiating v t with discontinuity at tO left limit v1 and right limit v2 results in the following value of the derivative at t0 v to v v to 11 3 Therefore differentiating the two equations above results in der v_rel der v2 der v1 der v_ rel 1 e pre v_rel dirac t0 148 Chapter 11 Re initialization of Continuous Variables Draft Jan 21 2009 Applying the dummy derivative method results in a set of 8 equations in 8 unknowns at the time instant when edge x2 gt x1 is true input xl x2 vl pre v_rel dder v rel lte pre v_ rel dirac t0 der v1 m2 dder v_rel m1 m2 dder v2 dder v_ rel der v1 m2 dder v2 v_rel e pre v_rel v2 PS Bel tyb der x1 vil der x2 f 23 6699 where dder v is
40. n gt 0 The returned array has the same type as s linspace x1 x2 n Returns a Real vector with n equally spaced elements such that v linspace x1 x2 n v i x2 x1 i 1 n 1 forl lt i lt n It is required that n gt 2 The arguments x1 and x2 shall be scalar Real or In teger expressions xl Matrix and vector algebra functions transpose A Permutes the first two dimensions of array A It is an error if array A does not have at least 2 dimensions If A is a matrix transpose A is the trans pose of the matrix outerProduct v1l v2 Returns the outer product of vectors v1 and v2 matrix v transpose matrix v symmetric A Returns a matrix where the diagonal elements and the elements above the di agonal are identical to the corresponding elements of matrix A and where the elements below the diagonal are set equal to the elements above the diagonal of A i e B symmetric A gt B i j Afi j if i lt j B i j A j i if i gt j Draft Jan 21 2009 21 6 Array Operators 235 Operator Description cross x y Returns the vector cross product of the vectors x and y i e cross x y vector x 2 y 3 x 3 y 2 x 3 y 1 x 1l yI 3l x l y 2 x 2 yl 1 skew x Returns the 3 x 3 skew symmetric matrix associated with a 3 vector i e cross x y skew x y skew x O x 3 x 2
41. number of event iterations becomes larger as a predefined value say a maximum of 20 iterations This is of course not acceptable when using a Modelica model in a real time system Therefore in the current con troller applications of Modelica the real time model is usually a sampled data system since event iteration does not occur here It would be useful to introduce a better solution in a future Modelica release to widen the application area of Modelica in the real time area Historical note The basic feature of Modelica to solve a system of differential algebraic and discrete equa tions at every time instant and at every event instant was developed by Hilding Elmqvist based on the same principle used for continuous equations in the Dymola language Elmqvist 1978 and for discrete equations in the SattLine language Elmqvist 1992 This basic idea was further developed and improved in Elmqvist et al 1993 Otter et al 1999 Elmqvist et al 2001 3Non linear equations may have multiple solutions However starting from appropriate defined initial conditions the desired solution is usually uniquely identified 132 Chapter 9 Finite State Machines Draft Jan 21 2009 Chapter 9 Finite State Machines In many applications a user would like to define a discrete algorithm graphically as a finite state machine e g as a Petri net a Sequential Function Chart SFC or a Statechart In this chapter hints are given h
42. priate place together with the initialization of the left hand side variables when invoking the algorithm section As another example the algorithm section above with the two while clauses has three variables identical i j that appear on the left hand side of the assignment statements Therefore this algo rithm section is treated as a set of three coupled equations of the form for simplicity the dependence on the pre values of the output variables is not shown identical i j A B where the part informally means the set of variables that is identical to the vector expression on the right hand side During sorting see Chapter 12 this set of equations is treated as three independent equa tions of the form 0 fl identical i j A B 0 f2 identical i j A B 0 3 identical i j A B This means that these three equations will appear as part of an algebraic loop and therefore remain togeth er during sorting If the evaluation of an algorithm section depends on the start value or on the pre value of a variable this usually indicates that the user has made a mistake or at least that it is possible to formulate this algorithm section in a cleaner and easier to understand way It would be helpful if a Modelica tool would indicate this in an information message if this situation can be detected during translation currently neither Dymola no
43. v t of v where is infinitesimally small e The values of Integer Boolean String discrete Real and enumeration variables can only be changed at event instants It is impossible to change their values during continuous integration Only non discrete Real variables can change their value during continuous integration Note a variable declared as Real is treated as discrete Real if it appears on the left hand side of an equation or statement in a when clause e An event is processed in zero simulated time i e Modelica has instantaneous communication If this is not desired e g because the actual computing time of a sampled data controller shall be taken into account it has to be explicitly defined e g by triggering two events Draft Jan 21 2009 Chapter 6 Discontinuous Equations 109 e It is not defined whether two or more events occur at the same time instant If time synchronization of events is needed it has to be explicitly programmed e g by accessing the same Boolean variable that signals an event or by explicitly implementing a counter if say a slow sampling event shall occur at every 5th fast sampling event e It is not possible to determine the time instant of the last accepted step of the integrator In some special cases such a feature might be useful but it is always a hack with the danger of severe errors and there are usua
44. when sample 0 T then yd a pre yd b ud end when uc pre yd By this construction an infinitesimal small time delay is introduced that as a side effect removes the alge braic loop The basic idea is that the equations of the controller are evaluated once at the sample instant and that the computed actuator signal yd is afterwards applied to the plant Event iteration takes place to restart the continuous integration but during this iteration the controller equations are no longer evaluated Let us analyze this situation in more detail by inspecting the sorted system of equations eliminating again the two trivial connection equations Sorted equations with input yc ref xc pre yd uc pre yd der xc xc uc yc g xc uc c sample 0 T yd if edge c then a pre yd b yc_ref yc else pre yd A time event is triggered by the sample operator The sorted model equations from above are evaluat ed at the event instant As actuator input uc the output pre yd of the previous sample instant is used and the continuous equations of the plant model are evaluated especially the measurement signal yc g xc pre yd Since sample triggered an event c true edge c true and therefore the controller equation is evaluated using the measurement signal yc as an input Since yd pre yd at the end of the model evaluation the sorted model equations are again evaluated after setting pre yd
45. 0 fi 3 1 In order that the above balance equations are generated from connect equations the corresponding variables have to be defined as flow variables in the connector Examples connector XXX import SI Modelica SIunits flow SI Current i flow SI MagneticFlux Phi flow SI Force f flow SI Torque tau Draft Jan 21 2009 3 1 Connector Design 43 flow SI MassFlowRate m_ flow flow SI HeatFlowRate Q flow flow SI MomentumFlux G flow end XXX When designing a component interface an abstraction of the real physical interface has to take place Espe cially it has to be decided which physical effects shall be mathematically described From this decision fol low the boundary and balance equations that shall be fulfilled The most fundamental balance equation is the energy balance Therefore the energy balance must al ways be fulfilled directly or indirectly The energy balance is fulfilled when the energy flow rate is already used as flow variable or when corresponding potential and flow variables are used in the connector For example a suitable connector for 1 dimensional translational mechanical systems is connector TranslationalFlange Modelica SIunits Position S flow Modelica SIunits Force f end TranslationalFlange Assume that 3 components c1 c2 c3 with this interface are connected together This leads to the following connection equations cls c2s cls c3 s 3 2 0 cel f c2f c
46. 0 2 0 2 0 2 0 2 0 2 0 Boolean active f ill true 3 active true true true The first argument defines the value that should be used for all elements and the type of the array is identical to the type of this first argument The remaining arguments define the desired dimensions of the array to be constructed For two often occurring cases special operators are introduced for convenience zeros is the same as i11 0 and ones is the same as il1 1 So in the first case an array with zero elements and in the second case an array with one elements is generated With the inquiry operators ndims A the number of dimensions of array A are returned as Inte ger and with size A i the size of dimension i is returned again as Integer The keyword end as index of an array element characterizes the last element of a dimension Example Real v equation il v size v 1 sQQSM VOOCAM4 i2 v end i2 is identical to il File Edit Simulation Plot Rin Packages 2 Modelica Reference Classes model function Operators der size connect In the package browser of Dymola the library ModelicaReference is present as shown in the right figure This is a free library from the Modelica Association that contains a short refer ence to all Modelica language elements In particular sublibrary Op erators contains
47. 21 2009 19 1 Inversion of Linear SISO Models 207 19 1 6 Design of Prefilter It remains to be discussed how the prefilter R shall be constructed For minimum phase plants R is the refer ence transfer function T The optimal result would be 7 1 since then the output signal would follow exact ly the reference signal This is usually not possible because the relative degree of R must be larger or at least as large as the relative degree of plant P which is usually gt 0 A suitable choice is therefore often a low pass filter where the filter order is identical to the relative degree of P and the cut off frequency of the filter is as large as possible The plant model P used in the feedforward path is only valid up to a certain frequency Only up to this frequency a reasonable cancellation of the plant dynamics can be expected and therefore the cut off frequency of the filter should be smaller Practically the frequency range of the plant model is often known at least a rough order and the cut off frequency of the filter is adjusted around the plant frequency range so that the control behavior is improving In Figure 17 3 on page 181 Modelica models for filters are discussed From the step response of the fil ters see Figure 19 7 it can be deduced that only the Critical Damping and the Bessel filter are suitable as a reference transfer function since otherwise the reference signal would have unwanted vibrations Fig
48. 75 A matrix can be constructed with the matrix constructor 11 12 13 parameter Real M1 2 3 fit u2 133 21 22 23 SEMI Fe 22 Real M2 1 3 row matrix equation M2 time 2 time 3 time which means that elements in a row are separated by a required comma and different rows are separated by a required semicolon An array of any dimension including vectors and matrices can be constructed with the array construc tor This is achieved by assembling arrays of dimension n along a new first dimension and con structing thereby a new array with n 1 dimensions Examples use 2 scalars to build a vector parameter Real v 2 l 2 use 3 vectors to build a matrix parameter Real M 3 2 v 2 v 3 v 7 use 4 matrices to build an array with 3 dimensions parameter Real A 4 3 2 M 2 M 3 M 4 M In the example it is also shown that an array can be multiplied with a scalar according to the usual mathe matical scalar array operation where all elements of the array are multiplied with the scalar The array constructor can also be nested e g parameter Real M 3 2 1 2 2 4 3 6 1 27 2 4 3 6 In Modelica scalars vectors matrices arrays of 3 dimensions arrays of 4 dimensions etc are all different data types Arrays with different dimensions can only be used in an operation together if a cast operator gen erate
49. Beam beam data crossSection BeamChoices circle diameter 0 04 The different functions to define the cross section are provided in a separate sub library package BeamChoices function circle input Modelica SIunits Length diameter 0 02 input Integer numberOfPoints min 4 8 output BeamCrossSection crossSection redeclare Real polyLine numberOfPoints 2 import Modelica Constants pi protected Modelica SIunits Angle phi algorithm crossSection area pi diameter 2 4 crossSection Iyy pi diameter 4 64 crossSection Izz crossSection lyy crossSection It 2 crossSection lyy for i in l numberOfPoints loop phi 2 pi i 1 numberOfPoints crossSection polyLine i diameter 2 sin phi cos phi end for end circle end BeamChoices Function BeamChoices circle has diameter and numberOfPoints as input arguments and re turns an instance of the BeamCrossSection record In the function all elements of this record are com puted and assigned The record contains a vector of type Real numberOfPoints 2 i e the dimen sion of this vector changes according to the input argument numberOfPoints It is a bit tricky to define the actual dimensions of an array that is used in an output record of a function Currently the only possible way in Dymola is to use a redeclaration see section 5 10 where the declaration of the array is given as modifier to the output argument with t
50. CriticaDaming Bessel Butterworth Chebyshev Af FilterFIR Discrete finite impulse response low or high pass filters J Integrator Output the integral of the input signal continuous or discrete block f Derivative Approximate derivative continuous or discrete block t FirstOrder First order continuous or discrete transfer function block 1 pole J4 SecondOrder Second order continuous or discrete transfer function block 2 poles 1b PI Proportional Integral controller continuous or discrete block Delay the input by a multiple of the base sample time if discrete block or 4 i st UnitDela i y u if continuous block Sampler Sample the input signal if discrete block or y u if continuous block AA ADconverter Analog to digital converter including sampler 4p DAconverter Digital to analog converter including zero order hold TH Noise Generates uniformly distributed noise in a given band at sample instants if discrete and y 0 if continuous As an example block Sampled StateSpace is shortly described Its icon is shown in the figure at the left When double clicking on the block the parameter menu shown in Figure 17 11 pops up Draft Jan 21 2009 17 3 Continuous Discrete Control Blocks 189 Figure 17 11 Parameter menu of Sampled StateSpace block General Advanced Options Add modifiers stateSpace Component picon StateSpace A B Name stateSpace
51. Default packages in the File Libraries menu Dymola Dynamic Modeling Laboratory Diagram amp File Edit Simulation Plot Animation Commands Window Help e New pertu zu gt BB Open Ctrl O F A z z Demos gt Modelica Standard Library a Save Ctrl S Model Management Save As Design Save All Plot 3D Save Total Vehiclelnterfaces version 1 1 Version Modelica_LinearSystems A user can conveniently extend this menu with his her own definitions This is performed by storing file libraryinfo mos in directory lt package directory gt Scripts or in lt package directory gt For ex ample in the Modelica_LinearSystems library this file is stored as Dymola Modelica Library Modeli ca_LinearSystems libraryinfo mos When Dymola is started it determines the libraryinfo mos files of all packages in the search path and uses this information to build up the Library menu As an exam Draft Jan 21 2009 2 6 Model Libraries 37 ple the libraryinfo mos file of the free Modelica_LinearSystems library is shown File Modelica LinearSystems libraryinfo mos LibraryInfoMenuCommand category libraries text Modelica_LinearSystems referenc Modelica_LinearSystems version 0 95 isModel true description Modelica_LinearSystems Library 0 95 ModelicaVersion 3 0 pos 1001 LibraryIn
52. If this option is selected missing initial conditions are selected so that a subset of the equations der x 0 is added to the initialization equation system Again the equation subset is selected so that the initializa tion problem is not structurally singular This means that a model is initialized in steady state if no initial conditions are provided and if this option is enabled Example model FirstOrderInSeries parameter Real k parameter Real T RealInput u RealOutput yl start 1 fixed true RealOutput y2 equation T der y1 yl k u T der y2 y2 yl end FirstOrderInSeries This model has 2 variables appearing differentiated y1 y2 and therefore 2 initial conditions have to be provided we assume that the initial value of the input signal u is defined somewhere else In the model only 1 initial condition is defined and therefore 1 initial condition is missing If the above option is set Dy mola will add the initial equation der y2 0 Note it is not possible to select the other potential ini tial equation der y1 0 because otherwise the second differential equation cannot be properly initial ized and since this is a structural property it can be detected during the translation phase As a result the fol lowing 4 equations define the initialization problem yl 1 der y2 0 T der yl yl k u T der y2 y2 yl This system of equations has
53. In case an if branch is selected during translation i e when all conditions are parameter expressions the restrictions that every if clause must have an else clause and that the number of equations in every branch must be the same do no longer hold since only the selected branch is used for the symbolic pre processing If a basic model has different modeling levels it is often implemented with if clauses where the condi tions are parameters as in the previous example with parameter levelOfDetail see e g Kuhn et al 2008 Since Dymola selects the branch during translation and then performs symbolic preprocessing on this branch the resulting code is as efficient as it can be for every modeling level If the sketched special semantics is not desired an if expression can be used For example Dymola will not select the branch during translation if the above example is written as 0 if levelOfDetail 1 then u2 ul else u2 ul L der i Here the branch is selected during simulation and levelOfDetail might be a variable that changes dur ing time In this particular example the simulation will fail because Dymola is currently not able to handle such a drastic change of a model during simulation to switch between an algebraic and a differential equa tion The condition of an if expression clause consists of relations that are build with 13 39 66 gt gt lt lt greater greater or
54. T T 0 0000 0 0002 0 0004 0 0006 0 0008 0 0010 time s The upper red curve in Figure 5 3 belongs to y2 and is wrong The first action in such a case is to simulate with a higher tolerance i e the number of correct digits is higher Simulating y2 with a tolerance of le 6 gives the correct result of the lower blue curve in the figure The reason of this behavior can be seen when plotting the state of the y2 transfer function see Figure 5 1 Figure 5 2 The state x 1 of the y2 transfer function is in the order of 10 1 2E 4 state x 1 of y2 10000 s 10000 u 8 0E 5 4 0E 5 0 0E0 T T T T 0 0000 0 0002 0 0004 0 0006 0 0008 0 0010 time s The state x 1 is in the order of 1e 4 As will be explained in Chapter 13 the step size control of a variable step size integrator is controlled by the relative tolerance which defines the desired number of correct dig its and the absolute tolerance which defines when a variable is so small that the integrator can consider it as negligible When a variable is smaller as its absolute tolerance the step size control for this variable is switched off The relative tolerance is defined in the Simulation Setup General Tolerance menu of Dymola The absolute tolerance is defined implicitly by the product of the nominal value the relative tolerance and a simulator specific value e g absoluteTolerance r
55. abs a end else 1 Real bb vector zeros max 0 na nb 1 b Real d bb 1 a 1 initial equation der x zeros nx equation assert nb lt na Transfer function is not proper size b 1 lt size a 1 required if nx 0 then y d u else der x 1 a 2 na x x_scale u a 1 der x 2 nx x 1l nx 1 y bb 2 na d a 2 na x x_ scale d u end if end TransferFunctionWithScaling Note since a end might be zero the scaling is disabled if a end is too small 5 2 For Loops In equation sections for loops are used to iterate over equation sets In the following example a for loop is used to build up a filter block where the order of the filter is defined by a parameter block CriticalDampingFilter Modelica Blocks Interfaces RealInput u Modelica Blocks Interfaces RealOutput y parameter Integer n min 1 Order of filter parameter Modelica SIunits Frequency f Cut off frequency output Real x n Filter states protected parameter Real w 2 Modelica Constants pi f initial equation der x zeros n Steady state initialization equation der x 1 u x 1 w for i in 2 n loop der x i x i 1l x i w end for y x n end CriticalDampingFilter This block has an input signal u and an output signal y A set of differential equations is present that de fine a unnormalized critic
56. amp File Edit Simulation Plot Animation Commands Window Help Name of new rm CEI w oo saQS M NOOV AM 4 ah Z gt OBE E x New Package E Modeling v Simulaton _ packages pes Modelica Reference pA tis la 0 Modelica Unnamed Unnamed A Open Library Window aende option ServoSystem Open Class in This Window Maor o Open Class in New Window Insert in package Unload ServoSystem Check Rename Cancel e g Search Duplicate Class m Close Extend From Component Info New Class in Package gt New Model TServoSyste Parameters Order gt New Connector New Record New Block New Function A package can be restructured by again right clicking on the corresponding package and selecting Edit The Edit menu contains all necessary operations to restructure a package depending on the context espe cially Rename Duplicate Class and Remove Note with Rename it is possible to move a class from one package to another package or from the top level in to a package Classes are shown in the package browser according to the definition order This order can be changed by clicking on a class in the package browser and moving it up or down with the arrow keys of the keyboard provided the lt ctrl gt key is pressed Alternatively the class can be moved by selecting it with the right mouse 3
57. and the user gets diagnostics for this case If an explicit start value would be provided in model Inertia then no warning message would be printed if the variable is selected as iteration variable Most likely this default start value would not be suited as guess value for the iteration variable and initializa tion might fail If it is not possible to change the library to introduce a start value dialog of the iteration variable in the parameter menu or if the selected iteration variable is too special for the problem at hand the start value has to be provided as modifier in the model to be simulated Such a modifier can be provided in the Add modi fiers tab of the parameter menu of the corresponding model instance see the example in Figure 4 12 Figure 4 12 Adding a start value as modifier in the Add modifiers tab in a parameter menu Genera Add modifiers Add new modifiers here e g phi start 1 w start 2 OK Info Cancel This introduces an input field for the start value of variable y in this parameter menu and gives the start value a default of 2 the value of 2 will then be used as guess value for the iteration variable The next time when this parameter menu is opened it is possible to directly change the start value in the parameter menu Above we have discussed how to provide guess values of iteration variables in order to define a reasonable setup of the initialization problem If initia
58. as in if then elseif clauses the number of equations in every branch of a when elsewhen clause must be identical Additionally the references on the left hand side of must be identical in all branches If every branch has n equations such a when clause is conceptually mapped to n equations and therefore the equation system becomes well defined For example the when clause above is basically mapped to one equation for openValve bl hl gt levell b2 h2 lt level2 openValve if edge bl then true else if edge b2 then false else pre openValve Currently elsewhen branches in equation sections are not yet supported in Dymola and therefore the above code fragment does not work in Dymola However Dymola supports el sewhen clauses in algorithm when h1 gt levell then openValve true elsewhen h2 lt level2 then openValve false end when This code fragment is again conceptually mapped to one equation in openValve and therefore the equa tion system becomes well defined We will now discuss the mapping of an algorithm section to a set of equations in more detail by means of the following example Integer vl Real v2 Real v3 algorithm vl exprl v2 expr2 when condition then v3 expr3 end when This algorithm section is conceptually mapped in a first phase to Integer vl Real v2 Real v3 Draft Jan 21 2009 7
59. clause v must be a discrete variable This defini tion implies that pre v is the left limit v t of variable v at event instant te e 0 if no event iteration takes place edge b Returns true when the Boolean variable b changes from false to true i e edge b b and not pre b change v Returns true when variable v changes its value i e change v v lt gt pre v and v is not of base type Real The sample operator triggers time events in a regular fashion It can e g be used to implement the Pulse block a little bit simpler model Pulse import IF Modelica Blocks Interfaces import SI Modelica SIunits IF RealOutput y parameter SI Time T discrete SI Time TO start 0 fixed true U equation when sample 0 T then TO time end when y if time lt TO T 2 then 1 else 0 end Pulse The equations in a when clause are discrete equations that are only evaluated at an event instant but not during continuous integration Therefore only the already explained restricted form of equations v expr are allowed in a when body where at the left hand side of the equal sign a variable reference must be present The variable computed by a discrete equation has to be explicitly identified contrary to a contin uous equation When the when clause is not active all discrete variables computed in the when body all variables referenced o
60. common properties in this case the declaration of an input and an output signal In fact such a super class is available as Modelica Block s Interfaces SISo partial block SISO Single Input Single Output block Modelica Blocks Interfaces RealInput u input Modelica Blocks Interfaces RealOutput y output end SISO The transfer function is then implemented as block TransferFunction extends Modelica Blocks Interfaces SISO parameter Real b 1 Numerator coefficients parameter Real a 1 Denominator coefficients output Real x size a 1 1 State of transfer function protected parameter Integer na size a 1 teger nb size b 1 na l zeros max 0 na nb 1 b parameter Int parameter Integer nx Real bb vector bb 1 a l Real d initial equation der x zeros nx equation assert nb lt na Transfer function is not proper size b 1 lt size a 1 required if nx 0 then y d u else der x 1 a 2 na x u a 1 der x 2 nx x 1l nx 1 y bb 2 na d a 2 na x d u end if end TransferFunction The model is defined as block in order that it can only be used in a connection structure according to Draft Jan 21 2009 5 1 Arrays 79 block diagram semantic The coefficients of the numerator and denominator polynomial are defined with pa rameter vectors b and a t
61. complex pairs in order that their product re sults in a polynomial with Real coefficients A description with zeros and poles is problematic For example a small change in the imaginary part of a conjugate complex pole pair leads no longer to a transfer function with real coefficients If the same zero or pole is present twice or more then a diagonal state space form is no longer possible This means that the structure is very sensitive if zeros or poles are close together Performing arithmetic operations on such a de scription therefore leads easily to transfer functions with non real coefficients For this and other reasons the constructor transforms this data structure and stores it internally in the record as first and second order poly nomials with real coefficients T stay TY s ny 5 2 y k u 17 7 1 s au TL s dy st d5 All functions operate on this data structure It is therefore guaranteed that an operation results again in a transfer function with real coefficients It is possible to construct a ZerosAndPoles record directly with these coefficients by using function fromFactorized This data structure is especially useful in applications where first and second order polynomials are nat urally occurring e g as for filters With function filter this factorized form is directly generated from the desired low and high pass filters of type CriticalDamping Bessel Butterworth or Chebyshev The filter optio
62. controller kernel is feedback linearization in aerospace applications also known as Nonlinear Dynamic Inversion NDI see for example Isidori 1995 and Enns et al 1994 The basic structure is given in Figure 19 11 The principal difference compared with the feedforward Figure 19 9 and the compensation controller Figure 19 10 is that the states in the inverse model are obtained from the actual plant via measurement and estimation Contrary to the compensation controller the methodology can also be applied to unstable plants Figure 19 11 Feedback Linearization Controller feedback inverse u Z yO gt Lik i i controller plant model piant controller When deriving feedback linearizing control laws manually the outputs to be controlled are differentiated un Draft Jan 21 2009 19 5 Inverse Model in Feedback Linearization Controller 213 til an analytical relation with a control input is found For this process usually Lie algebra is used The number of required differentiations is the so called relative degree of the specific output If the system model is available in Modelica the derivation of the control laws can be automated using a similar procedure as described above However instead of a filter of appropriate relative degree a set of inte grators is added see above y gt i 19 18 where v is the i new model input corresponding wit
63. criticalTemperatur 647 096 criticalPressure 22064 0e3 criticalMolarVolume 1 322 0 0 018015268 triplePointTemperature 273 16 model MediumBaseProperties Modelica SIunits SpecificHeatCapacity R Gas constant equation R Modelica Constants R fluidConstants molarMass 100 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 end MediumBaseProperties gt end Water The record instance waterConstants above is a constant record where every element of the record is assigned a value The elements of a record instance can be accessed by dot notation e g fluidCon stants molarMass The above construction shows how to build up structured data that is defined once and cannot be modified afterwards since defined as constant 5 5 2 Record constructor function Every record definition has an associated function called record constructor function It has the same name and is in the same scope as the record definition All components of the record are input arguments of the function with exception of components defined as constant or as final parameter and the sin gle output argument is an instance of the record where all input arguments are assigned to the record ele ments Example record Complex Complex number extends Modelica Icons Record Real re Real part Real im Imaginary part end Complex function add input Complex ul u2 o
64. desired For example in order to fill a tank usually several actions are necessary e g to close one valve and to open another one In a SFC all actions to fill a tank would be defined as actions to a fill a tank step and this might be more convenient for the user For example copying or deleting a fill a tank step would require only a change at one place in a SFC whereas it would require changes at several places ina StateGraph 18 2 1 Example In Roussel and Faure 2002 an example of a simple safety related program is given that is verified with a new method The implementation is shown as a PLC program with function blocks from IEC 61131 3 and a newly introduced language In both cases the implementation is difficult to understand because flip flops are used that are very low level description elements This example shall be modeled with a StateGraph The specification given in the article is The aim of this program is to monitor the safe operation of two pushbuttons used to operate presses and similar dangerous machinery It ensures that both hands of an operator are kept outside the danger zone during machine operation The required properties are 1 A cycle can only be initiated by pressing the two pushbuttons simultaneously within 0 5 s 2 A cycle is interrupted by releasing one or both buttons to stop the output 3 The output signal can only be re initiated after both inputs have been released and the
65. display the name of the connector instance see right part of Figure 2 7 below In the Modelica Standard Library the convention is used to have always two identical connector defini tions that have however different icons It is then easier to identify different connector instances of a com ponent For example there are two pin definitions PositivePin and NegativePin where the first one has a filled and the second one has a non filled square icon in the Modelica Standard Library We will use these two pin definitions to build up a Modelica model of an inductor the Modelica decla rations are identical to connector Pin above Dragging the PositivePin and NegativePin of the Modelica Electrical Analog Interfaces library from the package browser into the Diagram layer of a new model results in Figure 2 7 Dragging PositivePin and NegativePin in the Diagram layer of a model p ioi x EDETEN i iolxi Modelica Text model Inductor Modelica Electrical Analog Interfaces PositivePin pin_p 4 Modelica Electrical Analog Interfaces NegativePin pin_n 4 end Inductor Dragging the two pins in to the Diagram layer will automatically also introduce them in the Icon and the Text layer of the Inductor see figure above The Text layer of the inductor can then be edited to add the remaining declarations and equations Note when the connector definition would be just manually de
66. document e It shall provide an in depth description to use Modelica in control applications such as digital control systems safe state machines and advanced controllers based on non linear inverse plant models CAD Computer Aided Design 3 PLM Product Life cycle Management 9 Draft Jan 21 2009 This document does not explain all details of the Modelica language or of the Dymola software The Modeli ca language is formally defined in the Modelica Language Specification available at http www modeli ca org documents ModelicaSpec30 pdf or at Dymola Modelica Documentation ModelicaSpec30 pdf Fur thermore the book of Peter Fritzson Fritzson 2003 gives a very good description and explanation of all the details of the Modelica language The details of the Dymola software are described in the Dymola manuals available in a Dymola distribution at Dymola Documentation Dymola User Manual Volume 1 pdf and Dy mola User Manual Volume 2 pdf The emphasis of this document is on using Modelica to model and simulate technical systems on the ba sis of many examples All examples are provided in Modelica version 3 0 and the Dymola screen shots have been produced with Dymola 7 1 on Windows XP with the classic style option Gilching January 2009 Martin Otter 11 Part A Modeling of Continuous Systems In this part the most important Modelica language elements are introduced and explained to model continu ous physical systems
67. e g if components have very steep characteristics such as diodes in electrical circuits or pressure drop components for turbulent flow in fluid networks The problem is that in some region very small changes of the iteration variables lead to very large changes of other variables During initialization this is quite critical because the guess values of the iteration variable are not yet good enough and they might be too far away from the solu tion During simulation this is less critical because good initial guesses for the iteration variables are avail able from the solution of the last integration step and the integrator has just to decrease the step size in order to find a solution One solution strategy in such a case is to start the model to be simulated in an operating point where ini tialization is possible After initialization the sources are controlled such that they drive the system to the de sired operating point For example electrical circuits may be initialized with zero voltage sources and after initialization the voltage sources are ramped up to the desired voltages Fluid systems might be initialized with zero mass flow rates and after initialization mass flow sources or pumps are ramped up to the desired mass flow rates 4 5 3 Inconsistent redundant equations The algebraic equations appearing during initialization might be redundant without having a solution A very simple constructed example of this type is shown in the nex
68. eigenValues size A 1 2 Modelica Math Matrices eigenValues A Real realEigenValues numberOfReals eigenValues algorithm realEigenValues realNumbers2 ecigenValues end testRealNumbers2 The drawback is the inconvenient rewriting of the code and that function numberOfReals is called twice in testRealNumbers2 and in realNumbers2 and not once as in the first variant 5 4 5 Functions with internal memory Modelica functions are mathematical functions i e functions that return always the same result values when called with the same input arguments In other words Modelica functions do not have a memory This is enforced by the semantics of an algorithm section see section 5 3 All left hand side variables of the assignment operator are initialized with their start values before the algorithm section is executed if no start value is defined the default start value of 0 0 for Real 0 for Integer false for Boolean and an empty string for String types are used It is therefore not possible to access the value of a variable from the last function call If an internal memory is really needed a Modelica block has to be used instead but then a function call syntax is not available and a Modelica block cannot be used inside another function Alternatively an external C function in form of an ExternalObject can be used see section 5 6 There are the following
69. elements to synchronize events because the synchronization is automatically performed by data flow analysis since all active equations are treated as a system of equations that has to be solved concurrently and in a first step the equations are sorted Therefore if information about a triggered event has to be passed to other parts of a model or to a different model this is simply performed by passing Boolean or Integer variables e g Boolean limitReached activateController 122 Chapter 7 Instantaneous Equations Draft Jan 21 2009 equation limitReached x gt xLimit triggers event activateController edge limitReached passes event info 7 4 1 Miulti rate controller There is no guarantee in Modelica that two relations or operators that trigger an event trigger it at the same time instant For example a multi rate controller may have a base sampling rate for some fast parts of the controller and a multiple of this sampling rate for slow parts of the controller The modeler would expect to define this as fastSampling sample 0 basicSamplingPeriod slowSampling sample 0 5 basicSamplingPeriod In most cases this will work as expected However round off errors and the particular epsilon hysteresis strategy at an event instant of a Modelica simulation environment might not give the expected result that fastSampling and slowSampling are both true at every Sth instant of the basic sampling peri
70. equal less less or equal e g xl gt x2 or ul lt u2 and that are combined by the standard Boolean operators and or not Draft Jan 21 2009 2 4 Conditional Equations 29 Hereby the standard precedence holds i e the highest precedence has a relation followed by not fol lowed by and and finally by or Example if u gt 0 or not y lt 1 and u gt 1 then identical to if u gt 0 or not y lt 1 and u gt 1 then end if The operator can be used to check equality of Integer Boolean and String variables e g Integer gear equation if gear 1 then end if For Real variables the operator is only allowed in Modelica functions see section 5 4 even in a func tion it should be avoided since the result is usually very sensitive Outside of functions e g in an equa tion section this operator is forbidden The reason for this restriction is that every relation in Modelica trig gers an event see section 6 1 in order that the model is guaranteed to be continuous during integration which is a pre requisite of every numerical integration algorithm In order that event handling is possible a relation is transformed to a zero crossing of a variable together with a hysteresis around zero As a result a relation of the form x 0 is transformed to a zero crossing function z x 0 i e whenever x crosses zero an ev
71. fluids in pipe networks such as air conditioning systems batch processes or machine cooling e generation distribution and consumption of electric power Modelica is also very well suited for advanced non linear control systems based on non linear inverse plant models Details are discussed in Part E of this document Modelica is designed such that it can be utilized in a similar way as an engineer builds a real system First trying to find standard components like motors pumps and valves from manufacturers catalogs with appropriate specifications and interfaces and only if there does not exist a particular subsystem a component model would be newly constructed based on standardized interfaces Models in Modelica are mathematically described by differential algebraic and discrete equations Modelica is designed such that available specialized algorithms can be utilized to enable efficient handling of large system models having more than hundred thousand equations It is suited and used for hardware in the loop simulations and for embedded control systems Modelica is not designed for the direct description of partial differential equations as e g performed by Finite Element FE or Computational Fluid Dynamics CFD programs However results of finite element computations are utilized in Modelica e g for the de scription of flexible bodies The Modelica language is a textual specification that describes details of a model on a high le
72. function with actual values for the input arguments If a function has only one variable declared with the output prefix the function can be called at all places where an ex pression is allowed The actual values for the input arguments can be either given in the order of the declara tions positional arguments or the actual values can be provided with binding equations named ar 88 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 guments Therefore all of the following function calls are equivalent yl Limiter u 2 2 y2 Limiter u uMax 2 uMin 2 y3 Limiter u uMin 2 uMax 2 y4 Limiter uMin 2 uMax 2 u u y5 Limiter u uMax 2 uMin 2 Actual values of input arguments need not to be provided as demonstrated with y5 Limiter u if the input arguments have default values as for uMax and uMin above If a function has several output arguments the actual output arguments must be provided in parenthesis in the same order as the corresponding variables are declared in the function Here only positional argu ments are allowed Example A function f with 2 input and 3 output arguments is called as ol o2 03 il 12 If one or more of the output arguments are not needed when calling the function they need not to be provid ed Therefore the function f could also be called in the following forms 01 02 11 12 02 03 i1 12
73. hierarchies can be constructed Assume that the Motor model above is used as object in another model e g in the drive train of Figure 1 1 When clicking on the Motor icon the parameter menu is opened The menu will be empty because no parameter is defined in the Motor model It is always possible via a hierarchical modifier to change parame ter values at every level of a hierarchy e g Motor motor inertia J 0 002 In the graphical user interface of Dymola this is performed by right clicking on _ Parameters motor and selecting Show Component see figure to the right This shows EESMAA the diagram level of motor Clicking on inertia opens the parameter menu of inertia and a value of J can be given After clicking OK the above modification is introduced at the highest model level For important parameters it is more convenient for a user to propagate all or the most important parameters from the lower to the highest hierarchy when defining the Motor model For example the Motor model above could be changed to Open Class in New Window Attributes Change class Info Edit annotation model Motor import SI Modelica SIunits parameter SI Inertia J 0 001 Motor inertia parameter SI Resistance R 0 5 Armature resistance parameter SI Inductance L 0 05 Armature inductance paramet
74. i e it rotates with the shaft The absolute angle phi between the x axis of the coordinate system fixed in ground and between the x axis of the coordinate system fixed in the shaft is the potential variable of the flange connec tor The resultant cut torque t tau in the cut plane of the shaft is the flow variable This means that at a connection point the boundary condition of identical angles as well as the torque balance is fulfilled A simi lar analysis as for the connector of the Translational library see equation 3 3 shows that additionally also the energy balance is fulfilled at a connection point When needed in a component derivatives of the flange angle can be easily derived by applying the der operator For example the angular velocity speed of the shaft is w der phi Two connector definitions are available in the Rotational library Flange a has a filled circle icon and is usu ally used for the left end of a shaft Flange _b has a non filled circle icon and is usually used for the right end of a shaft Besides the slightly different icons the remaining parts of the Modelica connector def initions are identical Angular velocity and torque are vector valued quantities Operations on vectors from different compo nents can only be carried out if the coordinates of the vectors are resolved in the same coordinate system Therefore conceptually every Rotational component ha
75. if end for x rem seedOut 1 30269 0 seedOut 2 30307 0 seedOut 3 30323 0 1 0 end random This function is a variant of the Wichmann Hill random number generator Wichmann and Hill 1983 It pro duces a uniformly distributed output argument x in the range from 0 0 to 1 0 The period how many numbers it generates before repeating the sequence exactly of this generator is 6 953 607 871 644 The built in function rem is the Integer reminder of the division of two Real numbers see Table 21 7 on page 230 The function gets the internal memory seedIn from the last call as input argument changes it and returns it as output argument seedOut In a function this random number generator might be utilized in 98 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 the following way function testRandoml input Integer seed _start 3 23 87 187 protected Integer seed 3 seed start Real x algorithm for i in 1 10 loop x seed random seed Modelica Utilities Streams print x String x significantDigits 16 end for end testRandoml In the local variable seed the internal memory of the random number generator is stored The ran dom function is called in a for loop For convenience the same variable is used for the input and out put seed In an algorithm section this can be performed because the right hand side expressio
76. if the trick from Range limited sampling in section 7 4 is used It is often a good approximation to model it in such a way that it reacts only in particular situations defined by state events thereby neglecting the sampling time An ABS algorithm is described in terms of when this happens do something when that happens gt do something else kind of procedure A straightforward model would therefore use an algorithm section with when elsewhen clauses to model the controller A very simplified model of the sketched system is shown in the next example 10 This example is from Michael Tiller Draft Jan 21 2009 7 5 Algebraic loops between continuous and instantaneous equations 127 Wrong Modelica model model ReactiveControllerl Real w a tau initial equation tau 10 equation a tau w der w a algorithm when a lt 4 then tau 0 end when end ReactiveControllerl On first view this model is uncritical because the discrete variable tau is set to zero in a when clause and then applied to the continuous equation a tau w Mapping the when clause to an instantaneous equa tion results in c a lt 4 tau if edge c then 0 else pre tau a tau w der w a However it is not possible to sort these equations in a forward sequence because a is needed to computed tau and tau is needed to compute a i e an algebraic loop is present
77. in testRandom2 is executed every 0 2 seconds once As input seed the value of the seed vector from the previous sample instant is used If noise for several mea surement signals is needed different seed vectors with different start values have to be defined since oth erwise the noise of different measurement signals would be correlated to each other would produce the same values The above test model produces the following output to Dymola s message window when simu lated for 1 second 0 672061 0 367860 time time 1 0 2 0 x QO X Draft Jan 21 2009 5 4 Functions 99 time 0 4 x 0 604109 time 0 6 x 0 166737 time 0 8 x 0 462959 time 1 0 x 0 928989 As demonstrated with the example above functions with memory can be emulated in Modelica Besides the slight inconvenient usage since the internal memory has to be passed as input and as output argument to a function call the major drawbacks are that the internal memory can be accessed in the calling environ ment and that the whole internal memory is copied for every function call As a result this approach is lim ited to cases where the internal memory has a limited size since copying large data structures for every function call would be too inefficient 5 5 Records 5 5 1 Basic properties of records Modelica records are used to structure data by name similarly to a struct in C or
78. instead of model the developer of this component gets early diagnostics when implementing the block accidentally not as component that has only input and output variables For a user of such a block it is sufficient to inspect that the block keyword is used in order to be sure that the Modeli ca translator guarantees that this component behaves as an input output block of a block diagram 3 5 Connectors for Signal Buses Signal interfaces become quickly unmanageable if many signals have to be communicated between compo nents Therefore buses are used in control systems of vehicles aircrafts drive trains power plants etc to communicate a large amount of data between components This behavior can be mimicked in an idealized way in Modelica A typical example is shown in Figure 3 8 from the free VehicleInterfaces library this library is available in Dymola under File Libraries Figure 3 8 Communication of signals in a vehicle system model of the VehicleInterfaces library driverEnvironment controlBus This is a system model of a vehicle with engine transmission drive line chassis brake accessory and driver models The many signals between these components are communicated via the connector controlBus In principal it would be possible to just define one large connector in which all desired signals are present However this gives several practical problems e If only some
79. is not known which variable x or y keeps its value during event instants and is therefore a discrete variable Slightly rewriting this equation results in correct Modelica code Correct Modelica code 2 xk 3 y time when sample 0 1 then x 4 y end when The interpretation is that during continuous integration x is known the value from the previous event in stant and that y is computed from the continuous equation as y time 2 x 3 equations solved during contin integration At an event instant also the discrete equation is active and we have a linear system of equations for x and y 2 x 3 y time equations if when clause is active x 4 y 0 Note the current version of Dymola Dymola 2008 cannot handle algebraic loops involving continuous and instantaneous equations so the above example will not translate in Dymola On the basis of some examples the semantics of a when clause has been explained The semantics is defined precisely in the Modelica specification by the following conceptual mapping rule that defines how a when clause is mapped to a set of equations when condition then c condition vl exprl ev edge c and not initial if ev then exprl else pre vl v2 expr2 D vl v2 if ev then expr2 else pre v2 vN exprN end when vN if ev then exprN else pre vN This means that every equation in a when clause is mapped to a standard equation using an if expre
80. is obtained as with the input filter of second order with the feedforward controller Starting from the ideal response k 19 31 r s s k s k 19 31 the feedback controller may also be shaped as T UN i 19 32 a S r s 1 S kys whereby in the numerator s has been added since the input of the inverse model is effectively the second time derivative of cas The over all closed loop system is depicted in Figure 19 19 Component in verseMixingUnit is the manually derived inverse model with 2 integrators according to Figure 19 18 Draft Jan 21 2009 19 7 Example Application 219 Figure 19 19 Closed loop system of mixing reactor and feedback linearization controller inverseMixingUnit gaint feedback2 Figure 19 20 shows the response of the closed loop system to the same command as in Figure 19 17 The command input has been smoothed with a first order filter When the parameters of the mixingUnit are changed by 50 as in Figure 19 17 the closed loop system becomes unstable Therefore in Figure 19 20 only parameter variations of 20 are shown Figure 19 20 Simulation results of mixing unit of Figure 19 19 Blue lines same model parameters for plant and inverse plant model Red lines model parameters of plant are enlarged by 20 PlantVyvith20PercentLargerParameters mixingUnit c NominalPlant mixingUnit c 0 4 o E 0 0 0 100 200 300 400 500 NominalPlan
81. iteration continues Useful assumptions on relation values are for example d Use the relation values computed in the last iteration and perform a fixed point iteration the convergence can be enhanced by some algorithmic improvements e Try all possible combinations of the values of the relations systematically exhaustive search In the above example both approaches can be simply applied because there are only two possible values s gt 0 is false or true However if n switches are coupled there are n relations and therefore 2n possible combinations which have to be checked in the worst case A more complicated example is shown in the next figure consisting of a rectifier circuit with 4 idealized diode models 138 Chapter 10 Variable Structure Systems Draft Jan 21 2009 Figure 10 3 Rectifier circuit with 4 ideal diode models resistor v2 R R2 aheyoa capacitor ground The diodes are connected together such that direct current is flowing through the load R2 C1 driven by an AC voltage source This circuit has essentially two operational modes If the source current i0 is positive diodes 1 and 4 are on switches are closed and diodes 2 and 3 are off If the source current is negative diodes 2 and 3 are on and diodes 1 and 4 are off In both cases the current flows from left to right through the load R2 C1 i e always in the same direction A typical simulation result can be seen in the figure be
82. just add a Boolean equation for flying that describes this behavior The result of a simulation is displayed in the next figure which shows the expected behavior 146 Chapter 11 Re initialization of Continuous Variables Draft Jan 21 2009 Figure 11 2 Simulation results of BouncingBall2 satisfactory result h 1 5 1 0 0 5 m 0 0 The above model has the slight problem that many events occur at the end before the ball remains on the ground This effect is called chattering and could be solved using the method from Fillipov Mathemati cally it is well known that this problem has a different solution An infinite number of events occur but there is a time barrier that the ball cannot cross In reality such a behavior of course does not occur The model is too idealized In reality the coefficient of restitution is not constant but depends on the impact velocity If the impact velocity is becoming smaller the coefficient of restitution also becomes smaller 11 2 Experimental operators to describe impulsive behavior In the EU project Realsim Real time Simulation of Multi physics systems Dynasim has experimented with alternatives to re initialize DAEs From a user point of view two built in operators are introduced in Dymo la Table 11 1 Experimental operators to describe impulsive behavior DiracImpulse condition Returns an infinitely large value at the time instant when con dition
83. low where the source voltage u0 and the voltage drop over the capacitance C are shown Figure 10 4 Simulation result of rectifier circuit with 4 ideal diode models Cv Vy 0 0 01 0 02 0 03 0 04 0 05 0 06 Collecting the equations of all components and connections and transforming this set of equations with the algorithms sketched above results in the state space form Draft Jan 21 2009 10 1 Parameterized Curve Descriptions 139 input t u output m s lt 0 m s lt 0 m lt 0 m s lt 0 10 1 0 m 0 m Sy u m m mM M S2 _ 0 m R 1 m m R m 1 0 a3 uo R m 1 m l 1 m 1 my S4 0 Note that s1 s2 s3 s4 are the curve parameters of the corresponding ideal diodes and m1 m2 m3 m4 are the Boolean variables that characterize the off structures of these diodes It is assumed that the value true of a Boolean variable is represented by 1 and the value false by 0 This System has the structure explained above and can be solved with one of the sketched method The technique of parameterized curve descriptions was introduced in Clauss et al 1995 and a series of related papers However no proposal was given how to implement such models in a numerically sound way In Modelica the solution method follows logically because the equation based system naturally leads to a system of mixed continuous discrete equations which have to be solved at event instants Otter
84. many domains that are based on standardized interface definitions Some typical examples are shown in the next figure Constants fea Electrical Icons Vectors Matrices AIMC1 uction have especially a look at Open Class in This Window Jerview provides an overview of the Modelica Standard Library inside the User s Guide N R lease Notes summarizes the changes of new versions of this package Open Class in New Window ftact lists the contributors of the Modelica Standard Library Edit p pdelicaStandardLibrary pdf is the complete documentation of the library in pdf format Examples packages in the various libraries demonstrate how to use the ponents of the corresponding sublibrary EilModelica Z Check i Wig Library Modeli Sirane Meche Q Search of the Modelica Standard Library consists of v Close gt Info Modeling Dymola opens then a menu in which all input arguments of the function together with the description texts are shown The actual arguments to the function can be provided in the input fields e g A 1 2 3 4 see Figure 5 9 N A Figure 5 9 Automatically constructed menu when calling function Modelica Math Matrices solve Modelica Math Matrices solve 21x f solve Description Solve real system of linear equations 4 x b with a b vector
85. of a operator when clause v must be a discrete variable This definition implies that pre v is the left limit v t of variable v and v is the right limit v t at event instant te 0 ifno event iteration takes place Other discrete operators are expressed as function of pre g edge v v and not pre v We are now in the position to state the operational model of Modelica informally Operational model of Modelica A Modelica model is mapped to a set of differential algebraic and discrete equations The differen tial and algebraic equations are solved with a numerical integration method During this phase the values of discrete variables do not change and the discrete equations are not taken into account Whenever a relation e g x1 gt x2 that is not part of anoEvent operator changes its val ue the integration is halted The halt of the integration is called event At the event instant the continuous states and the pre variables and external input variables if they appear are treated as known and their values do not change during the event All other vari ables are computed such that the differential algebraic and discrete equations are fulfilled concur rently This is usually performed by sorting the equations and by solving remaining implicit alge braic systems of equations by an appropriate numerical method If there is at least one discrete var
86. of the emf component drives the inertia of the motor In the text layer the above model is described by the following Modelica model model Motor Modelica Mechanics Rotational Interfaces Flange b flange Modelica Blocks Interfaces RealInput i telj Modelica Mechanics Rotational Inertia inertia J 0 001 Modelica Electrical Analog Basic Resistor resistor R 0 5 Modelica Electrical Analog Basic Resistor inductor L 0 05 equation connect inertia flange b flange connect inertia flange a emf flange b connect resistor n inductor p end Motor The first two declarations define the two interfaces flange and i_ ref Then all used components are de clared in the form lt model class gt lt object gt lt modifier gt where lt model class gt is the name of the model to use e g Resistor or Inertia lt object gt is the name of the local instance of this component such as resistor or inertia and lt modifier gt is a modification of the model class In the simplest form new values for parameters are provided For exam ple inertia J 0 001 means that parameter J of object inertia gets the value 0 001 In the equation section every connection line in the diagram layer is represented by a corresponding con nect statement defining the connectors that are connected together In a similar way further
87. operator s or the shift operator z For simplicity we only discuss the continuous case The relative degree D is defined as D deg d deg n 19 2 where deg p is the degree of polynomial p We assume that the plant is causal i e Dp gt 0 which means that the transfer function is proper A polynomial p is called stable if all zeros of the polynomial are inside the left half part of the complex plane i e Re zeros p lt 0 Otherwise the polynomial is called unstable A transfer function P is called stable if all zeros of the denominator polynomial dp are inside the left half part of the complex plane i e Re zeros dp lt 0 A polynomial p can be factorized in a product of the form p p p 19 3 Draft Jan 21 2009 19 1 Inversion of Linear SISO Models 203 where p contains all elementary divisors that are unstable and p contains all elementary divisors that are stable If a polynomial has no unstable zeros p 1 The zeros of the numerator polynomial p s of a trans fer function are also called the zeros of the transfer function and the zeros of the denominator polynomial de s of a transfer function are also called the poles of the transfer function A transfer function is called minimum phase if the numerator polynomial is stable A transfer function is called non minimum phase if the numerator polynomial is unstable The above well known standard definitions are demon
88. p C3 p of course also 3 equa tions have to be provided for C1 n C2 n C3 n either by connections or by the automatically intro duced default equation that flow variables are zero if the connector is not connected With this explanation the final rule follows naturally 40 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 Rule 3 Restrictions on Modifiers and Connectors a Modifiers are only allowed on variables declared with the constant parameter input prefix or a declaration equation b Modifiers must be provided for variables with the input prefix outside of a connector c For every connector of a component that is not connected and has N variables with the input prefix exactly N equations must be provided no modifiers The basic idea is that whenever a component is used all missing equations must be provided either a by appropriate connections for connectors b by providing appropriate modifiers for variables defined with the input prefix outside of a connector c by providing appropriate equations for variables defined with the input prefix inside a connector which is not connected A tool can easily check these rules and can therefore easily pin point precisely which component declaration does not follow these rules and therefore leads to a model that is not balanced It is now also clear that a model is automatically balanced if all subcomponents follow this rule because
89. possible as long as all array dimensions in the model can be determined during translation For example if the following function is called in a model ev erything should be fine since array dimensions in the model are not influenced by this function call function testRealNumbers1 input Real A size A 1 1 2 0 4 5 6 7 8 9 protected Real eigenValues size A 1 2 Real realEigenValues algorithm eigenValues Modelica Math Matrices eigenValues A realEigenValues realNumbers2 eigenValues for i in 1 size rEval 1 loop Modelica Utilities Streams print realEigenValues String i J String realHigenValues i end for end testRealNumbers1 However in Dymola 7 1 this is not yet supported and Dymola reports a warning during translation Variable realEigenValues was declared with dimension That is not yet supported in dsmodel c and the function will fail if called in the model Dymola fails when simulating this model as announced in this warning message It is possible to rewrite the function so that translation and simulation is successful in Dymola by calcu 96 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 lating the dimension of realEigenValues to get rid of the dimension declaration function testRealNumbers2 input Real A size A 1 1 2 0 4 5 6 7 8 9 protected Real
90. pseudo code that can be easily implemented in a desired programming language output set assignment function assign Integer Assigned n input Integer i Boolean Visited n output Boolean success Boolean success algorithm algorithm if a variable j of equation i exists Assigned zeros n such that Assigned j 0 then success true for i in 1 n loop Assigned j i Visited 1i11 false n else success assign i success false if not success then for every variable j of equation i error Singular with Visited j false loop end if Visited j true end for success assign Assigned j if success then Assigned j i return end if 160 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 end for end if There are two global arrays that hold the current status and they are available in both functions i Assigned j Equation i is solved for variable j If i 0 there is not yet an assignment for variable j Visited i if false then variable i is not yet visited otherwise it was already visited The main function analyzes every equation once Before starting the analysis of equation i the Visited array is initialized to false Then the central function assign is called to perform an assignment for this equation It is first tried to perform a simple assignment i e trying whether any of the unknown variables of
91. ratip 2 J 10 startTime 1 The drive train consists of a rotational inertia inertial that is connected via an ideal gear to another ro tational inertia inertia2 The driving torque is a ramp that has a discontinuous change at time 1 as shown in the figure below Note discontinuous changes of driving forces or torques occur naturally with sampled data systems 166 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 Figure 12 7 Dymola simulation results of drive train of Figure 12 6 torque tau 20 E Z 0 20 0 0 0 4 0 8 12 16 2 0 inertial a inertia2 a a v T inertial w inertia2 we w T p 15 1 0 0 5 0 0 m 0 5 0 0 0 4 0 8 1 2 1 6 2 0 Simulating this model with Dymola or another Modelica simulator is unproblematic and leads to the result shown in the Figure above Note that at time 1 s the accelerations of the two inertias change discontinu ously due to the discontinuous change of the driving torque Let us now analyze this system under the as sumption that no symbolic preprocessing would take place The system above is described by the following equations the equations of the angles are ignored for simplicity J O Taye T equation of inertial w iw equation 1 of idealGear 12 26 T3 iTi equation 2 of idealGear l J W T equation of inertia2 where are the angular velocities of inertial and inertia2 Tarve is the pre defined dri
92. system in state space form x k 1 7 A x k T B u k 7 ylk T C x k 7 D ulk 7 17 9 x k T x k T B u k T where e tis the time e T is the sample time e kis the index of the actual sample instance k 0 1 2 e u is the input vector e y t is the output vector e x t is the state vector of the continuous system from which the discrete block has been derived e x Z is the state vector of the discretized system If the discretization method e g the trapezoidal integration method accesses actual and past values of the input u e g u Z k u T k 1 u T k 2 a state transformation is needed to arrive at the difference equa tion above where only the actual value u 7 is accessed If the original continuous state vector at the sample times shall be computed from the discrete equations the matrices of this transformation have to be known For simplicity and efficiency library Modelica_Lin earSystems supports only the specific transformation as needed for the provided discretization methods As a result the state vector of the underlying continuous system can be calculated by adding the term B u to the state vector of the discretized system In fact since the discrete state vector depends on the discretization method it is a protected variable that cannot be accessed outside of the block This vector is also not needed by the user of the block because the continuous state vector is available b
93. t of the desired model at time instant t given t and the states X at this time instant This basic structure is shown in the next figure Figure 12 2 Basic structure of integration algorithms Xo Integrator t X t Model x f x t From the integrator point of view the initial states xo are provided as inputs and the integration algorithm computes the states x t at desired time instants in the future This means that the states x are the unknown variables and they are computed from the integrator From the model function point of view time instants t and the states x at this time instant are inputs to the model function This means that the states x are known variables The Model function computes the state derivatives X at this time instant To summarize Depending on the view point the states x are either known or unknown variables The main goal in this chapter is to explain how the model function can be automatically generated from a Model ica model For this reason in the rest of this chapter we take the view point that the states x are known vari ables provided from the integrator 154 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 The set of equations of the simple electrical circuit from above is mathematically described as a set of differential algebraic equations abbreviated as DAE 0 f x x y z u t p t tER xEIR yeER ueR
94. that define a rapid change of a characteristic do not have to be determined by measurements if this rapid change is approximated by a discontinuity Modelica is very well suited to describe discontinuous and variable structure systems in a numerically reli able way Still before trying to express the desired behavior with native Modelica language elements it is easier to first try to use the model classes already available in the Modelica Standard Library such as shown in Table 5 1 Table 5 4 Modelica libraries with discontinuous and or variable structure systems Modelica library Features components odelica Blocks Logical Logical blocks and or etc timer Modelica_ LinearSystems Sampled Sampled data systems discrete controllers Modelica StateGraph Hierarchical finite state machines Modelica Mechanics Rotational Coulomb friction clutches brakes Modelica Electrical Analog Ideal Ideal electrical switches ideal diodes Modelica Electrical Digital Logical blocks with 9 valued logic 108 Chapter 6 Discontinuous Equations Draft Jan 21 2009 Chapter 6 Discontinuous Equations Numerical integration algorithms approximate the solution of differential equation systems by n th order polynomials that are attached continuously and smoothly together It is therefore impossible to describe dis continuous systems precisely because polynomials do not have discontinui
95. the desired reference transfer function from y a to ye Under the assumption that the desired and the actual plant behavior is identical the inverse and the actual plant model cancel each other and the transfer function from yea to ye is identical to 1 r s i e r s of the feedback controller defines the desired closed loop behavior Note that it is assumed that y is measurable in this case y Ym Alternatively the procedure as described above may be applied integrators are added to the inverse model input before design ing the feedback controller This structure has the disadvantage that it can be applied to stable plants only Also the inverse plant model needs to be stable For a linear plant model this is obvious since otherwise an unstable pole zero can cellation occurs resulting in an internally unstable system For multi input multi output systems it is nearly always possible also for unstable plants to construct the inverse of a stationary desired plant model Once the control error e has reached a stationary value the inverse plant model leads to a decoupled control loop In other words the different outputs might be controlled independently from each other by simple PID like single input single output controllers and the stationary inverse plant model is used to decouple the control loops from each other 19 5 Inverse Model in Feedback Linearization Controller A complete theory to use nonlinear plant models as the
96. the following solution yl 1 y2 k yl der yl kl u der y2 0 4 5 Initialization fails In the previous sections it was explained how to formulate well defined initialization problems in Modelica using Dymola This is unfortunately sometimes not enough because the numerical solution of the initializa tion equations may fail In this section we will analyze reasons for the failure and will discuss what to do in order that initialization becomes successful Modelers will usually be upset if a tool claims that it cannot solve the initialization problem In many cases the tool is right that a numerical solution of the initialization problem is either very difficult or is impossible to obtain Therefore the basic remedy is to change or to im prove the definition of the initialization problem changing the tool will not help in such a case 4 5 1 Guess values for iteration variables missing Problems often occur because non linear algebraic equations are solved during initialization Especially for 70 Chapter 4 Initialization Draft Jan 21 2009 steady state initialization large non linear algebraic equation systems appear For example if the ODE 4 1 is initialized in steady state we have a system of nx coupled non linear algebraic equations with xo as un knowns iteration variables 0 f X t 4 10 Every numerical solver for non linear algebraic equations requires guess values for the used iteration vari a
97. the following way e The displayUnit attribute is displayed in the parameter menu the plot variable list and is used as part of a label in a plot window if it is defined Otherwise the unit attribute is displayed In a parameter menu the desired displayUnit can be selected from a scroll down list An example is shown in Figure 2 2 Figure 2 2 Selection of displayUnit here deg or rad in parameter menu Initialization phi start 90 gt deg w start gt rad s Pe Initialization phi start 90 gt rad w start gt The displayUnit attribute can only be selected if a literal value e g 90 is given in the input field If an expression is given e g a parameter name the unit defined in the unit attribute is used as unit If this unit is not already shown it will be shown when closing the parameter menu and opening it again The possible display units shown in the scroll down list are configured when starting Dymola by executing the script file Dymola insert displayUnit mos The user can modify this file to change the unit scroll down lists e When connecting components together the corresponding connector variables must have the same quantity and unit definitions provided they are not empty e g a connection between variables is allowed if one variable has a unit defined and the corresponding one has an empty unit e Dymola checks that
98. the pin ii A Current flowing into the pin Connection connect resistor p plug phase JV Label connection to bus connector CL Cancel After pressing OK the connection definition is finalized and results in the following Modelica code in the text layer of the test model Figure 3 17 Text layer of model in which the pin of a resistor is connected to a plug model PinToPlug Modelica Electrical Analog Basic Resistor resistor R 100 3 Plug plug 3 a r equation connect resistor p plug phase 3 end PinToPlug In a similar way a hierarchical connection from a connector to any hierarchy inside another connector is pos sible The only restriction is that the connected sub connectors must have identical definitions For reasonable simple hierarchical connectors an alternative is to provide an adapter model where the hi erarchical connector is present on one side and all sub connectors to which connections are possible are provided on the other side of the adapter model For example an adapter model for the Plug connector might be defined as shown in Table 3 10 Draft Jan 21 2009 3 6 Hierarchical Connectors 59 Table 3 10 Adapter model for Plug connector Icon layer Diagram layer name phase W phase E ground pra E Ce This adapter model consists just of a connection from phase to plug phase and from ground
99. the plant r is the reference signal i e the desired value of y e eis the control error e dis the disturbance and e nis the measurement noise The goal is to design the feedback controller C in such a way that the transfer function from the reference r to y is one and the transfer functions from the disturbance d and the measurement noise 7 to y vanish Obvi ously this is not possible There are in fact quite severe restrictions because the measurement noise and the Draft Jan 21 2009 19 1 Inversion of Linear SISO Models 205 reference signal have the same transfer function and yi as Gree 19 8 The sum of the two transfer functions is independently of C Therefore when C is constructed such that the reference transfer function T is optimized then the disturbance transfer function Ta is fixed and cannot be influenced any more This strong coupling means that the standard controller configuration of Figure 19 3 has only one structural degree of freedom The design of the controller with respect to the measurement noise n is not as critical because C can be constructed such that T has a low pass behavior with unit gain and then the effect of the high frequency measurement noise is considerably reduced Still this leads to structural limitations on the achievable performance for the reference transfer function independently what type of controller C is actually used 19 1 4 Controller with two structural degrees of freedom fo
100. this button is pressed after a stop or shut opera tion the process operation continues 2 Button stop stops the above process by closing all valves Then the controller waits for further input either start or shut operation 3 Button shut is used to shutdown the process by emptying at once both tanks by opening valve 2 and valve 3 When this is achieved the process goes back to its start configuration where all 3 valves are closed Clicking on start restarts the process A safe operation requires that the tanks never overflow One approach could be to state this proof goal for mally and then proof that the implementation of the above operations is always safe An alternative approach seems to be to define how a safe operation can be guaranteed and implement this directly The following rules guarantee that the tanks never overflow 1 When tank reaches its fill limit valvel is closed so that no new fluid enters tank1 Draft Jan 21 2009 18 4 Composite Steps 199 2 When tank2 reaches its fill limit valve2 is closed so that no new fluid enters tank2 Due to the single assignment rule in Modelica there must be exactly one equation for valvel and for valve2 respectively It is then sufficient to define this equation e g for valvel in the following form valvel lt normal operation gt and not levell gt maxLevel By this implementation it is guaranteed that the tank never overflows indepen
101. to olug ground Note that the plug connector in the icon layer has the double size of a pin icon since the Plug connector was defined with Scale 0 2 in the icon layer The small demo model from Figure 3 15 can be implemented with the adapter model in the following way Figure 3 18 Connecting the pin of a resistor to a Plug connector with an adapter model plugAdapter plug resistor 60 Chapter 4 Initialization Draft Jan 21 2009 Chapter 4 Initialization In this chapter it is discussed how models are initialized in Modelica and Dymola before a simulation is start ed 4 1 Mathematically defining the initialization problem A Modelica model is a declarative description of the desired abstraction of the physical world Such a model can be utilized for various purposes The most important usage but not the only one is the solution of an initial value problem For an ordinary differential equation in state space form abbreviated as ODE an ini tial value problem is mathematically defined as x 4 f x t 0 x t t x tER xelR 4 1 This means that the ODE x f x t is numerically solved starting at the initial time fp and from the given initial value of the state x If no initial values xo would be defined the ODE has an infinite number of solu tions and therefore the problem description is not well defined Consequently initial values xo must be de fined directly or indirectly before a simula
102. to 19 12 might differentiate equations see section 12 4 the known inputs of the inverse model may be differentiated too i e the derivatives of these inputs must exist and must be provided analytically up to a certain order These derivatives can be provided if e g the inputs are available as analytic functions that can be differentiated sufficiently often or by a desired reference mod el that in combination with the inverse DAE results in a DAE that does not require derivatives of inputs Of ten the reference model is selected as a filter such that a combination of the filter states yields the needed derivatives For linear systems this approach is well known see section 19 1 Take for example the follow ing linear system with one zero and two poles eee eres y s 2 s43 u 19 13 The inverse model together with a reference model might be constructed as _ s 2 s 3 1 ETT Peel ae In order that the model is causal i e can be implemented as an algorithm additional poles have to be added until the degree of the denominator is larger or at least as large as the degree of the numerator For this reason a filter 1 Tst 1 has been connected in series making the combined transfer function causal Another possibility is not to control y but one of its derivatives instead POENTE s s 1 The transfer function has now a relative degree of zero and may be inverted s 2 s 3 aes 19 1 i s s 1 f oe Conne
103. to the symbolic pre processing DASSL would be not suited for many Modelica models if it would be directly applied to the original equations before the symbolic pre processing takes place The reasons for this behavior will be dis cussed in the next sections 156 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 12 2 Sorting BLT In this section the most fundamental algorithm for Modelica models called Block Lower Triangular trans formation will be discussed this algorithm is abbreviated as BLT Before applying this algorithm a trivial simplification is first carried out by substituting alias variables of the form a b or a b This means equations of the form a b a b a b 0 etc are removed and variable a is re placed at all occurrences by b or b Modelica models contain a lot of such alias variables by construc tion because every connect definition leads to such equations Therefore the reduction of the num ber of equations by this trivial substitution is usually substantial Note that the number of operations is not increased by this type of variable substitution In order to simplify notation we collect the unknown variables together in vector z oe 12 9 The goal is to solve the non linear system of equations for the unknowns z under the assumption that time t and state variables x are known quantities and the Jacobian with respe
104. too many scalar conditions To correct it you can remove Initial equations equation der s 0 v 0 The reason is that v der s and therefore the same initial condition is given twice After removing one of the two equations a warning message will be printed that one initial condition is missing Therefore another round is needed to supply the last missing initial condition If the redundancy of the provided initial conditions is not a structural property it is not possible to de tect this defect during translation Note if redundant equations are present in most cases either an infinite number of solutions exists or no solution at all It might be that the numerical initial equation solver will still find a solution It is however likely that the numerical solver will fail because no unique solution exists The heuristics of Dymola to construct missing initial conditions operates roughly in the following way 1 If m initial conditions are provided Dymola adds nx m additional initial conditions by providing start values for a suitable subset of variables x i e the variables on which the der operator is applied Since there are nx variables x this selection is usually not unique In any case the selection is made such that the resulting initial equation system is structurally non singular this concept is explained in detail in Chapter 12 This is a necessary condition in order that a unique
105. word is only available in an algorithm section It cannot be used in an equation section In an algorithm section it is also possible to use a while loop it cannot be used in an equation section The example above can be implemented with while loops in the following way Determine whether matrices A and B are equal input Real A input Real B size A 1 size A 2 B has same sizes as A parameter Real eps min 0 0 0 Two numbers rl r2 are identical if abs rl r2 lt eps output Boolean identical Integer i Jj algorithm identical true i 1 while i lt size A 1 loop j 1 while j lt size A 2 loop if abs A i j B i j gt eps then identical false i size A 1 j size A 2 end if J es 44 1 end while i i 1 end while The while clause corresponds to while statements in programming languages and is formally defined as follows 1 The expression of a while clause shall be a scalar boolean expression 2 When entering a while clause the expression of the while clause is evaluated 3 If the expression of the while clause is false the execution continues after the while clause 4 If the expression of the while clause is true the entire body of the while clause is executed except if a break statement or return statement is executed and then execution proceeds at step 2 As with all Modelica language elements also an algorithm sec
106. www 3ds com news events press room release 1220 1 A central part will be behavioral modeling and simulation with Modelica and Dymola It is to be expected that Dassault Syst mes strategic decision will further accelerate the growth of the Modelica user community More details especially the actual language specification free Modelica libraries downloadable publi cations links to Modelica modeling and simulation environments as well as to Modelica consultants can be found at the Modelica homepage http www Modelica org Also books about Modelica are available such as Tiller 2001 and Fritzson 2003 This document is written for the following purposes e It shall serve as a script for the students of my course Simulation von elektromechanischen Systemen Objektorientierte Modellierung mechatronischer Systeme at the Technical University of Munich I give this course since 1997 once a year The content of the course is continually changing from year to year to take into account the latest developments In the past about 80 students per year passed the examination for this course e It shall provide an in depth description of the powerful features of Modelica to model discontinuous and variable structure systems This part of Modelica is described in a condensed form in the Modelica Language Specification and overviews are provided in articles and books However a user oriented description of the details is missing and is provided with this
107. yet available Stability Region not yet available Inline Integration not yet available Event Handling and Chattering not yet available Practical Considerations not yet available 171 Part D Libraries for Physical Systems This part contains a description of a selected set of libraries available in the Modelica Standard Library All libraries are first described from the view how to use the library and then internal details of the implemen tation are discussed 172 Chapter 14 Drive Trains Rotational library Draft Jan 21 2009 Chapter 14 Drive Trains Rotational library Draft Jan 21 2009 Chapter 15 Mechanical Systems MultiBody library 173 Chapter 15 Mechanical Systems MultiBody library In this chapter an overview of the Modelica Mechanics MultiBody library is given It is explained how to use the elements to build complex mechanical systems how the library is implemented and how a modeler can introduce his her own additional basic models such as special joint or force elements 174 Chapter 16 Thermo Fluid Systems Media Fluid libraries Draft Jan 21 2009 Chapter 16 Thermo Fluid Systems Media Fluid libraries In this chapter an overview of the Modelica Media and Modelica Fluid libraries is given to model thermo fluid flow in pipe networks 175 Part E Libraries for Control Systems This part provides an overview how to use Modelica in control applications in particular e
108. 06 Homepage http www OpenModelica org Otter M and F E Cellier 1996 Software for Modeling and Simulating Control Systems The Control Handbook by W S Levine editor CRC Press pp 415 428 Otter M H Elmqvist and S E Mattsson 1999 Hybrid Modeling in Modelica based on the Synchronous Data Flow Principle EEE International Symposium on Computer Aided Control System Design CACSD 99 Hawaii U S A Otter M K E Arz n and I Dressler 2005 StateGraph A Modelica Library for Hierarchical State Machines Proceedings of the 4th International Modelica Conference Hamburg Germany ed G Schmitz pp 569 578 http www modelica org events Conference2005 online_proceedings Session7 Session7b2 pdf Pantelides C 1988 The consistent initialization of differential algebraic systems SIAM Journal of Scientific and Statistical Computing pp 213 231 Roussel J M J M Faure 2002 An algebraic approach for PLC programs verification Proc of the Sixth International Workshop on Discrete Event Systems WODES 02 0 7695 1683 1 02 Pfeiffer F and C Glocker 1996 Multibody Dynamics with Unilateral Contacts John Wiley 238 Chapter 22 Literature Draft Jan 21 2009 Schumacher J M and A J van der Schaft 1998 Complementarity Modeling of Hybrid Systems IEEE Transactions on Automatic Control vol 43 pp 483 490 SIMPACK 2006 Homepage http www simpack com Snell A 2002 Decoupl
109. 1 For every branch of the Hysteresis block one discrete state high and low is introduced If on the up per branch state high the state machine switches to the lower branch state low ifu lt 1 When in state low it switches back to the upper branch when u gt 1 Applying the general formula from above leads to the following two Boolean equations high pre low and u gt 1 or pre high and not u lt 1 low pre high and u lt 1 or pre low and not u gt 1 Since low not high the two Boolean equations can be simplified to high not pre high and u gt 1 or pre high and not u lt 1 low not high For simplicity the second Boolean equation can be removed The remaining Boolean equation was used in the Hysteresis block in section 6 2 It is now clear how this equation was derived from a finite state machine Another example is shown in the next figure that describes a simple two point controller e g a tempera ture control Figure 9 3 Two point controller defined with state machine y_ref T2 yim gt y_ref b inactive T1 y_m lt y_reft b At the left side y ref is the reference signal and y_m is the measurement signal that should be close to its reference If y_m is smaller as y ref b where b is a pre defined bandwidth the controller switches from its inactive in its active step and the Boolean output u
110. 19 7 1 Inverse Model in Feedforward Path The inverse plant model is constructed from 19 12 by assuming that the variable to be controlled i e the concentration c f is a known time function and that the previously known input 7 shall be computed Draft Jan 21 2009 19 7 Example Application 215 from the inverse model By inspection or by using the Pantelides algorithm see section 12 4 it turns out that the first two equations of 19 12 have to be differentiated SHT y T 19 24 C ay ean Yy 19 22 and 19 24 are the inverse model of 19 22 A filter with an nth order pole on the negative real axis is used as reference model Since the second derivative of the input appears at least a filter of order 2 is needed such as 1 c Cates 19 2 s w l one with Caes the desired concentration w 27 f and f the cut off frequency of the filter A state space descrip tion of the filter is given by x c gs7 x W i 19 26 x c w The needed second derivative of c is obtained by differentiating the second equation of 19 26 x w 19 27 Equations 19 22 19 24 19 26 19 27 are the DAE of the inverse model of 19 22 with a prefilter of order 2 i e these are the connected blocks labeled as inverse plant model and as reference model in Figure 19 9 It turns out that this DAE has two states One possibility is to use the filter states x c as state vect
111. 2 ParameterPropagation 2 x General Add modifiers Component leon Name inductor Inductor eyo Comment is Model Path Modelica Electrical Analog Basic Inductor Comment Ideal linear electrical inductor Parameters L Edit 3 _ ___ Edit Text OK Copy Default R View Parameter Settings Propagate Reset display unit final Component Reference Select Class This pops up a window showing the declaration of the selected parameter here inductor L 32 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 Figure 2 13 Declaration of parameter via Propagate selection Declaration Type Prefix Annotations Type and name parameter elica Slunits Inductance L gt Description Inductance Modelica parameter Modelica Slunits Inductance L Inductance In the input fields all parts of the declaration can be modified The result is shown in the bottom field named Modelica For example the Description text might be changed from Inductance to Armature induc tance Clicking on OK introduces 1 the shown declaration in to the model and 2 the corresponding modifier In the example above the result is the underlined text is newly introduced after selecting Propa gate model Motor arameter Modelica SIunits Inductance L Armature inductance Modelica Electrical Analog Basic Indu
112. 2 initial conditions are missing and that Dymola added the initial equations s 0 and v 0 A useful strategy for initialization is 1 Define either no initial conditions or define initial conditions where it is clear that they have to be applied 2 Translate the model with the Log selected default initial conditions option enabled Dymola will print the number of missing initial conditions and will provide suitable proposals for the missing initial condi tions 3 Include the initial condition proposals of Dymola in the model or if necessary modified ones A new translation of the model should then no longer result in missing initial conditions It might be that some of the provided initial equations are redundant i e include exactly the same infor mation If this is a structural property of the initialization problem Dymola prints an error message due to a singular system or due to an overspecified system and rejects the model For example if the initial equations der s 0 and v 0 are provided for the SpringMassSystem above the error message of Figure 4 9 is printed 68 Chapter 4 Initialization Draft Jan 21 2009 Figure 4 9 Error Message in Dymola for a redundant set of initial conditions Translation of SpringMassSystem DAE having 2 scalar unknowns and 2 scalar equations Error The initial conditions for variables of type Real are overspecified There are 1
113. 3 1 4 0 2 rem x y Returns the integer remainder of x y such that div x y y rem x y x Result and arguments shall have type Real or Integer If either of the arguments is Real the result is Real otherwise Integer Note outside of a when clause state events are Additionally to the above conversion functions function String converts a Real Integer Boolean and enumeration variable in to its String representation This function call is not vectorizable The following Draft Jan 21 2009 21 3 Conversion Functions 231 function calls are supported arguments with a default value need not be provided when calling the function in such a case the default value is used String b_expr minimumLength 0 leftJustified true String i_expr minimumLength 0 leftJustified true String e_ expr minimumLength 0 leftJustified true String r_expr significantDigits 6 minimumLength 0 leftJustified true Streingir expr format The arguments have the following meaning Table 21 8 Optional arguments of operator String Arguments Description Boolean b _ expr Boolean expression Integer i_expr Integer expression type e_expr Enumeration expression enumeration Real r_ expr Real expression Integer minimumLength 0 Minimum length of the resulting string If necessary the blank character is used to fill up unused space Boolean leftJustified
114. 3 Prioritizing event actions 121 Boolean c algorithm Initialization of algorithm section vl pre vl can be always optimized away v2 v2 start can be usually optimized away v3 pre v3 can be always optimized away Transformed statements vl exprl v2 expr2 c condition if edge c then v3 expr3 end if The transformation of the when clause in to an if clause is similar as for a when clause in an equation sec tion The only difference is that the assignment to pre values is performed when entering the algo rithm section in order to handle all possible i clauses when the corresponding when clause is not ac tive Note that also non discrete Real variables are initialized by using their start value in order that a modeler cannot introduce a hidden memory For example the following code fragment Real Tlast stepSize algorithm if noEvent time gt Tlast then stepSize tim Tlast Tlast time end if would allow to compute the actual integrator step size if variable Tlast would not be explicitly initialized whenever the algorithm section is entered It would be of course better to forbid such models in order to give better diagnostics to the user However it seems difficult to figure out at translation time all such ob scure models and therefore the simple approach from above is used in Modelica since Tlast is initialized to its start value zero
115. 3f Therefore the force balance and the boundary conditions of equal positions are fulfilled The energy flow from the three components into the infinitesimal connection point is computed as under the assumption that no energy is stored in the connection point dE dt dcl s dc2 s dc3 s clf a2 f S E dt SOS elf te2f e3f dcl s 0 dt 0 P c3 f 3 3 Since cl s c2 s c3 s also the derivatives of the positions are identical and it is possible to move the posi tion derivative out of the parenthesis The term in parenthesis vanishes due to the zero sum equation of the forces and therefore the sum of the energy flows vanishes as well and the energy balance is fulfilled identi cally In the Modelica Standard Library see below elementary connectors for all important technical do mains are provided based on the general design principle explained above The definition of these connectors is summarized in the next table Table 3 3 Connector definitions in the Modelica Standard Library type potential flow connector name Modelica library icon variables variables physical connectors electric pot electric E electric analog Modelica Electrical Analog Interfaces PositivePin current el multi phase Vector of electrical pins p Modelica Electrical MultiPhase Interfaces PositivePlug 44 Chapter 3 Component Coupling by Ports Draft Jan 21 2009
116. 6 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 button and selecting a command under Edit Order If a package is stored in one file the definition order is preserved by storing all classes in the defini tion order in this file If a package is stored in several files there is the problem that the storage scheme does not allow to preserve the definition order As a result if such a package is loaded in to a Modelica en vironment no definition order is present and it is tool specific how the classes are ordered shown in the package browser This is a flaw in Modelica 3 0 and previous versions Most likely this will be fixed in the next language release Modelica 3 1 In Dymola classes stored in different files are shown in alphabetical or der in such a case classes in the same file are shown in the order in which the classes are stored in this file Furthermore Dymola provides the Dymola specific annotation __Dymola_classOrder to explicit ly define the ordering of classes in Dymolas package browser For example package Modelica Fluid annotation Dymola_classOrder UsersGuide Examples System Vessels Pipes Machines Valves end Modelica_ Fluid defines that classes are ordered according to the given sequence first UsersGuide then Examples etc and that classes not in the list are shown in alphabetic order at the end due to the las
117. 7 k k 0 1 2 remains constant provided the input vector u 7 k remains constant Most simulation systems support steady state initialization only for the continuous 186 Chapter 17 Continuous and Digital Control Systems LinearSystems library Draft Jan 21 2009 part of a model Due to the equation based nature of Modelica where basically everything is mapped to equations it is possible in Modelica to initialize also a mixture of continuous and discrete equations in steady state Via parameter methodType in the global block sampleClock or also individually for every block the following discretization methods can be selected e explicit Euler integration method e implicit Euler integration method e trapezoidal integration method e exact solution under the assumption that the input signal is piecewise constant e exact solution under the assumption that the input signal is piecewise linear The last method exact solution for piecewise linear input gives nearly always the best approximation to the continuous solution without frequency or phase distortion The effect of the discretization methods and of steady state initialization is demonstrated on the basis of a simple PT2 block that is initialized in steady state where the input signal is 0 2 at the beginning and jumps to 1 2 at 0 5 s In the next figure the blue curve is the solution of the continuous block and the red curve is the solution of the digital block using the implicit
118. 7 and the example with FluidPortA and FluidPortB since input output prefixes must be used in the connector only 1 1 connections are possible If a loop structure is present a special loop breaker element is needed in order that two FluidPortA or FluidPortB connectors are not connected together 2 Relevant boundary conditions and balance equations When connecting components together the relevant balance equations and the relevant boundary condi tions for the infinitesimal small connection point see Figure 3 1 below must be generated by the Modeli ca connection semantics of the corresponding connect equations variables without the flow prefix are equal and the sum of the variables with the flow prefix is zero This is a fundamental requirement in order that the balance equations and boundary conditions are fulfilled both in a component and when con necting a component together with other components Figure 3 1 Cut at an infinitesimal small connecting point Infinitesimal small connection point y 42 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 Examples for boundary conditions at an infinitesimal small connection point are given in the next table Table 3 1 Examples of connector boundary conditions at an infinitesimal small connection point Vi Vj Equality of electrical potentials V magi Vmagj Equality of magnetic potentials V nog J H ds H magnetic field strength
119. Connect to component of outside connector by selecting sub components below Name Unit Description y Nave nt De U U UUO i speedSensor pw Absolute angular velocity of flange lt Add variable gt Connection connect speedSensor w pee controlBus IV Label connection to bus connector 4 Clicking on OK introduces the connection connect speedSensor w controlBus speed and adds the bus variable as label on the connection line since Label connection to bus connector is en abled in the menu above Figure 3 12 Step 4 Result after clicking on OK in the menu controlBus speedSensor The label on the connection line is in principal a nice feature to see at once to which variable speedSen sor w is connected The current handling in Dymola is however limited If the connection line is drawn from a connector to the bus then the label is placed horizontally to the right side of this connector see Fig Draft Jan 21 2009 3 5 Connectors for Signal Buses 55 ure 3 12 If the connection line is drawn from the bus to a connector then the label is placed horizontally to the eft side of this connector Therefore the components that shall be connected to the bus must be appropri ately placed in order that the label is near to the connection line If this is not possible it is usually best to disable Label connection to bus connector in the menu above I
120. Edition McGraw Hill Isidori A 1995 Nonlinear Control Systems 3rd Edition Springer Verlag Isidori A 2006 Nonlinear Control Systems II Springer Verlag Kral C A Haumer M Plainer 2005 Simulation of a thermal model of a surface cooled squirrel cage induction machine by means of the SimpleFlow library Proceedings of the 4 International Modelica Conference Hamburg Germany ed G Schmitz pp 213 218 http Awww modelica org events Conference2005 online_proceedings Session3 Session3b1 pdf Kreisselmeier G 1999 Struktur mit zwei Freiheitsgraden Automatisierungstechnik at 6 pp 266 269 Kuhn M M Otter and L Raulin 2008 A Multi Level Approach for Aircraft Electrical Systems Design Proceedings of the 6 International Modelica Conference Bielefeld Germany ed B Bachmann pp 95 101 http www modelica org events modelica2008 Proceedings sessions session1d1 pdf Lotstedt P 1982 Mechanical systems of rigid bodies subject to unilateral constraints SIAMJ Appl Math vol 42 pp 281 296 Looye G 2001 Design of Robust Autopilot Control Laws with Nonlinear Dynamic Inversion Automatisierungstechnik at 49 12 p 523 531 Mah R S H 1990 Chemical Process Structures and Information Flows Butterworth publisher MathModelica 2006 Homepage http www mathcore com products mathmodelica Mattsson S E and G S derlind 1993 Index reduction in differential algebraic equations using dummy deri
121. Euler integration method for the block discretization As can be seen even for this simple block the results of the continuous and digital representation are quite different A much smaller sample time would be needed here in order that the solutions of the discrete and the continuous rep resentations would be closer together Figure 17 9 Step response of PT2 block discretized with the implicit Euler method P y continuous y implicit Euler integration method 2 0 J ett 0 8 2 0 4 4 0 0 1 mr r 1 t J mrri 1 Draft Jan 21 2009 17 3 Continuous Discrete Control Blocks 187 Figure 17 10 Step response of PT2 block discretized with the exact solution for piecewise linear input signals method rampExact DA y continuous y exact for piecewise linear input 2 0 r A 1 25 0 8 0 0 0 1 2 3 4 5 In the next figure the red curve is the solution of the digital block using the rampExact method i e the exact solution under the assumption that the input signal is piecewise linear The solutions of the continuous and of the discrete blocks are practically identical without any phase shift A better solution cannot be expected Continuous models in StateSpace form are transformed into discrete systems according to the sketched discretization methods A TransferFunction system description is implem
122. In such a case the original equation system is structurally singular This means that the equation system has no longer a unique solution independently how the equations are built up This can also be stated the other way round A system of equations must be not structurally singular in order that a unique solution is possible a necessary condition for uniqueness For example in the fol lowing system with 3 equations in 3 unknowns 0 f z Zz 0 f z 12 15 0 f z it is possible to assign either z or z to fi and z to f However it is impossible to make an assignment for f because Z is already assigned and a reassignment is not possible If is identical to f we have an infi nite number of solutions because either z or z can be selected arbitrarily and z2 can be computed either from fi or from fs If the two functions are not identical we have a contradiction and there is no z that fulfills both equations all the time i e there is no solution at all If the model equations are structurally singular it is still possible that a unique solution of the underlying differential algebraic equation exists by modifying the assumption that all variables appearing differentiated are states The details are discussed in section 12 4 The sketched approach of the output set assignment can be coded in a very compact form in a recursive function as proposed in Pantelides 1988 This algorithm is shown in form of a Modelica like
123. LargerParameters mixingUnit c NominalPlant mixingUnit c 0 6 0 4 PSS 0 2 0 100 200 300 400 500 NominalPlant mixingUnit T PlantvyithSOPercentLargerParameters mixingUnit T 320 lt 310 0 100 200 300 400 500 NominalPlant mixingUnit T_c PlantvvithSOPercentLargerParameters mixingUnit T_c 310 300 ho wo Oo 0 100 200 300 400 500 19 7 2 Feedback linearization The compensation control scheme and the feedback linearization cannot be applied directly to the example plant since the concentration c is not measurable One possibility is to use model knowledge in combination with estimation e g a Kalman filter but this is beyond the scope of this example For this reason it is as 218 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 sumed that the concentration is measurable In Modelica design of feedback linearization and the compensa tion controllers start in the same way For these controller types two integrators are added instead of an input filter see Figure 19 18 Figure 19 18 Inverse model of mixing unit for feedback linearization compare with Figure 19 15 integrator integrator2 der2_c The input der2_c of this inverse model is the second derivative of c_des In the case of the compensation controller the inversion work is done For feedback linearization the states in the inverse plant model must be replaced wit
124. M State state of the finite state machine has N input and M output conditions This situation can be described by the following Boolean equation state pre inStatel and inCondl or pre inState2 and inCond2 or pre inStateN and inCondN or pre state and not outCondl or outCond2 or or outCondM A finite state machine is now just defined by providing this type of Boolean equation for every discrete state The Boolean equation can be adapted to the peculiarities of the desired discrete formalism For example in Petri nets there must be an additional part that checks whether the following step does not have a token If the state machine is modularized in state and transition objects the above equation must be split into parts that are partly present in the state and partly present in the transition component The Boolean equation above can be also utilized to implement very simple finite state machines in order Draft Jan 21 2009 9 1 Implementation of Finite State Machines 133 to avoid some overhead associated for example with a StateGraph model Let us apply this technique on the simple Hysteresis block discussed in section 6 2 on page 111 which has the characteristic shown in the left part of Figure 9 2 below and can be described by the simple state ma chine visualized in the right part of the figure Figure 9 2 Hysteresis block defined by a simple state machine y T2 A u lt
125. Mathematical Data Structures cccccccesccessecseseeceeeeeneeesseeeeseeceeeensaeensaeeseseeeessentseeeeeee 176 17 2 Lincar System Descriptions si2i0 cit 5 cteitheinsienttieed ans deme he ee Laine accents 178 17 3 Continuous Discrete Control BIOCKS cccccesseeessceenseeeneeeeseeceseesseeeseeeneecseneeceseeseseeseeeeeeeess 182 Chapter 18 Hierarchical State Machines StateGraph Librarry csscccsssssscsssssecsscseessssceeesseees 190 18 1 Steps nd Transitions siie eienn a E id sedeens E EEE TEE RRES EEEa 190 18 2 Conditions and ACtOHS esseen erorien Ro oE ERE E a a aa aeia AERES 191 Draft Jan 21 2009 6 18 3 Parallel and Alternative Execution ccccccccccccsssccccceecessssceeccccessssseeeccccesssseeeeeecccessseeeeeeceeeess 195 18 4 Composite Steps assests scetedensslasieed a a A E Mediadl pled nh bee teed essed 197 18 5 Execution Model cs csvscesssccdsctssees secede sestisitecdauteusiikcectbeteaiin dal dutederideaied Qiveedi A E 199 18 6 Infinite Unsafe and Unreachable StateGraphs cccceessccesseceteeeeseeeeseeeesneeeeeeesnaeeeeeeseenseeeees 200 Chapter 19 Nonlinear Inverse Models for Advanced Controlllers ssccssccssssscsssssssscssssseesessees 202 19 1 Inversion of Linear SISO Models c cccccccesssscccceceeesssccecceecccsssseeeccccesssseeececeeececeececeeeeeeeeess 202 19 2 Inversion of Nonlinear Models cccccccccccesssscccccccsssssseeccccessssceececeessssseecccessss
126. Modeling Simulation and Control with Modelica 3 0 and Dymola 7 Preliminary Draft January 21 2009 Martin Otter AIMC1 Copyright 2006 2009 Martin Otter all rights reserved You are allowed to download the pdf version of this document print it for your use or view it on your com puter All other usages are explicitly forbidden without written prior permission of Martin Otter Especially it is forbidden to distribute this document to copy or print this document for others to make it publicly avail able e g on the Web on a CD ROM as printed book to sell it and or charge a fee for it This document is provided as is without warranty of any kind either expressed or implied The copyright holder assumes no responsibility for its contents what so ever Please send typos and other error reports as well as improvement suggestions to Martin Otter DLR de Preliminary Draft January 21 2009 Business Address Prof Dr Ing Martin Otter Deutsches Zentrum fiir Luft und Raumfahrt e V DLR Institut fiir Robotik und Mechatronik D 82234 Wessling Germany Email Martin Otter DLR de Web _ http www robotic dlr de Martin Otter http www Modelica org In this document the following trademarks are referenced e Modelica is a registered trademark of the Modelica Association e MATLAB Simulink Stateflow and Real Time Workshop are registered trademarks of The MathWorks Inc e CATIA is a registered trad
127. Similarly to the previous ex ample this indicates a modeling error The reason is that again in reality a small time delay is present that is needed to compute tau For an idealized controller this time delay can be approximated by an infinitesi mal small delay but it cannot be just completely removed The correct model which has again no longer an algebraic loop is represented by the following model Correct Modelica model model ReactiveController2 Real w a tau tau controller initial equation tau controller 10 equation tau pre tau controller a tau w der w a algorithm when a lt 4 then tau controller 0 end when end ReactiveController2 A similar analysis as previously shows that at the event instant when a became smaller as 4 the continu ous equations are evaluated with the value of tau_controller that was present just before the event oc curred The new value of tau_controller is then set to zero and since pre tau_controller tau_controller at the end of the model evaluation the model is re evaluated after setting pre tau_controller tau_controller In this second evaluation phase the continuous equa tions are evaluated with tau 0 The equation in the when clause is not active because a lt 4 was true in the previous evaluation and is true in this new evaluation and therefore edge c false c a lt 4 Therefore pre tau_controller tau_contro
128. Simulation and design of continuous and digital control systems with the Modelica_LinearSystems library e Defining state machines and logical controller functions with the Modelica StateGraph and the Modelica Blocks Logical library e Automatically constructing inverses of non linear Modelica models and using them in advanced control systems This topic is closely related to the discussed symbolic transformation algorithms and is a unique feature of Modelica and of Modelica tools to design controllers of non linear plants in a systematic fashion 176 Chapter 17 Continuous and Digital Control Systems LinearSystems library Draft Jan 21 2009 Chapter 17 Continuous and Digital Control Systems LinearSystems library The free Modelica_LinearSystems library provides basic data structures for linear control systems as well as operations on them and includes blocks that allow quick switching between a continuous and a discrete rep resentation of a multi rate controller It is useful to have both description forms of a controller in a consistent way The continuous controller approximation allows much faster simulations and is often sufficient for pa rameter design studies whereas the discrete controller model is needed for more detailed analysis This Mod elica library is shipped with Dymola since some time and is available via the library menu File Libraries Modelica_LinearSystems It is planned to include this library in the Modelica Standa
129. This can be defined in the following way in Modelica block InertialDelay Modelica Blocks Interfaces IntegerInput u Modelica Blocks Interfaces IntegerOutput y parameter Modelica SIunits Time delayTime parameter Integer y start 0 Initial value of y protected Integer u_delayed start y start fixed true Draft Jan 21 2009 7 4 Time synchronization multi rate delays range limited sampling 123 Integer u_old start y_start fixed true discrete Modelica SIunits Time tNext start delayTime fixed true algorithm when change u then u_old u tNext time delayTime elsewhen time gt tNext then u_delayed u end when equation assert delayTime gt 0 Positive delayTime required y u_delayed end InertialDelay 66 99 66 99 Whenever the input u is changing the value of u is remembered and a time event is planned after time delayTime When u is not changing during delayTime the planned time event is triggered and the value of u is remembered in u_ delayed This value is used as output y When u is changing the previously planned time event is replaced by a new time event A typical example of the InertialDelay block with a delay time of 1 s is shown in the next figure Figure 7 3 Simulation results of InertialDelay block inertialDelay u ine
130. This is performed with an algorithm section Example block Limiter Modelica Blocks Interfaces RealInput u Modelica Blocks Interfaces RealOutput y parameter Real uMax 1 Maximum value parameter Real uMin 1 Minimum value algorithm yr u if u gt uMax then y uMax elseif u lt uMin then y uMin end if end Limiter Block Limiter provides the limited input signal u as output signal y Contrary to an equation section in an algorithm section only assignment statements are allowed They have the form variable expression The operator is used to define that the right hand side here expression is assigned to the vari able defined on the left hand side here variable This is performed by first evaluating the expression on the right hand side and then assigning the result to the variable on the left hand side All statements in an algorithm section are evaluated in the defined order In an algorithm section if expressions if clauses and for loops can be used in a similar way as in an equation section The only difference is that the body of if clauses and of for loops must have as signment statements and no longer equations and that the restrictions from the equation sections do no longer hold For example an if clause needs not to have an else clause and the number of statements in the different branches of an if clause need not to be the same Furthermo
131. Tietze and Schenk 2002 have been helpful for the design and the actual implementation Draft Jan 21 2009 17 3 Continuous Discrete Control Blocks 183 17 3 1 Introductory Example In the left part of Figure 17 6 below a screenshot of the Sampled library and a simple example in using it is shown in the right part of the figure The example is a simple controlled flexible drive consisting of a motor inertia gear elasticity and load inertia The angle and the angular velocity of the motor inertia are measured and are used in the position and speed controller The output of the speed controller is directly used as the torque driving the motor inertia The multi rate controller consists of all blocks in the red square together with component sampleClock The latter defines the base sampling time together with defaults for the blockType Continuous or Discrete the methodType the used discretization method for a discrete block and the block initialization none initialize states initialize in steady state All Sampled blocks on the same hierarchical level as sampleClock and all blocks on a lower hierarchical level use the sample Clock setting as a default Every block can override the default and can use its individual settings Figure 17 6 Blocks of Sampled library left figure and introductory example right figure sampleClock Sampled controller JExamples SampleClock mee i ram filter sampler3 i PH Sampler
132. Transformation to State Space Form 153 0 Ci Ci Litli c Cu ees L Lu S Ci Co dL i 0 Sitsi A Su Svi S v G 0 G v Gv C Sv RI v a 7 oo 1 Sv R2 2 a E v Sv pea Res 0 ShtCLILG GI 3 R1 v Cy 4 R2 v Lv 0 R1i Ci 0 R2i Li The result is a set of 27 equations that describe the electrical circuit above in a unique and declarative form independently which type of analysis is performed on the circuit Our goal is to transform this set of equations into a set of ordinary differential equations in state space form or shortly also called state space form abbreviated as ODE X t f x t x t t x tER xelR 12 2 The variables x are called continuous states of the system The reason is that a lot of efficient and reliable numerical integration algorithms are available to solve this initial value problem Especially most algorithms for real time simulation require an ODE form A short introduction and an overview of these algorithms are provided in 13 1 Another reason is that it is then possible to import the Modelica model also in other model ing and simulation environments e g Simulink from MathWorks because these environments provide in terfaces to import ODE descriptions of models but usually not more general descriptions such as a DAE form Numerical integration algorithms for ODEs need a function that computes the state derivatives X
133. a description of all built in operators such as the op erators sketched above 5 1 2 Array operations Most standard mathematical operations on arrays are available in Modelica They are demonstrated in the following examples Real 3 2 4 Al A2 A3 Real sl s2 equation Addition Subtraction of arrays Al A2 A3 Scalar multiplication Al s1 A2 s2 A3 The and operator on arrays with the same sizes have the standard interpretation to add or subtract corresponding array elements Multiplication of a scalar with an array results in the multiplication of every element of the array with this scalar Draft Jan 21 2009 5 1 Arrays 77 Real A 3 4 B 3 5 C 5 4 Real v1 3 v2 4 s equation Scalar product vector vector scalar s vl vl Matrix multiplication matrix matrix matrix A B C Matrix vector multiplication matrix vector vector vl A v2 Multiplication of two vectors is a scalar product and results in a scalar Multiplication of two matrices is in terpreted as the mathematical matrix multiplication Multiplication of a matrix with a vector is defined as matrix multiplication where the vector is interpreted as a column matrix The resulting column matrix is im plicitly casted to a vector 5 1 3 Empty Arrays It is allowed to define arrays with zero dimensions and to use such arrays in equations provided the equation has no effec
134. a serious issue and modelers should always provide start values for the variables shown in such warn ing messages since in most cases the default start value of zero is not a good choice as a guess value Whenever a start value of a variable is set in a model it appears in the parameter menu of this model and can be easily set in the parameter menu see section 4 2 Dymola tries to select variables as iteration vari ables for which start values are provided If there are several choices Dymola selects the variable for which the start value is given on the highest level of the model hierarchy Still there are cases where Dymola selects an iteration variable for which no start value is defined In such a case the start value does not appear in the parameter menu of the corresponding model If the user has access to the underlying Modelica library and providing a start value for the corresponding variable makes sense in general then the best action is to include a suitable start value in the library in order that a different start value can be conveniently set in the parameter menu If it is not possible to provide a start val ue in general it is best to just add a start value dialog to the parameter menu with the Dymola_ini tialDialog annotation For example the Modelica Mechanics Rotational Components Inertia component is defined as model Inertia import SI Modelica SIunits parameter SI Inertia J min 0
135. able in the Modelica Standard Library as summarized in Table 3 3 on page 43 In Modelica it is possible to build up hierarchical connectors i e connectors consisting of other connectors It is usually best to use the available connectors in the Modelica Standard Library as basis for such hierarchical connectors For example a user might define the following connector Plug to describe cables with two electrical lines by selecting File New Connector and defining the connector in the text layer by dragging from the package browser the corresponding physical connectors in to the text ayer resulting in connector Plug Modelica Electrical Analog Interfaces PositivePin phase Modelica Electrical Analog Interfaces NegativePin ground end Plug Finally an appropriate icon is drawn in the icon and diagram layer and using the text Yoname displayed as name in the diagram layer in order that the connector instance name is displayed when dragging the con nector in a diagram layer Table 3 9 Icon and diagram layer of Plug connector icon layer diagram layer name As usual the icon in the diagram layer should be about 40 50 of the size of the icon in the icon layer in order that the connector looks nice when dragging it in to the diagram layer of a model When dragging this connector in to a model then the diagram layer of the Plug connector is shown in the dia
136. absolute angular velocity as it is often suggested in literature The reason is that there are applications where the angle of a drive train shall be controlled e g the angle of a robot joint or the angle of a motor that drives an elevator In these ap plications the angle must be measured and is then used in the control system A typical simple configuration is Shown in Figure 3 4 Figure 3 4 Simple drive train to measure the angle of an inertia angleSensor inertia J 1 Basically this simple system is described by the following Modelica models model Servo model Inertia Inertia inertia Flange b flange_b AngleSensor sensor Sgi end Inertia equation connect inertia flange b model AngleSensor sensor flange Flange_a flange end Servo end AngleSensor We will now discuss two implementation possibilities Draft Jan 21 2009 3 2 Connectors for Drive Trains Rotational library 47 Variant 1 Angle in Flange Connector model Inertia model AngleSensor parameter Modelica SIunits Inertia J Flange a flange Flange_a flange_a RealOutput phi Flange b flange b equation Modelica SIunits Angle phi phi flange phi Modelica SIunits AngularVelocity w 0 flange tau equation end AngleSensor phi flange _a phi phi flange b phi w der phi J der w flange_a tau flange_b tau end Inertia When the inertia and the angleSensor components are connected together
137. activities can be defined by component StateGraph Alternative An example for both components is given in the next figure Figure 18 7 Example for parallel and alternative execution paths in a StateGraph transition1 step0 transition2 true true Here the branch from step2 to step6 is executed in parallel to step1 A transition connected to the output of a parallel branch component can only fire if the final steps in all parallel branches are active simul taneously The figure above is a screen shot from the diagram animation of the StateGraph model in Dy mola Whenever a step is active or a transition can fire the corresponding component is marked in green col or This diagram animation is automatically available after a simulation has been performed once the Dia gram and the Runs the animation button in the tool bar of the simulation window have been pressed as sketched in the next figure 196 Chapter 18 Hierarchical State Machines StateGraph Library Draft Jan 21 2009 Figure 18 8 Diagram animation in Dymola by selecting the Diagram button O x File Edit Simulation Plot Animatian Commands Window Help JB e vron g FS2R DOF ExecutionPaths 1 stepO transition step transition2 step2 transition3 The three branches within step2 to step6 are executed alternatively depending which transition con dition of transition3 transition4 transition5 fires first S
138. aic loop that can be reached by sort ing the equations The other variables can be computed in a recursive sequence For example zz is computed from fs Then z can be computed from f using the already computed z2 etc The final result of this analysis is shown below i e the BLT form of the example 0 LAS Pes Saz i 0 f2 Z1 6 Z7 OES toy 12 16 O fizy O Ff ilzs z4 0 f 3 z3 Zs The worst case complexity of the overall algorithm to transform to BLT form is the sums of the two used al gorithms i e it is O n m On a standard PC a suitable implementation can transform 100000 equations and more in a few seconds to BLT form 12 3 Solving Algebraic Equations Tearing In the last section it was discussed how the model equations are sorted and algebraic loops of minimal di mensions with respect to sorting are identified Applying the BLT transformation to block diagrams gives in most cases the optimal result since often no algebraic loops appear Applying the transformation to physi cal systems e g electrical circuits mechanical or fluid systems results nearly always in algebraic loops Surprisingly the minimal dimensions of the algebraic loops in the BLT form are usually still quite large for physical systems and a direct solution would be still inefficient Fortunately it is possible to reduce the effort to solve the algebraic loops by a special variable substitution technique that is called tearin
139. al damping filter For a filter of order n there are n first order differential Draft Jan 21 2009 5 2 For Loops 83 equations which are constructed using a for loop Every for loop in an equation section is expanded during translation in to a set of equations by iterating over the loop variable in order that symbolic processing see Chapter 12 is possible In the example above the loop variable is i and it takes the values defined by the vector after the in keyword Above this vec tor is defined via the colon operator 2 n i e 2 3 4 n Below the expanded code is shown for n 3 n 3 der x 1 u x 1 w der x 2 x 1 x 2 w der x 3 x 2 x 3 w y x 3 All types of vector expressions can be used to define the values of the loop variable as long as it is a parame ter expression in order that the for clause can be expanded into equations during translation This vector expression is evaluated once for each for clause and is evaluated in the scope immediately enclosing the for clause Example parameter Real x 1 1 2 2 3 3 4 4 equation for il in 1 10 loop il takes values 1 2 3 10 end for for i2 in 1 3 6 7 loop i2 takes values 1 3 6 7 snd for for rl in 1 0 1 5 4 5 loop rl takes values 1 0 2 5 4 0 5 5 gad for for r2 in 2 x loop r2 takes values 2 2 4 4 6 6 8 8 ahd for The loop variable is implicitly intro
140. al event iteration if event then loop condition time gt pre tNext tNext if edge condition then pre tNext samplePeriod else pre tNext break if pre tNext tNext pre tNext tNext end loop end if which in turn implies that the model equations need to be evaluated only once at every sample instant 136 Chapter 10 Variable Structure Systems Draft Jan 21 2009 Chapter 10 Variable Structure Systems 10 1 Parameterized Curve Descriptions If a physical component is modeled macroscopically precisely enough there are no discontinuities in a sys tem When neglecting some fast dynamics in order to reduce simulation time and identification effort dis continuities may appear As a typical example consider modeling of a diode where i is the current through the diode and u is the voltage drop between its pins Figure 10 1 Detailed and idealized diode characteristic detailed diode idealized diode model model Uknee u Uknee A detailed nonlinear characteristic is shown on the left part of the figure If the detailed switching behavior around the origin is negligible compared to other model phenomena it is often sufficient to approximate the characteristic by the ideal diode model on the right part of the figure This corresponds to an ideal switch This abstraction typically gives a simulation speedup of 1 to 2 orders of magnitude with respect to the de tailed diode characteris
141. al y 2 x 3 x 1 is defined as Polynomial 2 3 1 Besides arithmetic operations functions are provided to compute the zeros of a polynomial via the eigen values of the companion matrix to differentiate to integrate and to evaluate the function value its deriva tive and its integral Find below a typical example from the scripting environment of Dymola import Modelica_ LinearSystems Math Polynomial p Polynomial 6 4 3 Polynomial String p 6 x 2 4 x 3 int p Polynomial integral p Polynomial String int _p 2 x 3 2 x 2 3 x der p Polynomial derivative p Polynomial String der_p 12 x 4 Polynomial evaluate der p 1 8 r Polynomial roots p printRoots true 0 333333 0 62361j 0 333333 0 623614 With function fitting a polynomial can be determined that approximates given table values Finally 178 Chapter 17 Continuous and Digital Control Systems LinearSystems library Draft Jan 21 2009 with function plot the interesting range of x is automatically determined via calculating the roots of the polynomial and of its derivative and plotted The plotting is currently performed with the plot function in Dymola Once a Modelica built in plot function is available this will be adapted A typical plot is shown in the next figure Figure 17 1 Plot of a polynomial with automa
142. alization i e pre v v start before initialization starts The alternative possibility to re quire v v start after initialization would usually lead to nasty initial equation systems involving pre v as unknowns For example the equations of the Hysteresis block would not have a solution if high true and u 2 would define the initial value problem and it would be in general difficult to compute pre v once v is given The question arises in which order discrete and continuous equations are evaluated The fundamental principal of Modelica is to map all language elements to continuous and or discrete equations and to re quire that the unknown variables are determined such that all equations are fulfilled concurrently i e a set of n equations in n unknowns has to be solved at every time instant and at every event instant A first step is usually to sort the equations The result is an explicit evaluation order of equations and or of systems of alge braic equations During continuous integration the discrete variables cannot change their values because relations can not change and therefore only the continuous equations have to be solved using the values of the discrete variables from the previous event instant At an event instant including the initialization that is treated as initial event the continuous and the discrete equations have to be solved concurrently under the assump tion that the
143. all missing equations of a component must be provided when it is instantiated Before Modelica 3 0 modifiers have been allowed for all variables in a component The above rule sig nificantly restricts modifiers The big advantage is however that a tool is able to precisely detect the source of an error Examples model Test wrong Modelica 3 0 model was previously correct Capacitor C1 C le 6 fine since modifier on parameter Capacitor C2 u sin time wrong since modifier on variable without allowed prefix VoltageSource V1 u sin time fine since modifier on input VoltageSource V2 wrong since no modifier on u end Test There are some additional rules to define balanced models Some of them will be discussed when introduc ing the corresponding new language elements All these rules are in the same spirit as above Whenever a component is used all missing equations must be defined at this place Draft Jan 21 2009 Chapter 3 Component Coupling by Ports 41 Chapter 3 Component Coupling by Ports 22 22 An important and difficult design decision is the definition of component in terfaces called connectors in Modelica For signal blocks the interface de O sign is simple The modeler defines input and output signals and that s it For physical components that interact with other components the interface de sign is more difficult In this chapter a systematic approach t
144. an nx independent additional conditions are required The details why this is the case and how this is algorithmically handled by a tool is explained in Chapter 12 For the moment we as sume that the Jacobian is regular and that nx additional conditions are needed The most simple initialization problem is defined by providing initial values for the variables x The re maining unknowns are then computed from the model equations known ty Xo solve 0 f o Xo Yoto for Xo Yo a This means that a non linear system of nx ny equations has to be solved to compute the unknowns X Yo Draft Jan 21 2009 4 1 Mathematically defining the initialization problem 61 In most practical cases many equations can be explicitly solved and only a rather small subset of the equa tions will be really non linear requiring a numerical non linear equation solver As will be explained in Chapter 12 this is exactly the same equation system that has to be solved at every time instant too There fore this is a rather unproblematic situation for a Modelica tool It is often not practical to require from a modeler to define initial values for all variables on which the der operator is applied Instead it is often easier to provide initial values on other variables It might be even necessary to provide additional initial equations In the most general case nx additional initial equa tions g have to be provided to formul
145. an error Due to the fundamental prop erty of Modelica that all language elements are mapped to equations and that every model evaluation re quires to solve n equations in n unknowns a Modelica translator can easily detect this type of non determin ism and therefore forces the modeler to resolve this problem by defining an explicit priority of the event ac tions One solution is to rewrite the equations as cl hl gt levell 120 Chapter 7 Instantaneous Equations Draft Jan 21 2009 c2 h2 lt level2 when cl c2 then openValve if edge cl then true else false or alternatively openValve edge cl end when However this is not very easy to understand A better alternative is to use el sewhen clauses equation when h1 gt levell then openValve true elsewhen h2 lt level2 then openValve false end when The when elsewhen construct is treated similarly as an if then elseif construct i e if the first condition h1 gt leveli becomes true then the first branch openValve true is active and the other branches are ignored Otherwise the second branch openValve false is active provided the second condition h2 lt level2 becomes true There might be additional elsewhen branches but no else branch The above implementation therefore introduces a priority so that the relation h1 gt lev e11 has a higher priority as the relation h2 lt level2 Similarly
146. and Operators Draft Jan 21 2009 Real Boolean String String String String ia b sr sb sr2 sr3 1 23456789 false r String r sr r 1 23457 h 4 String b shy b feaise String r SignificantDigits 16 minimumLength 15 leftJustified false f f 1 23456789 String r format 15g 1 23456789 21 4 Derivative and Special Purpose Operators with Function Syntax Table 21 9 Derivative and special purpose operators with function syntax Operator Description der expr The time derivative of expr If the expression expr is a scalar it needs to be a subtype of Real The expression and all its subexpressions must be differentiable If expr is an array the operator is applied to all elements of the array For non scalar arguments the function is vectorized For Real parameters and constants the result is a zero scalar or array of the same size as the variable delay expr delay delayMax delay expr delay m Fs Time Time Returns expr time delayTime for time gt time start delayTime and expr time start for time lt time start delayTime The arguments i e expr delayTime and delayMax need to be sub types of Real DelayMax needs to be additionally a parameter expres sion The following relation shall hold 0 lt delayTime lt delayMax otherwise an error occurs If delayMax is not supplie
147. andard function div must be used Result and arguments shall have type Real or Integer If either of the arguments is Real the result is Real otherwise Integer Note outside of a when clause state events are changes discontinuously Examples floor 2 3 2 0 floor 2 3 3 floor x Returns the largest integer not greater than x Result and argument shall have type Real Note outside of a when clause state events are triggered when the return value result has type Integer changes discontinuously Examples integer 2 3 2 integer 2 3 3 integer x Returns the largest integer not greater than x The argument shall have type Real The Note outside of a when clause state events are triggered when the return value Integer e Not vet supported by Dymola 7 0 Returns the ordinal number of the enumeration value e Example the return value changes discontinuously Examples type Day enumeration Mo Tue Wed Thur Fr Sa Su Day day Integer w Integer day Wed 3 mod x y Returns the integer modulus of x y i e mod x y x floor x y y Result and arguments shall have type Real or Integer If either of the arguments is Real the result is Real otherwise Integer Note outside of a when clause state events are triggered when mod 3 1 4 0 2 mod 3 1 4 1 2 mod 3 1 4 1 2 triggered when the return value changes discontinuously Examples rem 3 1 4 0 2 rem
148. anslated Modelica code in dsmodel mof and translating the model Dymola generates a file dsmodel mof where the symbolically processed equations are present in a Modelica like syntax These equations are also stored in file dsmodel c but the C file is by far not as readable as the dsmodel mof file The dsmodel mof file for the model above is dsmodel mof file for model InconsistentRedundantEquations Linear system of equations Matrix solution Original equations 0O bl all xl al2 x2 0 b2 a21 x1 a22 x2 Calculation of the J matrix and the b vector but these calculations are not listed here To have them listed set Advanced OutputModelicaCodeWithJacobians true before translation May give much output because common subexpression elimination is not activated x Solve J b J x b Xl x2 x Torn part End of linear system of equations From the description it is apparent that Variable x1 is computed from a linear system of equations Taking in to account the actual values of the parameters gives the following equation system 2 x1 3 x2 1 2 x1 3 x2 3 It is now obvious that no value of x1 or x2 can fulfill this equation system and therefore a modeling error is present and the modeler has to re formulate the initialization problem In a concrete application the equations in the dsmodel mof file might be too large in order to a
149. anslation time all equations are known which in turn means that the number of variables and the sizes of dimensions need to be known This implies that Modelica tools have currently the restriction that array sizes in models need to be known implicitly or explicitly during translation An exception are functions that might be translated without knowing the array sizes of the input or output arguments A typical scenario for a model is that array dimensions are defined with the operator and when using the model the actual array dimen sions are provided An array element is accessed by using as the access operator For example M 2 3 is the el ement in the second row and the third column of matrix M The first index of an array dimension is one as in FORTRAN or Matlab for example Real v 3 defines the thr lements v 1 v 2 v 3 equation v 1 1 v 2 2 v 3 3 Modelica makes a clear difference between the access of an array element such as v 2 and a function call such as v 2 to call function v with argument 2 In FORTRAN or Matlab the two operations are not distinguishable which means that the possibilities for error diagnostics are better in Modelica for this case A vector can be constructed with the array construction operator y parameter Real v1 3 1 2 3 Real v2 2 equation v2 time sin vl1 3 time Draft Jan 21 2009 5 1 Arrays
150. ant P resulting in Dr Z Dp Since R is the reference transfer function the structure of Figure 19 1 is a very simple way to design an open loop controller by just providing the de sired reference transfer function and multiplying it with the inverse plant model How to provide a suitable desired reference transfer function R is discussed at the end of this section on page 207 19 1 2 Open loop controllers for stable non minimum phase plants We will now analyze open loop controllers F s for causal and stable plants P s that might be non minimum phase This can be achieved with the controller structure of Figure 19 2 204 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 R arbitrary with u _ Mp Np Ea ERI e R stable and dp g D gt Dp deg n Figure 19 2 Open loop controller F for a stable plant P P might be non minimum phase For a non minimum phase plant P only the stable part of P can be utilized in the open loop controller be cause otherwise the unstable zeros of P would become unstable poles of F and then F would be unstable As a result the unstable zeros n of P remain in the reference transfer function and therefore the restrictions on the design part R of F are larger D Dy Dptdeg n gt 0 gt Dz gt Dp deg n 19 6 Still the construction of the open loop controller is conceptually very simple because the controller is de fined by multiplying t
151. anual control laws for a fighter aircraft The software code for the automatic landing system was automatically generated with Dymola and successfully flight tested on a small passenger jet Bauschat et al 2001 19 6 Inverse Model in Robust Controller Disturbance Observer All previous controller structures require that the plant model used as inverse system in the controller match es the real plant sufficiently accurate The controller structure in Figure 19 12 uses an inverse model to achieve a more robust design It was developed for linear systems with the goal to enhance robustness against disturbances and model errors This structure is called disturbance observer in the literature al though the name is misleading because it is actually an additional component for a controller It can be de signed independently from the main control loop Figure 19 12 Forcing a desired plant behavior by using an inverse desired plant model in the feedback path non linear disturbance observer plant must be minimum phase u t ji gt plan y j controller part for robustness filter p Yaf Ynfo Ying gt ay inverse of desired plant model 214 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 In Figure 19 12 the generalization for nonlinear systems is shown One important part is an inverse model of a desired plant beha
152. aph Transition Comment Transition where the fire condition is set by a modification of variable condition m Fire condition condition x gt true if transition may fire time varying expression Timer enableT imer j true x gt true if timer is enabled waitTime l 1 s Wait time before transition fires OK Info Cancel In the input field condition any type of time varying Boolean expression can be given in Modelica no tation this is a modification of the time varying variable condition Whenever this condition is true the transition can fire Additionally it is possible to activate a timer via enableTimer see menu above and provide a waitTime In this case the firing of the transition is delayed according to the defined waitTime The transition only fires if the condition remains true during the waitTime The transition condition and the waitTime are displayed in the transition icon In the above example the simulation starts at initialStep After 1 second transition1 fires and step1l becomes active After another second transition2 fires and initialStep becomes again ac tive After a further second step1 becomes active and so on In Grafchart Grafcet and SFC the network of steps and transitions is drawn from top to bottom In StateGraphs no particular direction is defined since Modelica models do not depend on the placement of components and connection lines Usually it is
153. apping of Modelica strings to native FORTRAN 77 code is not possi ble because the FORTRAN 77 data type CHARACTER is actually mapped to 2 arguments of a function call One argument for the address of the CHARACTER variable and one hidden argument for the length of the CHARACTER variable How the hidden argument is passed for a function call depends on the FORTRAN 77 compiler at hand e g passed as call by value as call by reference as INTEGER 2 IN TEGER 4 as last argument to the function call as argument followed directly the CHARACTER argument Dymola does not support an automatic mapping e g by introducing automatically the appro priate hidden arguments for strings and therefore the user has to manually introduce the hidden argu ments for strings depending on the used FORTRAN 77 compiler There is also the slight problem that the FORTRAN 77 name of a function is mapped to another name by the compiler when generating object code Often a FORTRAN 77 name like SOLVE is mapped to solve in the object code and therefore the FORTRAN 77 reference in the Modelica definition needs to be solve and not SOLVE To summarize directly using a FORTRAN 77 compiler is complicated and not portable if strings are passed as arguments This approach is therefore not recommended Instead it is recommended to use the free f2c translator from http www netlib org to transform the FOR TRAN c
154. ark yellow color as shown in the figures above The reason is that every connection line of any connector is drawn with a default color and default thickness This default is determined from the line attributes of the first graphical object defined in the icon layer of the connector at which the connection line starts For this reason the SignalBus and 56 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 SignalSubBus connectors in library Modelica Icons have a small dark yellow rectangle defined as first graphical object in the icon layer This rectangle is not directly visible because the bus icon is placed on top of it When drawing a connection line from the bus to another connector the color and thickness of this rectangle is used as color and thickness of the connection line Due to this feature it is recommended to copy the annotation of the Modelica Icons SignalBus SignalSubBus component in your own con trol bus connector Note in this special case you should not use inheritance using extends Modeli ca Icons SignalBus see section 2 3 because the connection line attributes are not influenced by su perclasses and they are only determined from the first graphical object directly defined in the icon layer of the corresponding class 3 6 Hierarchical Connectors It is quite difficult to design physical connectors Fortunately for the most important domains these connec tors are avail
155. ate the initialization problem known t sive 0 F X Xo Yor to g o Xo Yos to for Xo Xo Yo to This means that a non linear system of 2 nx ny equations has to be solved to compute the unknowns X Xo Yo at the initial time fo This means that a completely different equation system has to be solved at the initial time as at all other time instants In Dymola this is handled by generating special symbolically pre processed code for the initialization problem An important special case of this general initialization formulation is stationary initialization also called steady state initialization Intuitively this means that the system shall start in rest Mathematical ly this means that all derivatives are zero 1 e the additional equations g are simply 0 x 4 7 In many cases this is what the modeler would like to have Unfortunately it is sometimes not possible to set all state derivatives to zero Reasons are e A Pl controller is present This controller is described by the equation u T y k x u If during initialization x 0 is required then this implies that the input signal u 0 However u is defined somewhere else and then during initialization there are two equations for u In such a case it is not possible to include the equation x 0 as initialization equation e An aircraft flying in the air is basically described by the equations h v v f h v where
156. ated before algorithm section 1 The question arises when algorithm sections should be utilized instead of equation sections The simple answer is Only use algorithm sections if absolutely necessary The reason is that the symbolic pre processing is considerably reduced in an algorithm section a Modelica translator can only symbolically process an algorithm section if the result is identical as if the algorithm section would have been eval uated without any symbolic pre processing Since this is difficult a Modelica translator might even decide to not perform any symbolic pre processing in an algorithm section this is currently the case for Dymola As a consequence the code is usually more reliable and more efficient if equation sections are used Usually algorithm sections are utilized if a while loop is needed if complex nested if clauses be come too cumbersome to define them with equations or if a discrete algorithm is implemented a controller where the evaluation order is defined by the implementor Finally algorithm sections are always used in Modelica functions see next section 5 4 Functions 5 4 1 Basic properties of functions Modelica supports functions that are similar to functions in programming languages like C For example the Limiter block from the previous section could also be implemented as the following function function Limiter input Real u input Real uMax 2 input Real uMin 2 output Real y
157. ater Proceedings of the IEEE Vol 91 No 1 pp 64 83 Berry G 1999 The Constructive Semantics of Pure Esterel http www esterel technologies com files book zip Carpanzano E and R Girelli 1997 The Tearing Problem Definition Algorithm and Application to Generate Efficient Computational Code from DAE Systems Proc of 2nd Mathmod Vienna IMACS Symposium on Mathematical Modeling Vienna pp 1039 1046 Clauss C J Haase G Kurth and P Schwarz 1995 Extended Amittance Description of Nonlinear n Poles Archiv fiir Elektronik und Ubertragungstechnik International Journal of Electronics and Communications vol 40 pp 91 97 Dymola 2008 Dymola Version 7 1 Dynasim AB Lund Sweden Homepage http www dynasim se Enns D D Bugajski R Hendrick and G Stein 1994 Dynamic Inversion An Evolving Methodology for Flight Control Design In AGARD Conference Proceedings 560 Active Control Technology Applications and Lessons Learned pages 7 1 7 12 Turin Italy May 1994 Elmqvist H 1978 A Structured Model Language for Large Continuous Systems Dissertation Report CODEN LUTFD2 TFRT 1015 Department of Automatic Control Lund Institute of Technology Lund Sweden 1978 Elmqvist H 1992 An Object and Data Flow based Visual Language for Process Control ISA 92 Canada Conference amp Exhibit Toronto Canada Instrument Society of America Elmqvist H F Cellier and M Otter 1993 Object Oriented Mode
158. athematically described as 0 falz Z falz 12 11 It is then easy to analytically solve for z during translation of the model _ F az 1 Fala een Of course we have to assume that f4 z 0 in order that a division by zero does not occur Sorting does not change the rank of the algebraic equation system equations are just re arranged in a different order Since there is only one meaningful order to solve the sorted equations due to the block diagonal form of the inci dence matrix the original system of equations would be rank deficient if fa z2 would be zero As a result if faa Z2 becomes zero during simulation then the original non transformed system of equations does no longer has a unique solution and therefore the model is erroneous If a function is not linear in the variable for which it shall be solved then a scalar non linear algebraic equation is present and this equation has to be solved numerically during simulation for the unknown The sketched approach is demonstrated on the basis of the following simple electrical circuit Figure 12 3 Simple electrical circuit to demonstrate BLT transformation In order to follow more easily the transformation process the equation system is not built from the Modelica model which would result in 27 equations in 27 unknowns but Kirchhoff s voltage and current laws are ap plied manually resulting in the following 7 equations 0 u R i f 0 u R i
159. ation phase Together with the stationary initialization equation pre y y we have therefore a linear system of equations during initialization to compute the unknowns y and pre y y a pre y b u pre y y that has the solution b u 1 a y y pre y This means that the output y remains constant as long as the input u does not change because pre y is initialized in such a way that every evaluation of y a pre y b u results in pre y 7 3 Prioritizing event actions We will now analyze an often occurring situation by means of the following example Wrong Modelica model when hl gt levell then openValve true end when when h2 lt level2 then openValve false end when The intention of the modeler is to open a valve when h1 gt level1 and to close the valve when h2 lt level2 This definition is problematic because the value of openValve is not well defined when both conditions become true at the same time instant accidentally or by purpose Mapping the above when clauses to instantaneous equations leads to cl hl gt levell c2 h2 lt level2 openValve if edge cl then true else pre openValve openValve if edge c2 then false else pre openValve It is now clear that we have two equations for one unknown openValve and this system of equations does not have a unique solution A Modelica translator will therefore report
160. ation does not change the rank of the system Therefore such a transformation can be carried out once during translation of the Modelica model Within Dymola a very effective unpublished al gorithm of this kind is available that produces in many practical cases the minimal or close to the mini mal number of iteration variables For example in the example above Dymola selects z as iteration vari able leading to the following tearing transformation known Zi input Zp iteration variable output r residue eae Z2 2 algorithm r Z Z sin z As can be seen a division by zero cannot take place in the function and the residue is always well defined Therefore the tearing transformed and the original system of equations have the same rank and both equa tion systems have the same solution Due to the above general requirement for any symbolic tearing algorithm a modeling specialist can in fluence the tearing transformation by the way how the model equations are defined note this is in contrast to the BLT form that is independent of the original equation ordering or the way an equation is defined For example a media model might contain the idea gas law p p R T 12 19 where p is the absolute pressure p is the density T is the thermodynamic temperature and R is the gas con stant The only way a symbolic tearing algorithm can use this equation is to substitute the pressure p at other places
161. ay u u inertialDelay y y onDelay u u pressButtons B1 and B2 and not onDelay y This might be improved in a future Modelica version by introducing syntax to call a block as a function 7 5 Algebraic loops between continuous and instantaneous equations As already mentioned the Modelica language allows algebraic loops between continuous and instantaneous equations However this feature is not supported by Dymola Dynasim 2008 Usually this is not a critical issue because models can be rewritten so that an algebraic loop is removed and is replaced by event iteration In many cases this is a better model of the physical world and should be performed even if a tool would sup port algebraic loops The basic idea is explained on the basis of some examples Let us reconsider the simple digital controller that is connected to a continuous plant The only differ ence is that the output equation of the plant shall now be explicitly a function of its input signal uc this was not the case previously Equations of continuous plant der xc f xc uc yo g xc uc Equations of discrete controller when sample 0 T then yd a pre yd b ud end when Equations that connect plant and controller together uc yd ud yc ref yc It is not possible to sort the equations of this system in an explicit forward sequence as it was possible previ ously In order to make this
162. be used in a controller 19 8 2 Equations that cannot be inverted A plant may have equations that cannot be inverted Examples are time delays backlash friction hysteresis This can be fixed by approximating the problematic elements in such a way that the resulting equation leads to a unique inverse Figure 19 22 Backlash y1 and approximate backlash y2 A typical example is shown in Figure 19 22 The original backlash characteristic y f u is not invertible because for y 0 there are an infinite number of solutions u 1 1 In Figure 19 22 an approximation y f u is shown that is strict monotonic and therefore the inverse function has a unique solution It might also occur that tables have to be inverted Formally a table in one dimension is defined as a function y flu Inversion of this function means to solve a non linear equation This can be often quite eas ily avoided by providing already the inverse tabulated values u g y in the plant before inverting the plant model The advantage is that the solution is faster and more robust This problem was e g encountered in Steinhauser et al 2004 where control surface effectiveness of a military jet tended to have a local maxi mum as a function of the deflection This was solved by adapting tables and internal limitations of control commands 222 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 The inverse plant mode
163. becomes true Otherwise the return value is zero The integral over this signal is 1 PositiveImpulse condition v Returns an infinitely positive large value at the time instant expr when condition becomes true Otherwise the return val ue is zero The integral over this signal is defined such that at the same time instant variable v is re initialized to expr when the PositiveImpulse value is applied The major differences to the reinit operator are that the two operators can be used to re initialize any variable not only states and that the discontinuous change of one variable may result in the discontinu ous change of other variables due to the propagation of the impulse So this concept is more powerful and more useful as the reinit method On the other hand there are several unresolved problems e g a mathe matically and physically well defined description of several impulses that occur at the same time instant handling of the contact constraint in a convenient way so that the colliding parts remain in contact to each other e g that the bouncing ball stays on the ground handling of the probably more realistic contact hy pothesis of Poisson where not the velocity after and before the impulse is related to each other with the im pact coefficient but the impulse of the decompression phase with respect to the impulse of the compression phase see e g Pfeiffer Glocker 1996
164. bel connection to bus connector canes Draft Jan 21 2009 18 3 Parallel and Alternative Execution 197 This is inconvenient There have been discussions to improve the Modelica language to handle such situa tions more conveniently 18 4 Composite Steps A StateGraph can be hierarchically structured by using a component that inherits from StateGraph PartialCompositeStep An example is given in the next figure Figure 18 11 Example in which a CompositeStep is used compositeStep initialStep transition transition2 suspend resume 7 transition3 transition4 true true The CompositeStep component contains a local StateGraph that is entered by one or more input transi tions and that is left by one or more output transitions Also other needed signals may enter a Compos iteStep The CompositeStep has similiar properties as a usual step The CompositeStep is active once at least one step within the CompositeStep is active Variable active defines the state of the Com positeStep Additionally a CompositeStep has a suspend port Whenever the transition connected to this port fires the CompositeStep is left at once When leaving the CompositeStep via the suspend port the internal state of the CompositeStep is saved i e the active flags of all steps within the Compos iteStep The CompositeStep might be entered via its resume port In this case the internal state from the suspend transition is re
165. bles In Dymola the number of iteration variables is nearly always much smaller as the total number of un known variables for details see the tearing method in section 12 3 A modeler has to provide suitable guess values for these iteration variables otherwise it is highly likely that the initialization problem will fail with any numerical solver Guess values are provided with the start attribute and either not providing a value for attribute fixed or setting fixed false Ifa guess value is not provided the corresponding itera tion variable starts with zero Initialization may fail because e g the solver converges to a minimum from a given guess value or converges to the wrong solution if the system of equations has several solutions It is not known in advanced which iteration variables Dymola will select during code generation and it might even be that a small change of a model will result in different sets of iteration variables and therefore guess values for other variables need to be provided If no guess value is provided for an iteration variable Dymola prints a warning message during code generation Example Figure 4 10 Warning message from Dymola if no guess values are provided for iteration variables Warning The following variables are iteration variables of the initialization problem y but they are not given any explicit start values Zero will be used Finished WARNING 1 warning was issued This is
166. brary is contained For example if Modelica Mechanics is referenced in the within clause and the package is called MultiBody then the full name of this package is Modelica Mechanics MultiBody The root level package uses the statement within 1 e an empty within clause As a last ingredient directory names can be stored in the environment variable MODELICAPATH for example MODELICAPATH C user StandardLibraries C user project vehicles We are now in the position to describe the search mechanism in Modelica When a Modelica translator needs the definition of class A B C and this class is not yet loaded into the Modelica environment the following search is performed e Is file A mo present in the current directory e Is directory A and file A package mo present in the current directory e Is file A mo or otherwise file A package mo present in one of the directories defined in the environment variable MODELICAPATH When class A is found in one of the locations above the rest of the name i e B C is searched under the location of A If A B C is found in the file system it is automatically loaded into the Modelica environ ment otherwise an error occurs that the Modelica class could not be located Every Modelica environment has Modelica packages in its distribution It is tool specific how these li braries are located In Dymola the
167. cannot do this if there are other choices Therefore if this equation appears in a larger algebraic loop it is not possible to simply eliminate p and therefore the number of itera tion variables will most likely be larger Note the discussed property does not only hold for the tearing algorithm used in Dymola but for every suitable symbolic tearing algorithm so it is a tool independent property for reliable tools Once the tearing variables and residue equations are determined Dymola analysis whether the equations from an algebraic loop depend only linearly on the iteration variables If this is the case Dymola transforms the teared non linear system of equations symbolically in a system of linear equations This is performed in the following well known way Assume that the equations of the algebraic loop are partitioned into tearing variables z and other variables z L Jy Z b 12 21 Ja f 4 Since z are tearing variables the remaining variables z can be computed in a forward sequence This is only possible if L is a lower triangular matrix Since a rank preserving symbolic tearing transformation is used the diagonal elements of the Z matrix are guaranteed to be non zero and therefore L is a regular lower trian gular matrix Due to this property the inverse of L exists and it is possible to solve a linear system of equa tions in L in at most O n operations where n is the row and column dimension of L without t
168. cientists and by modeling specialists from the mechanical electrical electronic hydraulic pneumatic fluid and control domain 1 Discrete equations are either equations that are active at distinct points in time only e g at a sampling instant or equations that are used to compute Integer or Boolean variables Draft Jan 21 2009 8 Figure 1 Modelica Language and Modelica Simulation Environment EE eee aici g x SUIS AES FO EL B EL L TA cmm o Sche matic editor for a model a Tah a are a Modelica simulation environment HHH such as Dymola acd pj TR alele r s Bromre mien Textual definition of a model equations graphical layout etc defined by free Modelica language aaee SiS Ee ro neS aox Simulation environment for a model se lee o a Modelica simulation environment such as Dymola Translates Modelica code to C and provides simulation plotting animation and scripting features The non profit Modelica Association was formed in 2000 with Martin Otter as chairman to manage the con tinually evolving Modelica language and the development of the free Modelica Standard Library In the same year Modelica was used the first time in actual applications As of 2009 there are more than 1000 users of Modelica In June 2006 Dassault Syst mes a world leader in CAD and PLM announced its new product line CATIA Systems see http
169. clearer the equations that are active at a sample instant are shown and the two simple connection equations at the end are substituted Equations at an event instant with input yc ref xc pre uc der xc f xc uc yc g xc uc uc a pre uc b yc ref yc 3 3 The last two equations are a non linear system of equations to compute yc and uc from xc pre uc yc_ref This algebraic loop is a result of an over idealization of the digital controller because the computing time of the controller is not taken into account Even if we want to model an idealized controller where the comput 126 Chapter 7 Instantaneous Equations Draft Jan 21 2009 ing time is neglected the above model does not express our intention The reason is that it is impossible to measure a signal compute the actuator signal and apply the actuator signal instantaneously From a numeri cal point of view several evaluations of the plant and the controller equations might be necessary to compute the solution of the non linear system of equations via say a Newton method However in reality the con troller equations are only evaluated once at the sample instant and not several times Therefore the solution of the non linear system of equations above might not be a solution to our problem A closer model to reality is the following implementation of the controller Improved equations of discrete controller
170. constructed and the CompositeStep continues the execution that it had before the suspend transition fired this corresponds to the history ports of Statecharts or JGrafcharts A CompositeStep may contain other CompositeSteps At every level a CompositeStep and all of its content can be left via its suspend ports actually there is a vector of suspend connectors i e a CompositeStep might be aborted due to different transitions The CompositeStep can be used in the same way as a superstate in Statecharts In a superstate it is possible to enter the state in different ways ending up in different internal states This can be modeled in a StateGraph or a Graphchart by having multiple input transitions each leading to a different internal step In a superstate it is possible to exit a superstate in different ways depending on which internal state that is ac tive This is modeled in a StateGraph or Graphchart by associating different output transitions to the dif ferent internal steps In a superstate it is finally also possible to exit the state independently from which in ternal state that is active This is achieved with the suspend port here The conditions connected to the tran sitions attached to the suspend port can also be conditioned by the status of the internal steps of the Com positeStep In this way it is possible to suspend the step if a certain condition holds and unless a certain internal step is active The history arcs in Statecharts
171. correspond to the resume port Superstates with paral lel subparts so called XOR superstates can be modeled using parallel constructs inside the Compos iteStep In addition to using CompositeSteps for modeling hierarchical states they can also be used to simply 198 Chapter 18 Hierarchical State Machines StateGraph Library Draft Jan 21 2009 aggregate a part of a larger StateGraph This can be useful to improve the structure 18 4 1 Example The figure below is a screen shot of one of the examples of the Modelica_Fluid library Figure 18 12 Control system of a two tank system W ambient oT tankController valve stop valve valve b 30K nee level2 4 tank2 shut F X evel tank1 level f 30C level2 J tank2 level f X ambient Two tanks are present that are controlled by the tankController which in turn is controlled by 3 push buttons The basic operation is to fill and empty the two tanks Valve 1 is opened and tank 1 is filled When tank 1 reaches its fill level limit valve 1 is closed After a waiting time valve 2 is opened and the fluid flows from tank 1 into tank 2 When tank 1 reaches its minimum level valve 2 is closed After a waiting time valve 3 is opened and the fluid flows out of tank 2 D a eoo N gt When tank 2 reaches its minimum level valve 3 is closed The above normal process can be influenced by three buttons 1 Button start starts the above process When
172. creteFirstOrderBlockVersion2 Modelica Blocks Interfaces RealInput u Modelica Blocks Interfaces RealOutput y Modelica Blocks Interfaces RealOutput y delayed parameter Modelica SIunits Time T Sample time parameter Real a b equation when initial sample 0 T then y a pre y b u y_ delayed pre y end when initial equation pre y y end DiscreteFirstOrderBlockVersion2 A plot of the same situation as before is shown in the next figure Figure 7 2 Simulation results of model DiscreteFirstOrderBlock2 for a step input DiscreteFirstOrderBlock2 T 0 1 a 0 8 b 0 2 u delayed 2 5 X A y 2 0 1 5 1 0 0 5 T T T Draft Jan 21 2009 7 2 Sampled data systems and initialization 119 The initial vibration of y is no longer present and the computational delay of one sample instant is clearly shown in variable y delayed A when clause can be triggered by several conditions that are defined as a vector of conditions of the form when conditionl condition2 conditionN then end when The semantics is that whenever one of the conditions changes from false to true the when clause is acti vated The definition in the discrete first order block when initial sample 0 T then end when states that the equations in the when clause are active both at sample instants and during the initializ
173. ct to the unknowns is regular 0 f z lt isregular z IR f ER 12 10 The basic idea shall be sketched by means of the following simple example of 5 equations f in 5 unknowns Zj f z z4 F a i i Incidence matrix S Sj 1 if variable j i ti tion i 0 7 A269 S l0 1 1 0 1 j if variable j is present in equation 0 falzi Z2 1 1000 0 fl Zi Z3 Z 1 0 1 0 1 In the left column the structure of the 5 equations is displayed i e it is defined which variable appears in which equation In the middle column the incidence matrix S of this system of equations is shown This ma trix consists of zeros and ones and defines formally whether variable j is present in equation i Note the inci dence matrix is just a conceptual matrix that is used to clarify the operations to be carried out When actu ally performing the BLT algorithm the incidence matrix is never actually constructed because this would be a vast of memory for large systems e g for 10 equations the incidence matrix has 10 10 10 elements i e about 10 Gbyte of memory would be needed if an element would be stored in one byte The BLT algorithm sorts the equations in such a form that the unknowns can be computed in a recur sive way Equivalently this can be stated as Permute columns and rows of the incidence matrix so that it is transformed to block lower triangular form In the simple example above this equation sorting
174. cting this controller with the plant in series results in integrator behavior A simple feedback loop may be added to place the integrator pole at a desired location With a Modelica simulation environment such as Dymola the practical derivation of inverse models is straightforward even for complex systems 1 Define the plant as Modelica model and include input and output signals of the plant over which the in version shall take place 2 If necessary provide a reference model or input filter of appropriate relative degree The relative degree may be known from physical knowledge of the plant dynamics or can be automatically derived by Dy mola as described below 3 Connect the u1 inputs of a Modelica Blocks Math InverseBlockConstraints block to the plant outputs the u2 inputs of this block to the outputs of the reference model and the input of this model to an input signal connector Modelica Blocks Interfaces RealInput that defines the desired plant outputs asymptotically stable For DAE systems there seems to be no publication with a corresponding definition The definition introduced here for DAEs is simple and reduces to the standard definition for linear systems if the DAE is linear 8In the Modelica Standard Library 2 2 2 and previous versions inversion is defined with block Modelica Blocks Math TwoInputs This block is not a balanced model and was therefore replaced in Modelica Stan
175. ctor inductor L L end Motor 2 6 Model Libraries To ease usage models are collected together in libraries A library is called package in Modelica For ex ample the Modelica Standard Library is defined as package Modelica package Mechanics package Rotational model Inertia end Inertia model Spring end Spring end Rotational end Mechanics end Modelica A package can contain packages models connectors and other types of classes Note a class is the generic term for all structuring elements in Modelica such as packages models or connectors It is not possible to store instances of a class in a package with the exception of constants A class stored in a package is accessed from the outside of the package by referring to the full name such as model Motor Modelica Mechanics Rotational Inertia inertia J 0 001 end Motor A class stored in a package is accessed from the inside of the package by searching the first name of the desired classes from the inside hierarchically up For example package Modelica package Mechanics Draft Jan 21 2009 2 6 Model Libraries 33 package Rotational package Interfaces connector Flange a end Flange a end Interfaces model Inertia Interfaces Flange a flange a Modelica Mechanics Rotational Interfaces Flange_ a flange b end Inertia model Spring end Spring end Rotational end Mecha
176. cts Draft Jan 21 2009 Figure 5 4 Defining an arithmetic expression in Dymolas command window S Dymola Dynamic Modeling Laboratory 2 0 x File Edit Simulation Plot Animation Commands Window Help Bho JIB ven ola ERK BUH M 44 A I gt gt Tineo H Speed 7 A fv 2 3 sqrt Z 4 When pressing the return key the expression on the right hand side is evaluated assigned to the variable at the left hand side of the sign and the operation and its result are logged in the window above the com mand window see Figure 5 5 Figure 5 5 Result of executing a statement in Dymolas command window B Dymola Dynamic Modeling Laboratory i loj x File Edit Simulation Plot Animation Window Help v 2 3 sqrt 2 Declaring variable Real v Bimodeing YP Simulation i When typing the variable name and pressing return the value of the variable is displayed as shown in Fig ure 5 6 Draft Jan 21 2009 5 4 Functions 91 Figure 5 6 The value of a variable is displayed in the command window when typing its name v 2 3 sqrt 2 Declaring variable Real v 6 24264068711929 Modein Y Simulation It is also possible to call any Modelica function in the command window For example a linear system Ax b shall be solved with function Modelica Math Matrices solve In order to verify the result vector b is first constructed by a given A matrix and x vector and then th
177. d constructor function Complex of the Complex number The two dif Draft Jan 21 2009 5 5 Records 101 ferent calling forms are shown either by positional or by named arguments This demonstrates that a record constructor function can be used at all places where a function call is allowed to construct a new instance of a record Record constructor functions allow to build up data libraries where the data entries can still be modified contrary to the medium record example above This is demonstrated on the basis of a small library of elec trical motors package Motors record MotorData Data sheet of a motor parameter Real inertia parameter Real nominalTorque parameter Real maxTorque parameter Real maxSpeed end MotorData model Motor Motor model using the generic MotorData parameter MotorData data equation end Motor record MotorTypel MotorData inertia 0 001 nominalTorque 10 maxTorque 20 maxSpeed 3600 Data sheet of motor type 1 record MotorType2 MotorData inertia 0 0015 nominalTorque 15 maxTorque 22 maxSpeed 3600 Data sheet of motor type 2 end Motors model Robot import Motors Motor motorl data MotorTypel just refer to data sheet Motor motor2 data MotorTypel inertia 0 0012 data can still be modified Motor motor3 data MotorType2 end Robot Library Motors has record MotorData defining the most i
178. d in the argument list delayTime need to be a parameter expression For non scalar argu ments the function is vectorized cardinality c This is a deprecated operator It should no longer be used since it will be removed in one of the next Modelica releases Returns the number of inside and outside occurrences of connector in stance c in a connect equation as an Integer number posit semiLinear x tives tives Lope negat Lope Returns if x gt 0 then positiveSlope x else negativeSlope x The result is of type Real For non scalar arguments the function is vec torized There are specialized symbolic transformation rules defined if x 0 Due to the new method to handle fluid flow with stream variables this operator is no longer needed and should no longer be used 21 5 Event Related Operators with Function Syntax Table 21 10 Event related operators with function syntax operators noEvent pre edge and change are vectorizable Operator Description initial Returns true during the initialization phase and false otherwise thereby trigger ing a time event at the beginning of a simulation terminal Returns true at the end of a successful analysis thereby ensuring an event at the end of a successful simulation Draft Jan 21 2009 21 5 Event Related Operators with Function Syntax 233 Operator Description noEvent ex
179. d up system models it is important that libraries of the most com 14 Chapter 1 Introduction Draft Jan 21 2009 monly used components are available ready to use and sharable between applications For this reason the Modelica Association develops and maintains a growing Modelica Standard Library called package Modeli ca This is a free library that can be used in open and in commercial Modelica simulation environments Furthermore other people and organizations are developing free and commercial Modelica libraries For in formation about these libraries and for downloading the free libraries see http www modelica org library Version 3 0 of the Modelica Standard Library from January 2008 contains about 780 models 550 func tions and 1200 media descriptions in form of packages in the following sublibraries Table 1 1 Structure of Modelica Standard Library Example system Name of sublibrary Modelica Electrical Analog Analog electrical and electronic components such as resistor capacitor transformers diodes transistors transmission lines switches sources sensors Modelica Electrical Digital Digital electrical components based on VHDL with nine valued logic Contains delays gates sources and converters between 2 3 4 and 9 valued logic Star2 m m Modelica Electrical Machines Uncontrolled electrical machines such as asyn chronous synchronous and direct current motors and ge
180. dard Library 3 0 by Modelica Blocks Math InverseBlockConstraints 210 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 A typical example is shown in Figure 19 8 Figure 19 8 Definition of inverse model in Modelica left part the 4 basic components right part the 4 basic components connected together NE lt On the left part of the figure the four basic components are shown 1 Modelica Blocks Interfaces RealInput is the input u to the inverse model 2 The input signal is filtered with a corresponding filter block e g Modelica Blocks Continuous CriticalDamping or Modelica_LinearSystems ZerosAndPoles filter 3 The output of the filter should be connected to the output of the plant This is not directly possible because signal connectors can only be connected according to block diagram semantics and in block diagrams it is not allowed to connect two output signals with each other For this reason block Modelica Blocks Math InverseBlockConstraints is present It has two input connectors ul u2 and two output connectors y1 y2 and is described by the trivial equations ul u2 yl y2 When connecting a block to the inner connectors u2 y2 the input output signals are exchanged 4 The plant model to be inverted On the right part of Figure 19 8 it is shown how the components are connected together N
181. del translation r Evaluate parameters to reduce models improves simulation speed r Use new state selection can give dynamic choices r Generate listing of flat Modelica code in mof file r Generate listing of translated Modelica code in dsmodel mof r Include a variable for elapsed CPU time during simulation v Warn about parameters with no default Logging r List continuous time states selected Log selected default initial conditions information when di iati tor index reduction r Output read classes to screen during parsing If enabled which is not the default setting a warning message is printed listing the initial conditions that have been automatically added by Dymola if the user did not provide enough initial conditions Assume for example that no initial conditions are defined for the SpringMassSystem above and that Dymola translates this modified system with the enabled Log selected default initial conditions option In the Messages window the warning message of Figure 4 8 is printed Figure 4 8 Warning message from Dymola if not enough initial conditions are defined Translation of SpringMassSystem DAE having 2 scalar unknowns and 2 scalar equations Warning The initial conditions for variables of type Real are not fully specified Assuming fixed default start value for the continuous states s start 0 v start 0 In the message it is stated that
182. delica Mechanics Rotational y zy zyl 1 dim rotational mechanical systems such as drive inertia2 iz Se p e o7 trains planetary gear Contains inertia spring gear ey ms box bearing friction clutch brake backlash torque ratio 100 40 ratio 100 20 etc e T SpringPlateA Modelica Mechanics Translational CI 1 dim translational mechanical systems such as gt gt mass stop spring backlash force damper Modelica Mechanics MultiBody 3 dim mechanical systems consisting of joints bod ies force and sensor elements Joints can be driven boxBody AN boxBouy2 by elements of the Rotational library Every element n 0 0 1 r 0 5 0 0 ravines eT has a default animation x revolute1 sie Modelica Blocks Modelica_LinearSystems controler Continuous and discrete input output blocks Con es tains transfer functions linear state space systems A filter mathematical non linear logical table source blocks The Modelica_LinearSystems library allows to quickly change between a continuous fast con troller model and a sampled description detailed controller model duration 2 f_cut 5 reference Modelica StateGraph intitStep transtiont step transtion2 Hierarchical state machines with similar modeling i power as Statecharts Modelica is used as syn chronou
183. dently how the normal opera tion is implemented This implementation is shown in the next figure Figure 18 13 Tank controller with safety rule so that tanks never overflow setya normal fillTank1 actife and not level levelMax normal level2 lt minLevel suspend resume emptyTanks setValve2 normal fillTank2 active or emptyTanks active and level gt minLev setValve3 normal emptyTank2 active or emptyTanks active and level2 gt minLevel level level2 If one of the safe guards and not level2 gt maxLevel becomes active it could be that the state machine for the normal operation hangs at some step since a transition can no longer fire For exam ple tank2 might be filled until the transition levell lt minLevel fires instead of levell lt min Level or level2 gt maxLevel This transition will never fire if valve2 is closed due to the safeguard valve2 and not level2 gt maxLevel In the current case this is uncritical because by clicking on button stop or shut the composite step is terminated independently which step was active Still it would be useful to detect such a situation that the state machine can hang at a step because a tran sition does no longer fire this would be detected by a reachability analysis that must be however somehow defined which is currently outside of the scope of Modelica 18 5 Execution Model The
184. designed based on one linear plant model have the significant drawback that linear plant models nearly never represent the reality in the whole operating region good enough and there fore such controllers often operate only reasonable close to the operating point for which they have been de signed In the remaining sections of this chapter we will generalize the approach by using inverse systems of non linear plant models in order that the controller operates satisfactorily in a much wider region The de scribed approach is based on a lot of practical experience in designing industrial control systems for aircrafts robots and vehicles at the DLR Institute of Robotics and Mechatronics since the end of the nineties 208 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 The essential idea is as follows 1 Take any controller structure for linear systems that utilizes a linear inverse plant model 2 Replace the linear inverse plant model by a more detailed nonlinear inverse plant model 3 Determine the remaining part of the control system by appropriate techniques e g by tuning controller coefficients via parameter optimization Several different controller structures according to this technique will be discussed below A difficult part is issue 2 The nonlinear plant model should be constructed in a convenient way and the inverse model should be directly derived from the plant model It turns out tha
185. duced in the scope inside the loop construct and shall not be assigned to If a variable with the same name is declared the loop variable hides the declared variable in the or loop For example Modelica code that is correct but is not recommended Us different names for constant integer j and for loop variable j constant Integer j 4 Real x j equation for j in 1 j loop The loop variable j takes the values 1 2 3 4 x j j Uses the loop variable j so x 1 2 3 4 end for This also means that if the loop variable is declared explicitly then a translation error occurs because an equation for the declared variable is missing Example Wrong Modelica code an equation is missing for j parameter Integer nx Real x nx Integer j No equation for j present equation for j in 1 nx loop x j j end for To summarize Loop variables are never declared 84 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 For loops in equation sections are not often used since loops are traditionally written with assignment statements see section 5 3 In equation sections for loops are usually used to build up regular differential equation structures as shown for the filter above and for building up regular connection structures of compo nents see section 5 8 5 3 Algorithm Sections It is possible in Modelica to program in a procedural style similar as in a programming language
186. e electrical potential v and electrical current i are needed for electrical component interfaces Angle phi and cut torque tau are needed for drive train elements connector Pin connector Flange import SI Modelica SIunits import SI Modelica SIunits SI Voltage Vi Angle phi flow SI Current i flow Torque tau end Pin end Flange A connector definition contains a set of declared variables The syntax of the declarations is identical to the declaration of variables in a model The flow prefix in front of a declaration defines that this variable is a flow variable When connect ing flow variables together the sum of the flow variables is zero a more detailed explanation follows Flow variables have always an associated direction In order that correct equations are generated a con vention has to be established how a positive flow variable has to be interpreted in a component Usually if the corresponding physical quantity flows from the outside into the component a flow variable has a posi tive value and a negative value when it flows out of a component For the Pin connector above this means that a current flowing in to an electrical component has a positive value the definition of the positive direction of a mechanical connector is more complicated and will be explained later Variables without the flow prefix are sometimes called potentia
187. e 19 9 are identical here In such a case the feedforward controller can be removed and can be replaced by the result of the trajectory optimization 5 and fia 7 moat m vi aa Sia For the trajectory optimization problem an fmax Should be used that is say 10 20 smaller as the actual limit in order to provide some margin for the feedback controller If the desired control variables y f are not known in advance but generated online e g by an operator online optimization techniques have to be used The operator request is reduced such that the plant con straints are fulfilled in the next sample time instant A well known measure in flight control is the so called daisy chain In case a control input saturates a secondary redundant control input is brought in that provides the remaining required control power In Steinhauser et al 2004 for example lateral deflection of the thrust vector is used to yaw the aircraft in case the rudder saturates 19 8 4 Real time implementation If inverse plant models are part of the controller linear and non linear systems of equations as well as non linear differential equations might have to be solved in every sample interval of the controller The tech niques developed for hardware in the loop simulations can be also applied for such an application The methods described in section 13 4 are available in Dymola with the Dymola real time option and can be ap plied by selecting the appropriate
188. e defined variants for an input field of a parameter menu For example in the commercial FlexibleBodies library flexible beams are defined via a hierarchical data record The geometry of a beam is defined by its cross section area and its length In Figure 5 12 part of the beam data record menu is shown Figure 5 12 Menu to define the cross section area of a flexible beam in the FlexibleBodies library Parameters O rectangle crossSection circle Type of cross section beam area ol squarepipe l gt m Length of beam rho kg m3 Mass density E N m2 Young s modulus G N m2 Shear modulus LJ U_pipe L _ beam T T_beam eneral Inputs a g diameter m outer contour dimension numberOfPoints gt number of points to visualize cross section Parameter crossSection consists of a pop down menu in which different cross section shapes can be selected such as rectangle circle I beam etc When one particular shape is selected then a menu pops up to define exactly the data for the selected shape e g diameter for a circle shape or width and height for a rectangle shape The above menus are defined in the following way record BeamCrossSection Modelica SIunits Area area Cross sectional area Real polyLine 2 Visual appearance for animation Real Iyy final unit m4 Geometrical moment of inertia Iyy
189. e different To summarize variant 2 has the dis advantages that a an additional initial condition has to be given for angleSensor phi and that b the two independent initial conditions must agree in order that the angles of the inertia and of the angle Sensor are identical Therefore variant 2 is not suited for component oriented modeling because connecting components to gether may easily lead to errors since the states are independent from each other but not the initial condi tions 3 3 Connectors for Heat Transfer HeatTransfer library Library Modelica Thermal HeatTransfer has components to model heat transfer with lumped elements by conduction convection and radiation These elements are derived from the integral form of the partial differ ential equation of heat transfer by using a simple one dimensional discretization formula see e g Holman 48 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 2001 In many cases this allows to model heat transfer in a rough way especially if the parameters of the lumped elements such as the heat capacity of a part can be calibrated with measurements see e g Kral et al 2005 It is often not practical to calculate these parameters from basic analytic formulas if complex ge ometries with different materials are present as it is e g the case for electrical motors More detailed mod els especially for complex geometries require usually to use a finite element p
190. e exactly the required initial conditions for a larger model If too many initial conditions are provided there exists usually no solution of the initialization problem because there are more equations as unknown quantities and therefore a Modelica tool has to reject such a model For example we might add the equation s s0 to the initial equation section of the SpringMassSystem model above During translation Dymola prints then the following error message Figure 4 5 Error message of Dymola if too many initial conditions are defined Translation of SpringMassSystem DAE having 2 scalar unknowns and 2 scalar equations Error The initial conditions for variables of type Real are overspecified There are 1 too many scalar conditions To correct it you can remove Initial equations equation der v 0 s s0 In the error message it is stated that there is one initial condition too much and that therefore one initial con dition has to be removed Dymola tries to identify which initial conditions are the critical ones Therefore in the above message only 2 of the 3 initial conditions are listed and one of the listed initial equations has to be removed Much more often the modeler defines none or not enough initial conditions In such a case there are usually an infinite number of solutions of the initialization problem Since Dymola 7 1 a warnings message is printed in the Messages window in such a case F
191. e function is called with matrix A and vector b to compute x The result of typing these commands in the command window are shown in Figure 5 7 Figure 5 7 Solving a linear system of equations in the command window 2 3 sqrt 2 Declaring variable Real v 6 24264068711929 1 2 3 4 x 1 2 b Declaring variable Integer A x b 2 21 A Declaring variable Integer x 2 b Declaring variable Integer 2 5 11 Modelica Math Matrices solve A b l 2 E Modeing W Simulation An alternative more convenient way is to select the corresponding function in the package browser right click with the mouse and select Call Function in the context sensitive menu as shown in Figure 5 8 92 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 Figure 5 8 Calling a Modelica function in the package browser by right clicking on the function odelica Modelica Read Only Documentation File Edit Simulation Plot Animation Commands Window Help jar EQS iN meow AE lth Z Z Modelica Standard Library Version 3 0 Modelica Reference _ Modelica HE User s Guide of Modelica library Information Package Modelica is a standardized and free package that is developed together with the Modelica language from the Modelica Association see http www Modelica org It is also called Modelica Standard Library t provides model components in
192. e parameter menu above The reason is that variable L is declared as Modelica SIunits Inductance and this pre defined type has unit H After dragging components such as the inductor model from above into a diagram layer connection lines can be drawn between connector instances Modelica supports only two point connections i e con nection lines that start at one connector instance and end at another connector instance It is not possible to connect say into a middle of a line Let us connect three inductors together in a Diagram layer of a new model Figure 2 9 Connecting three inductors together inductor and inductor2 and inductor1 and inductor3 are connected The circle in the connection point is automatically introduced by Dymola rr te D 3 3 inductor inductor2 L 0 3 5 5 k k inductor inductor2 L 0 1 L 0 2 L 0 3 oO te 5 5 B inductor inductor2 Draft Jan 21 2009 2 2 Basic Models 25 Although the diagram left part of Figure 2 9 looks as if inductor3 would be connected to the line connect ing inductorl with inductor2 this is actually not the case In the right part of the figure the two real connection lines from inductorl to inductor2 and from inductor to inductor3 are shown The circle in the middle of the three connected lines which looks graphically like an electrical node is automa
193. ead inputs update memory compute outputs end for update memory end for One essential issue is that there is a strict alternation between environment and program actions Once the in put signals are read the program starts and computes its output During that time any change to the inputs is ignored When all outputs have been computed or at a given time determined by a clock the output values are fed back to the environment and the program waits for the start of the next cycle The other essential in gredients of synchronous languages are e Instantaneous communication i e in the compute outputs part of the program no delay of any form is present i e all variables act instantaneously without any time delay or infinitesimal micro steps contrary to e g in VHDL e Deterministic concurrency by requiring that the parallel composition of two program parts is the conjunction of the reactions of the two parts This may lead to algebraic loops also called constraints that are handled in various ways in the different languages For example Lustre forbids algebraic loops We will now discuss the relationship of Modelica to synchronous languages It is clear that it cannot be a one to one relationship because standard synchronous languages operate on discrete entities whereas Mod elica operates on a combination of continuous and discrete entities The following definitions are needed Draft Jan 21 2009 Chap
194. ean expressions can be used when needed In Grafchart Grafcet SFC and Statecharts actions are formulated within a step Such actions are distin guished as entry normal exit and abort actions For example a valve might be opened by an entry action of Draft Jan 21 2009 18 2 Conditions and Actions 193 a step and might be closed by an exit action of the same step In StateGraphs this is not possible due to Mod elicas single assignment rule that requires that every variable is defined by exactly one equation Instead the approach explained above is used For example via the SetBoolean component the valve variable is set to true when the StateGraph is in particular steps This feature of a StateGraph is very useful since it allows a Modelica translator to guarantee that a given StateGraph has always deterministic behavior without conflicts In the other methodologies this is much more cumbersome As an example in the next figure a critical situation in Stateflow is shown Math works Stateflow is a Statechart type state machine and is integrated in Mathworks Simulink Figure 18 5 Parallel execution in Stateflow where a variable is assigned a value in parallel branches The upper definition results in openValve 0 whereas the lower definition results in openValve 1 entry openalve 1 entry openvalve 1 1 The two substates filll and fill2 are executed in parallel In both states the var
195. ear damping b 1e 5 rad Total backlash from Figure 1 1 is shown The left column contains the names of the parameters the middle column the actual values and the right columns the units and description texts e With symbolic algorithms the high level Modelica description is transformed into a form that is much better suited for a numerical evaluation Based on the natural description of composition diagrams large hierarchical systems can be constructed and simulated In Figure 1 3 below a typical screen shot of Dymola s schematic editor is shown Figure 1 3 Schematic Editor of Dymola G DriveSystem DriveSystem Diagram 5 xj File Edit Simulation Plot Animation Commands Window Help 18 x SUAS P 000A m o 2 Z gt BSH EZ 15 Modelica Reference Modelica Unnamed gr Motor Gear controller a Modeling Y Simulation 4 ee to build up a Modelica model The editor consists basically of 3 windows The upper left window is the package browser that shows the loaded Modelica libraries and models The large right window is the draw ing area in which component icons from the package browser are dragged with the mouse to construct the Modelica model The lower left window called Components above is the instance browser that shows the hierarchical structure of the model from the main window 1 2 Mbodelica Standard Library In order that a modeler can quickly buil
196. eedback lta plant P y controller You Year HOSS yP m R a VAC controller In Figure 19 9 the multi input multi output plant has inputs u measured signals Ym and controlled outputs ye In many cases Y Ym i e the controller outputs are a subset of the measured signals However there are also practically important cases where the two signal sets are different For example a standard electrical drive system has the speed of the motor inertia as measurement signal y and the speed of the load inertia as the signal y to be controlled For this controller structure the number of inputs must be identical to the number of controlled vari ables dim y dim u and the plant must be minimum phase but might be unstable In this case the in verse plant model with known inputs y and unknown outputs u is used in the feedforward path of the con troller to compute the desired actuator inputs ug to the plant A reference model defines the desired dynamic behavior of the closed loop system It is often most convenient to use a filter since the filter is parameterized by just the cut off frequency once the filter order and the filter type is fixed and because a filter provides the optimal reference model with transfer function 1 below the cut off frequency for details see page 207 There are also other useful choices of the refer ence model The outputs y a of the reference model are the inputs
197. eesesees 136 10 1 Parameterized Curve DeScriptions c cccescecsscsessesenseesseeceseeceeecseecsscecneneessseeseneeseseessnaeeeeees 136 10 2 Coulomb Friction not yet complete cccecsccssssecssecesneeesneeceeeeeseeesseeeseeeeeseeeseeeeseeessntaeeeeeees 139 Chapter 11 Re initialization of Continuous Variables sscsssscsssscssscssscsssecsssssssssssssscssssesees 143 11 1 Thereinit Operator ss ss ccesesecsseeeasadeeus has avebessndsedelitavtiai A E E E EAS 143 11 2 Experimental operators to describe impulsive behaviot cceccesceeeeeeeceeeeeeeeeeeeeeeeeeeneeeeaeees 146 Part C OMA Os aoe nce cae scececesndescecenccecececdccctag ccsseteseccensdaeceescedesoecie eee L4 Chapter 12 Symbolic Transformation Algorithms csscccssccsssscssscsssscssssssssssssssesesssssssesssnees 150 12 1 Transformation to State Space FOrm cccccccssceseseeeeseeceneeeeneeeseeeeeeceseeessaeeseeeessaeeseneeeeteeeeeees 150 12 2 Sortie BCT reei eevee canes bras sade E E E E E E ele T E ESETE 156 12 3 Solving Algebraic Equations Tearing cc cccssccsssccssceessseceseeceeseceeneeeeeecsseeeeeessneeensenteeeeesee 161 12 4 Singular Systems Pantelides Dummy Derivative not yet available cccceeeseeeeesteeeeeees 164 12 5 Hybrid DAEs need Symbolic Preprocessing c cccssccsesesesseceneeesneeesneceseeesseeeeaeesseeessaeensseenes 165 Chapter 13 Integration AIgOrithms ccsccccssscsssscs
198. efined Table 2 3 Unit related Modelica attributes quantity Type of physical quantity unit Unit in which the equations are defined displayUnit Unit that is by default used for input and output Quantities and units are defined as a Modelica string The syntax for units is defined by a grammar and fol lows an ISO recommendation Examples Table 2 4 Modelica syntax for units Unit Modelica syntax kgm s kg m2 s2 or kg m m s s rad s rad s I s 1 s or s 1 Besides units the following attributes can be defined in a declaration Table 2 5 Attributes of a Real variable Attribute Meaning min Minimum value of variable max Maximum value of variable start Initial value of variable fixed true Value of start is a fixed initial condition false Value of start is a guess value e g for iteration variable nominal Nominal value for scaling of numerical algorithms e g nominal 1e5 means that the usual value of the variable is around 10 stateSelect Influences the choice of the state variable of the differential equations details are given later Possible values StateSelect never avoid default prefer always A declaration of a Modelica model can therefore look like parameter Real m min 0 quantity mass unit kg 2 Draft Jan 21 2009 2 1 Simple Declarations and Equations 19
199. elativeTolerance nominalValue 0 01 where the nominal value is defined by attribute nominal in a variable declaration Since no nominal at tribute is set for x 1 i e the nominal value is 1 the absolute tolerance is in the order of the relative tol erance i e around 1e 4 Since x 1 is permanently quite small the step size control has not much effect and therefore the result is not reliably computed A quick fix of this situation is to provide a nominal value for the state vector x In our example this can be done by opening the parameter menu of the transfer function selecting the Add modifiers tab and declaring a nominal value of 1e 4 for x by the statement Draft Jan 21 2009 5 1 Arrays 81 Figure 5 3 The Add modifiers menu used to define the nominal value of a variable E transferFunction in TestTransferFunctionib 2 xi General Add modifiers Add new modifiers here e g phi start 1 w start 2 x nominal 1e 4 The defined nominal value is used for all elements of vector x According to the Modelica specification it would be possible to define a nominal value for every element of an array However Dymola supports cur rently only one nominal value for all array elements Simulating again with this change of the model gives the correct result even for the default relative tolerance of 1e 4 It is of course desirable to treat such a situation automatically without bo
200. elica Electrical Analog li brary For example a capacitor is defined by 3 equations 0 i ti u vv 12 1 odu ake 12 1 dt l The first equation states that the current flowing into the capacitor at the left connector is identical to the cur rent flowing out of the right connector i and i are positive if the current is flowing into the component The second equation computes the voltage drop as difference of the electrical potentials v and vz at the left and at the right connector Finally the third equation defines that the time derivative of the voltage drop is proportional to the current entering the left connector We will now use these elements to build the following simple electrical circuit 152 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 Figure 12 1 Simple electrical circuit to demonstrate code generation 1 2 G This circuit has been constructed by dragging the corresponding elements into the diagram layer of a new model and connecting them appropriately together In the text layer of Dymola the Modelica definition of this model is automatically available for clarity the graphical annotations that describe the visual appear ance in the diagram layer are removed below pins with a filled square have the instance name p and pins with a non filled square have the instance name n in the Modelica Electrical Analog library model SimpleCircuit
201. emark of Dassault Syst mes Draft Jan 21 2009 4 Table of Contents ad a E i CO scccscscssveisccenccccssescesscsessovsceuccsssstecsvsceusssstseusenssssssdeseseuesceveususcesesossencesensssssoesssscdeassceusouseseveasasscscueesvenes 7 Part A Modeling of Continuous SystemS essssesssesesssssocesssssossessssoeseossesssesssssse L Chapter 1 INtroductOniissssussscscccsssasscsceevessendsescecsvtessessecsescsnndedectscscensdacedecwasessndcssssuapnsdedecsssseneiccsevoesonss 12 Lal Composition Dia grams sssri resterai E E E A REEE E EEEE EE EESE 12 1 2 Mo delic Standard Libr t youenn E ENA E EE E E ia 14 Chapter 2 Basic Modelica Language Constructs eeesseeseeeseesseesseesseesoeesoessoeesoessoesceorsoorsosesosesosesosseeeseo 17 2 1 Simple Declarations and Equations cccccecssccsssecssseeesnecseneeceeeceeeseseeeneeceseeeecsesneeeesesenteeeeeees 17 22 Basie MOdeIS vvessivs secaedessscs sede catettasdavieceesin A E O E EEES 23 2 3 WnheritamCes sscetaseseeesaastecesess a E E E raat lace sasligtale nn eoeneene es 26 2 4 Conditional Equations iicccssisccscsicadeacstcatauesvoute cud aE EAE EO T E E E E AARE 28 2 9 Hierarchical Models sesser arra E E E OE E E Ta 30 256 Model LADAN eSa rerea Er EE ESEE EE E EPE EEEE E EaR 33 2 7 Balanced Models marierons iine aeaa EEEE EERE E OA AN AEEA Aa EE EEEE ERDE 38 Chapter 3 Component Coupling by Ports sssccsssssccsscssecsscssesssccessssccsecsssssessscsessssssesesscceee
202. ent is triggered Due to the hysteresis around zero which has to be introduced for numerical rea sons in nearly all cases x is either greater or less than zero when the event is triggered The situation that x is identical to zero can therefore nearly never occur If it is really necessary to check a Real variable for equality the recommended way in Modelica is to check for a small range of values e g Not allowed outside of functions yl if x 0 then 0 else 1 Use instead a small range or if any possible avoid it eps 1000 Modelica Constants eps y2 if abs x lt eps then 0 else 1 2 5 Hierarchical Models We will now use the basic electrical components defined in library Modelica Electrical Analog Basic to build up the electrical circuit diagram of the direct current motor in the next figure see also Figure 1 1 Figure 2 10 DC Motor model as example of a hierarchical Modelica model resistor inductor H R 0 5 L 0 05 i_ref W o inertia flange D DB im gt gt amp i a emf _ J 0 001 Josuagpeads ground The DC motor is modeled with a voltage source the resistance and inductance of the armature and the ideal ized emf electro motoric force component to transform electrical into mechanical energy without losses 30 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 The mechanical flange
203. ented as state space system in controller canonical form The transformation of a ZerosAndPoles system is more involved in order to reduce numerical diffi culties especially for filter implementations The basic approach is to transform the transfer function I s n T s Ny S Ny y k u 17 10 I std I s dystd 4 into a connected series of first and second order systems All these systems are implemented as small state space systems in controller canonical form that are connected together in series The output is scaled such that the gain of every block is one if this is possible in order that the inputs and outputs of the blocks are in the same order of magnitude The output of the last block is multiplied with a factor so that the original gain of the transfer function is recovered A simpler alternative implementation for both the TransferFunction and the ZerosAndPoles system would have been to compute the A B C D matrices of the corresponding state space system with available function calls and then use the general StateSpace model for their implementations The severe disadvantage of this approach is that the structure of the state space system is lost for the symbolic prepro cessing If e g index reduction has to be applied e g since a filter is used to realize a non linear inverse model then the tool cannot perform the index reduction anymore Example Assume a generic first order state space system is pr
204. entical false return end if end for end for end isEqual Here the function is at once terminated once an element of matrix A is not identical to the corresponding ele ment of matrix B This also means that the for loops are at once terminated 5 4 2 Calling functions interactively Modelica functions can be called in a model or a block and of course in another function so they are exe cuted during simulation Another possibility is to call them interactively In fact the command window in the simulation tab of Dymola is an interpreter for the algorithmic part of the Modelica language To be precise all Modelica language elements can be used that are also allowed in an algorithm section of a function In order to ease an interactive use there are a few simplifications e Variables in the interpreter are not declared Instead they are automatically introduced as the result of an operation this also means that no declaration is allowed in the interpreter e g Real x is not allowed e The interpreter supports only assignment statements since it is always an algorithm section To ease the usage the operator can be used but is interpreted as As a very simple example an arithmetic operation shall just be carried out Typing the assignment statement y 2 3 sqrt 2 in to the command window results in the situation displayed in Figure 5 4 90 Chapter 5 Advanced Modelica Language Constru
205. equations the new value of mode is computed by the last equation which is just a direct mapping of the state machine of the figure above Figure Coulomb friction characteristic The described procedure can be easily applied also for the more general friction element in the figure above where the sliding friction force has a nonlinear characteristic and there is a jump in the friction force from fmax to 0 when sliding starts The element equations of the simple friction element need only two changes 1 the linear equation of the sliding friction force has to be replaced by an appropriate nonlinear relationship usually realized by interpolation in a table and 2 the sliding conditions have to be modified to startFor pre mode Stuck and sa gt peak or pre startFor and sa gt 1 startBack pre mode Stuck and sa lt peak or pre startBack and sa lt 1 where peak fmax f0 2 1 All other equations are identical to the simple friction element It is straightforward to adapt this general friction element e g to model clutches or brakes The current implementation of friction models in Modelica allows to solved quite complex problem of coupled friction elements e g models of an automatic gearbox However the description is still too compli cated and it would desirable to find a simpler solution Additional Material Mass with friction and pulling force f ma f frp 10 2 Consider the tw
206. er Real k 1 05 Torque k current Modelica Mechanics Rotational Interfaces Flange b flange Modelica Blocks Interfaces RealInput i ref Modelica Mechanics Rotational Inertia inertia J J Modelica Electrical Analog Basic Resistor resistor R R Modelica Electrical Analog Basic Resistor inductor L L rest of the model as before end Motor Note that a modifier such as inertia J1 J2 means that variable J1 inside inertia iner Draft Jan 21 2009 2 5 Hierarchical Models 31 tia J1 is identical to variable J2 which is defined outside of inertia Clicking now on an instance of model Motor will open the following parameter menu in which the user can directly provide the motor data Figure 2 11 Motor model with propagated component parameters aixi Add modifiers Component leon Name motorWithParameters Motorvith Comment a Model Path BookModels PartA MotorWithParameters Comment Parameters J f 0 001 kg m2 Motor inertia R fo 000 OSR Ohm Armature resistance L f 0 055 H Armature inductance k 4 055 Torque k current OK Info Cancel Parameter propagation can be performed conveniently in Dymola by opening the parameter menu of the de sired component clicking on the small arrow at the right side of the input field and selecting Propagate Figure 2 12 Convenient definition of parameter propagation in Dymola inductor in BookModels Chapter
207. es by applying this model to the simple block on a rough surface shown in the right part of the figure above which is described by the following equation m der v u f Note that m is the mass of the block and u t is the given driving force If the element is in its for ward sliding mode i e s gt 1 this model is described by m der v u f v es il f f 0 f 1 s 1 which can be easily transformed into state space form with v as the state If the block becomes stuck 1 e 1 lt s lt 1 the equation v 0 becomes active and therefore v can no longer be a state i e an index change takes place Besides the difficulty to handle the variable state change there is a more serious prob lem Assume that the block is stuck and that s becomes greater than one Before the event occurs s lt 1 and v 0 at the event instant s gt 1 because this relation is the event triggering condition The element switches into the forward sliding mode where v is a state which is initialized with its last value v 0 Since v is a state s is computed from v via s v 1 resulting in s 1 i e the relation s gt 1 becomes false and the element switches back into the stuck mode In other words it is never possible to switch into the forward sliding mode Taking numerical errors into account the situation is even worse Figure Friction characteristic at v 0
208. es gs devs oauns roves bole sak cd E aa SE KN EE ectdaveodars 230 21 4 Derivative and Special Purpose Operators with Function Syntax cecceeceeseeseetteeeeeneeeeeaees 232 21 5 Event Related Operators with Function Syntax ccceecsecessceseceeeeseeeceeceeeeseeeseeeeureesneeesenaeeeees 232 21 6 Array Operator Seiren n EE AT ETEO E EO EEEE E 233 Chapter 22 Literature sicsisccesssccsssecssvsveccsccsnasssonsvesdesnsdsoecesesscsossncssesensesssnassssveasssccuscdasecenscscustsstsssascesctenes 236 7 Draft Jan 21 2009 Preface This document gives an introduction into object oriented modeling using Modelica performing simulations of Modelica models with the Modelica simulation environment Dymola Dynamic Modeling Laboratory and using Modelica and Dymola for control applications Modelica is a freely available language to model the dynamic behavior of technical systems based on schematics The most important language features Modelica libraries as well as symbolic algorithms needed to transform the high level description of Modeli ca into a form that is suited for a numerical integration algorithm are discussed Modelica is suited for multi domain modeling of large complex and heterogeneous technical systems for example e mechatronic models in robotics automotive and aerospace applications involving mechanical electrical hydraulic thermal and control subsystems e process oriented models with multi phase and multi substance
209. es the C storage layout for code generation When an array is passed to a FORTRAN function the storage layout must first be changed Dymola performs this change automatically if FORTRAN 77 is se lected as language This means every array with 2 or more dimensions is first copied before it is passed to a FORTRAN 77 function This procedure is also needed if the function was transformed with f2c For example the FORTRAN 77 functions from the LAPACK library are first transformed with f2c to C are translated with the C compiler and are stored in the object library Lapack 1lib Then function DGESV from LAPACK to solve a linear system of equations can be accessed in Modelica with the fol lowing Modelica definition function dgesv Solve A x b for x input Real A size A 1 input Real b size A 1 output Real x size A 1 b output Integer info protected Real Work size A 1 size A 1 A Integer pivots size A 1 external FORTRAN 77 dgesv size A 1 1 Work size A 1 pivots x size A 1 info annotation Library Lapack end dgesv The Library annotation tells Dymola to use the object library Lapack lib in the linking phase The ex ternal FORTRAN 77 definition tells Dymola to use the FORTRAN 77 storage layout when passing ar rays i e to copy Modelica arrays from a row wise storage in to a column wise storage scheme So even if dgesv is a C
210. esent x axtbu y extd u Gey and the values of the scalars a b c d are parameters that might be changed before the simulation starts If y has to be differentiated symbolically during code generation then y cktd i x a xt b u i As a result the input u needs to be differentiated too and this might not be possible and therefore translation 188 Chapter 17 Continuous and Digital Control Systems LinearSystems library Draft Jan 21 2009 might fail On the other hand if the first order system is defined to be a low pass filter and the state space system is generated by keeping this structure we have x 5 b x u 17 13 yS Differentiating y symbolically leads to y i 17 14 x bx u ee Therefore in this case the derivative of u is not needed and the tool can continue with the symbolic process ing 17 3 3 Block Components A short description of the input output blocks of the Sampled sublibrary is given in below Table 17 1 The blocks of the Modelica_LinearSystems Sampled library Name Description SampleClock Global options for blocks of Sampled library e g base sample time StateSpace Continuous or discrete state space system block TransferFunction Continuous or discrete single input single output transfer function F LL Be Continuous or discrete single input single output block described by ze ZerosAndPoles ros and poles Analog low or high pass IIR filters Ta Filter
211. et al 1999 Alternative approaches to treat ideal switching elements are a by using variable structure equations which are controlled by state machines to describe the switching behavior see e g Barton 1992 Elmqvist et al 1993 Mosterman and Biswas 1996 or b by using a complementarity formulation see e g Lotstedt 1982 Pfeiffer and Glocker 1996 Schumacher and van der Schaft 1998 The approach a has the disadvantage that the continuous part is described in a declarative way but not the part describing the switching behavior As a result algorithms with better convergence properties for the determination of consistent switching structure cannot be used Furthermore this involves a global iteration over all model equations whereas parameterized curve descriptions lead to local iterations over the equations of the involved elements The approach b seems to be difficult to use in an object oriented modeling language and seems to be applicable only in special cases e g it does not seem possible to describe ideal thyristors Note mixed systems of equations do not only occur if parameterized curve descriptions are used but also in other cases e g whenever an if statement is part of an algebraic loop and the expression of the if statement is a function of the unknown variables of the algebraic loop 10 2 Coulomb Friction not yet complete The simulation of components with ideal switch elements becomes difficult if switch
212. eter menu of Sampled PI controller EPIL in Modelica_LinearSystems Sampled Example Advanced Options Add modifiers 21x General Component leon Name PI1 i a Comment Model Path Modelica_LinearSystems Sampled Pl Comment Proportional Integral controller continuous or discrete block Parameters k kv Gain T Tv s Time Constant T gt 0 required OK Info Cancel In the Advanced Options tab see Figure 17 8 the remaining block settings are present especially to define whether it is a continuous or a discrete block and in the latter case define also the discretization method Since parameter sampleFactor is 1 the base sampling time of the sampleClock component is used In other blocks of the controller Samplerl Sampler2 parameter sampleFactor 5 which means that the inputs and outputs of these blocks are sampled by a sampling rate that is 5 times of the base sample time defined in the sampleClock component Note the samplerxX blocks in Figure 17 6 have only an ef fect if the controller is discrete For a continuous representation these blocks just state that the output signal is identical to the input signal Figure 17 8 Advanced Options parameter menu PI1 in Modelica_LinearSystems Sampled Examples SimpleControlledDrive 2 x General Advanced Options Add modifiers blockType gt Type of block Continuous Discrete methodType zb Type of discret
213. execution model of a StateGraph follows from its Modelica implementation Given the states of all steps i e whether a step is active or not active the equations of all steps transitions transition conditions 200 Chapter 18 Hierarchical State Machines StateGraph Library Draft Jan 21 2009 actions etc are sorted resulting in an execution sequence to compute essentially the new values of the steps If conflicts occur e g if there are more equations as variables of if there are algebraic loops between Boolean variables an error occurs Once all equations have been processed the active variables of all steps are updated to the newly calculated values Afterwards the equations are again evaluated The iteration stops once no step changes its state anymore i e once no transition fires anymore Then simulation contin uous until a new event is triggered i e when a relation changes its value 18 6 Infinite Unsafe and Unreachable StateGraphs It is possible to define networks with the StateGraph library that should not be allowed and should lead to an error since they give rise to infinite instantaneous looping or since they are unsafe or unreachable For ex ample the following StateGraph Figure 18 14 Erroneous StateGraph infinite looping intialStep transition transition2 true true starts at the initialStep fires at once at transition1 goes to step fires at once at transi tion2
214. f f at issingular 12 25 Since the basic requirement of a regular Jacobian is violated it is no longer possible to transform to the ODE form 12 24 Draft Jan 21 2009 12 4 Singular Systems Pantelides Dummy Derivative not yet available 165 XXXX 12 5 Hybrid DAEs need Symbolic Preprocessing There are a large number of publications how to solve DAE systems by numerical methods and there are sev eral simulation systems based on this approach In most engineering applications pure DAEs do not occur but usually in combination with discontinuities or other event based effects as described in Part B of this document These types of systems are called Aybrid DAEs In this section the limits of numerical integra tion methods shall be discussed when applied to hybrid DAEs without symbolic preprocessing The problem atic cases are singular DAEs i e DAEs that cannot be transformed algebraically to state space form without differentiating equations Although certain classes of singular DAEs can be directly integrated with a numeri cal method severe problems occur when variables change discontinuously This shall be demonstrated by means of the simple example of Figure 12 6 Figure 12 6 Simple drive train with discontinuous change of the driving torque that is difficult to handle by a numerical method without symbolic preprocessing ramp idealGear inertial inertia2 a UE _ gt _ aL D t J 1
215. feedback a feedback2 t StateSpace F FA ho obo e O A ttTransferFunction LL 7 z ion f_cut 5 k k ZerosAndPoles ation 2 IE 5 k o eference A 1 Filter E 9 pe FI B x B FilterFIR E b i G a 4 ntegrator Derivative i FirstOrder lt 38 26 4SecondOrder 3 3 API UnitDelay Z ADconverter torque spring a i O D O e 44DAconverter tau rT ii i J01 eben 03 plant flexible drive In the example of Figure 17 6 block filter is a continuous filter that filters the reference motion of the motor here ramp For this reason filter uses explicitly the blockType Continuous whereas all oth er blocks use the blockType UseSampleClockOption i e they use the setting defined in sample Clock By changing the blockType from Continuous to Discrete in block sampleClock the con troller is automatically transformed from a continuous to a discrete representation Every block of the Sampled library has a continuous input and a continuous output Inside the respec tive block the input and output signals might be sampled with the base sampling period defined in sam pleClock or with an Integer multiple of it For example the PI controller in Figure 17 6 has the following parameter menu to define the continuous parameterization of the PI controller i e gain k and time constant T 184 Chapter 17 Continuous and Digital Control Systems LinearSystems library Draft Jan 21 2009 Figure 17 7 Param
216. fined in the Text layer then the graphical information about the connectors would not be present and then it would not be possible to connect graphically to this component in a schematic Therefore connectors should always be dragged in to the Diagram layer The complete definition of the inductor model is model Inductor Ideal linear electrical inductor model parameter Modelica SIunits Inductance L Modelica Electrical Analog Interfaces PositivePin pin p annotation Placement transformation extent 110 10 90 10 Modelica Electrical Analog Interfaces NegativePin pin n annotation Placement transformation extent 90 10 110 10 equation OQ pin_p i pin_n i L der pin p i pin p v pin_n v end Inductor pin _p and pin_n are instances of the connector classes PositivePin and Neg ativePin respectively Variables in these instances are accessed by dot notation e g pin _ p i is the current i in pin_p The inductor is defined by two equations a The first equation states that the sum of the currents flowing in to the inductor via the two ports is zero Since by convention an inflowing current is positive this means that if a name 24 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 current is flowing in to pin_p then the current at pin_n has the same absolute value but is negative and theref
217. first part of a class name is first searched in the Dymola distribution di rectory under Dymola Modelica and Dymola Modelica Library Only if the name is not found the search strategy from above is used searching in the current directory and then in the directories of MODEL ICAPATH The advantage is that a user cannot accidentally shadow classes in the Dymola distribution which can lead to unexpected and wrong behavior Sometimes it is desired to search the Dymola library loca tion not in the first place Then the special string SDYMOLA Modelica Library has to be included in MODELICAPATH at the desired place exactly as it is given above Note that the Unix directory separator has to be used even if MODELICAPATH is defined on a Windows system A package could be manually constructed with a text editor in the file system It is however much more convenient to define it with a Modelica tool In Figure 2 15 it is sketched how a new package can be defined in Dymola Selecting File New Package opens the menu Create New Package Here basically the name of the desired package can be defined and whether the package shall be stored in one file or as a directory in the latter case deselect the check box Save contents of package in one file A final menu pops up where the user has to select Accept if the package shall be created as root level package not inside another package If the
218. foPreload reference Modelica LinearSystems The package is searched in the search path with the package name given under reference here Modelica_LinearSystems It is shown in the library menu with the string given under text here Modelica_LinearSystems and it is shown in Dymolas status bar with the string given under description here Modelica LinearSystems Library 0 95 Entry pos defines the position in the library menu Numbers 0 999 are reserved for Dynasim If command Library InfoPreload is given in the file this command is prepared in a comment above then the library is automatically loaded when Dymola is started and it is therefore shown in the package browser 2 7 Balanced Models In Modelica 3 0 the concept of balanced models is introduced in order to significantly improve the quality of Modelica models and to allow a tool to provide very precise error messages about the source of an error A detailed description and discussion of this concept is provided in Olsson et al 2008 This section is based on this article Basically restrictions are introduced in to Modelica in order that every model must be locally bal anced which means that the number of unknowns and equations must match on every hierarchical level It is then sufficient to check every model locally only once e g all models in a library Using these models in stantiating and connecting them redec
219. for providing the desired measurements Ym e An inverse plant model from y to u in the feedback path is used to make the closed loop system robust against model errors and disturbances e The feedback controller in the feedback loop is used to stabilize the system 19 7 Example Application In this section the feedforward and feedback linearization controller structures as discussed in the previous sections will be illustrated by means of a simple example The plant description is from F llinger 1998 page 279 A substance A is flowing continuously into a mixing reactor Due to a catalyst the substance reacts and splits into several base substances that are continuously removed The reaction generates energy and there fore the reactor is cooled with a cooling medium The cooling temperature T t in K is the primary actua tion signal Substance A is described by its concentration c t in mol l and its temperature 7 in K ac cording to the following DAE T y ckye a c ar yta 19 22 T ax T a y tany t b T with k 1 24 10 a 0 00446 a 0 0303 e 10578 a 0 0141 a 2 41 19 23 b 0 0258 a 0 00378 a 1 37 For the given input 7 these are 1 algebraic equation for the reaction speed y t and two differential equa tions for c t and 7 The concentration c t is the signal to be primarily controlled y and the tempera ture 7 f is the signal that is measured y
220. g The basic idea is well known under different names since the sixties It is sketched at the following simple example 162 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 Zz f 25 Z f z Z3 f3 Z1 Z2 12 17 Z4 fa Z2 Z3 Z f 5 Z4 2 This is a system of 5 algebraic equations in 5 unknowns zi It is not possible to reduce the dimension of the algebraic loop by sorting here The numerical solution of this non linear system of equations would be com puted by providing guess values for the 5 unknowns and then computing the residues of every equation The numerical solver modifies the unknown variables so that the residues become small Therefore the equation system must be provided to the numerical solver in form of the following function input Zi Z2 Z3 Z4 Zs iteration variables output Fi F2 F3 F4 rs residues rps Zi filz F2 i Zo falzi algorithm r 2 f3 Z1 Z3 r4 A falza z3 r5 Ze fs Z4 z Due to the special structure of the equations where some unknowns can be computed explicitly from other unknowns it is possible to reduce the number of iteration variables from 5 to 1 input Zs iteration variable output rs residue zi fi zs Z falzi algorithm z f3 Z z Z4 Size z3 F5 2 flay z Since a non linear numerical solver needs at least O n operations to compute the solution where n is the number of
221. g force Real s Position of block Real v Speed of block annotation Diagram Rectangle extent lt other definitions gt equation v der s m der v f end MovingMass1 Modelica keywords are written in bold A model component is defined within model lt Name gt end lt Name gt where lt Name gt is the user defined model name A Modelica model consists of sections that are separated by section keywords such as equation above Every variable or component needs to be declared when it is utilized in a model In the model above the needed variables m f s v are declared as real numbers due to the Real definition The keyword parameter defines that a the corresponding variable is constant and does not change its value during simulation and that b this declaration is included in the parameter menu of the model in order that a new value can be provided The value of a parameter can be either defined directly at its definition e g m 2 above or when using the model through the parameter menu The above model therefore defines two parameters m and f with pre defined values and two time varying variables s and v The order of the declarations does not matter Especially a variable can be used before it is declared The reason is that Modelica models are usually built up graphically by dragging components and then it would be a large burden if it would be im portant in which orde
222. g in a first step that pre values are unknown and that the equation pre v v is added BLT partitioning will then result in algebraic loops and these loops will identify the minimal parts for the event it eration In a second step BLT partitioning is applied again under the assumption that pre v is known The event iteration has then only to be applied on the identified minimal parts The effectiveness of this algo rithm can be checked at the following code fragment where a sampled data system is implemented not with the sample operator but by Boolean equations equation when time gt pre tNext then tNext pre tNext samplePeriod controller equations end when The above code fragment generates time events at every sampelPeriod time instant At every sample in Draft Jan 21 2009 9 2 Event iteration 135 stant the whole model equations need to be evaluated at least twice since after the first evaluation pre tNext tNext and therefore an event iteration has to take place The proposed approach adds the equation pre tNext tNext and identifies the mapped equation condition time gt pre tNext tNext if edge condition then pre tNext samplePeriod else pre tNext pre tNext tNext as algebraic loop since tNext and pre tNext are not used at any other place As a result a local event iteration can be used when generating code Pseudo code of loc
223. g initialization via the initial equation section approach as discussed in section 4 3 below should only be performed in special cases 4 3 Initialization with initial equations Arbitrary initial equations g can be defined in an initial equation section All equations present in such a section are only evaluated during initialization and are ignored otherwise This approach is demon strated with the first order system from above This time the goal is steady state initialization of the system model FirstOrder3 initialize always in steady state parameter Real k gain parameter Real T time constant RealInput u input signal RealOutput y output signal equation T der y y k u initial equation der y 0 end FirstOrder3 This model defines the following two equations for the initialization phase der y 0 T der y y k u Under the assumption that the initial value of the input variable u is computed somewhere else the two equa tions above have the solution der y 0 y k u and this is the solution of the initialization problem Let us analyze a slightly more complicated system a mass point in a gravity field that is attached with a spring to the environment The goal is to initialize the spring mass system in steady state i e after initial ization the system shall be in rest The Modelica model of this model is shown below together with an ani matio
224. g of normal ized has to be selected appropriately For example the signal processing toolbox of Matlab provides the fil ters in non normalized form and therefore normalized false has to be set if results shall be com pared In Figure 17 4 the magnitudes of the 4 supported low pass filters are shown in normalized form and in Figure 17 5 the corresponding step responses are given generated with the Modelica_LinearSystems Sam pled library Figure 17 4 Magnitudes of normalized low pass filters of order 4 Normalized low pass filters of order 4 critical damping Bessel Butterworth Chebyshev A_ripple 3dB 4 _ _ r mrar Frequency in Hz 182 Chapter 17 Continuous and Digital Control Systems LinearSystems library Draft Jan 21 2009 Figure 17 5 Step responses of normalized low pass filters of order 4 step Criticalbamping Bessel Butterworth Chebyshev 1 4 1 2 1 04 0 8 0 64 0 4 4 0 2 4 0 0 0 2 4 T T T T T T T T T T T T T T T T T T T Obviously the frequency responses give a somewhat wrong impression of the filter characteristics Although Butterworth and Chebyshev filters have a significantly steeper magnitude as the CriticalDamping and Bessel filters the step responses of the latter ones are much better since the settling times are shorter and no over shoot occurs This means for example that a CriticalDamping or a Be
225. goes to initialStep and continues Therefore at a given time instant the state machines loops forever Another unwanted situation is the case e g if two initial steps are introduced although only one should be present Figure 18 15 Erroneous StateGraph two initial steps initialStep transition transition2 true true It is not predictable what will occur in such cases Still another erroneous situation is displayed in the follow ing figure where the transition from step1 to step5 goes out of the parallel branches which should not be al lowed Figure 18 16 Erroneous StateGraph Branching out of a parallel execution component steps T3 T4 step6 true With the current StateGraph library all the cases above can be modeled without getting an error message Draft Jan 21 2009 18 6 Infinite Unsafe and Unreachable StateGraphs 201 The behavior is mostly undefined and the StateGraph will not work as expected As shortly mentioned in the introduction of this chapter a new state machine Modelica library called ModeGraph is currently under development By suitable enhancements of Modelica it is possible that either by construction unsafe state machines can not be build or a translation error occurs The new methodology has also significant other advantages 202 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 Chapter 19 Nonlinear Inverse Models for Advanced Contro
226. gram layer of the model and the icon laver of the Plug connector is shown in the icon layer of the model Usually the icon in the icon layer of a connector is too small if more details then just a square or a circle are shown The connector icon can be enlarged by selecting the icon layer of the Plug connector and then choose Edit At tributes Graphics see Figure 3 14 Draft Jan 21 2009 3 6 Hierarchical Connectors 57 Figure 3 14 Enlarging the default icon size when dragging a connector or model 2x General Version Attributes aye Coordinate system Min Max Horizontal 100 100 Vertical 400 hoo Grid Horizontal 2 Vertical 2 onent scale factor amp size of an inserted component is scale coordinate system Leave this field empty to use the default component size Aspect ratio k Preserve aspect ratio when resizing components of this class The default Scale factor is 0 1 This means that when dragging a component then the icon size is 0 1 times the size of the Coordinate system In order to enlarge the default icon size by a factor of two Scale is set to 0 2 above Additionally only the icon in the icon layer shall be enlarged not the icon in the dia gram layer For this reason the Copy attributes to Diagram layer is disabled Connecting two instances A and B of the Plug connector results in a set of equations where co
227. h is the height over ground and v is the forward velocity Additionally there are other equations describing e g the aerodynamics and the rotation of the aircraft Stationary initialization for such an aircraft model would require that h 0 v 0 i e the forward velocity of the aircraft is zero However this is not possible because an aircraft can only fly if the forward velocity is sufficiently large The only meaningful way of initialization is here to define the initial height h ho instead of setting the derivative of the height to zero e A drive system driven by an electrical motor shall usually not be initialized in steady state which would mean that the drive system is at rest but in such a way that the load drives with a pre defined speed For synchronous or asynchronous electrical motors this means that voltages and currents in the motor have a sine wave behavior which is obviously far away from steady state Changing the model variables by a non linear state transformation the so called Park transformation so that the transformed voltages and currents are defined with respect to the rotating coordinate system of the motor inertia gives the desired initialization by setting the derivatives of the transformed quantities to zero Whenever a non linear system of equations is solved guess values have to be defined for the iteration variables The reason is that non linear equation solvers are only able to compute a s
228. h measured ones This can be performed by setting the flag Advanced TurnStatesIntoInputs true in Dymola before translation to transform all states into inputs in the generated code This code can be incor porated with the Simulink export feature of Dymola in another environment such as Simulink from Math works Currently it is not possible to import this transformed system in to Modelica again Dynasim plans to support this in the future For the example the differentiated equations are added manually to the model and Cis and the plant states c T are selected as input variables This has been performed by 1 translating the model of Figure 19 18 in Dymola with option Simulation Setup Translation Generate listing of translated Modelica code in dsmodel mof 2 copying the equations of file dsmodel mof in to a new InverseMixingUnit model and 3 changing all der xx to new variables der xx This derivation is of course no practical for complex models The design is finished by adding the feedback controller e g 19 19 in case of feedback linearization For the model at hand 19 19 is defined as cl ki Came k 19 29 whereby c is available as measurement signal is computed with the inverse model and c is the input der2_c to the inverse model of Figure 19 18 By choosing bk n sy k 2 27rf 19 30 with f 1 300 Hz the same closed loop dynamics
229. h the i output with relative degree p The desired dy namic behavior of the closed loop system is then imposed by application of an additional feedback law like for example v kDa Yahi ar kon r 19 19 Note that this feedback law requires availability of the p 1 derivative of the controlled output This derivative may be obtained from measurements or less favorably from the computed value in the inverse model In aerospace applications first or no time derivatives are usually required since relative degrees of controlled variables tend to be low 1 or 2 One reason for this is that control laws are designed in the form of multiple cascaded loops In case the inverted model exactly represents the true system the closed loop system becomes ko s kg s H kis tho Note that the coefficients may be selected to match the reference model in Figure 19 9 In case time derivatives of the desired output y a are available the relative degree i e the phase lag of the response of this linear closed loop system may be reduced provided that these are not too fast as to re quire too large control inputs An important disadvantage of feedback linearization is that the state vector of the plant must be fully available from measurement and or estimation Automatic generation of feedback linearization control laws in Modelica will be illustrated in section 19 7 This procedure has been applied for an automatic landing system and m
230. h which definition is used we will assume that the reader applies the stability definition that is most appropriate for his her applications In the following we will therefore use the term stable DAE without defining it mathematically A DAE will be called unstable if it is not stable In the following only inverse models will be used in a controller if the DAE of the inverse model has a unique solution and if it is stable As discussed in section 19 1 for linear single input single output systems the latter requirement means that the plant must have stable zeros that is it must be a minimum phase sys tem To simplify notation we will generalize the term minimum phase system to non linear DAEs by the following definition Chin Isidori 1995 a non linear state space system is defined to be minimum phase if the zero dynamics is globally Draft Jan 21 2009 19 2 Inversion of Nonlinear Models 209 Definition A DAE is called a minimum phase system with respect to a subset of its inputs u and a subset of its alge braic or differentiated variables x ys with dim u dim x y if the inverse system from x ys to u has a unique solution and is stable In the following we will assume that models to be inverted are minimum phase systems according to this definition In section 19 8 it is discussed how to proceed if this requirement does not hold Since the transformation from 19 11
231. hat a modifier equation has to be provided for the corresponding variable when the component is utilized in order to guar antee a locally balanced model i e the number of local equations is identical to the local number of un knowns see section 2 7 Example model VoltageSource input Modelica SIunits Voltage u modifier required since input Pin p nj equation u p v nv 0 p i nails end VoltageSource model ElectricalCircuit VoltageSource source u sin time modifier required for u rlLectricalCircuit mT G end Finally in functions see section 5 4 the input output prefixes define the computational causality of the function body i e given the variables declared as input the variables declared as output are computed in the function body Let us now return to the definition of the input output connectors for components of a block diagram The in troduction of the input output prefixes in connectors RealInput2 and RealOutput2 above is the most im portant step However such a connector is inconvenient to use due to the unnecessary hierarchy For exam ple model FirstOrder NotNice RealInput2 inPort RealOutput2 outPort firstOrder gt parameter Modelica SIunits Time T Time constant equation T 1 T der outPort signal outPort signal inPort signal end FirstOrder NotNice As can be seen it is tedious to use a signal name such as outPort signal in a
232. hat have suitable defaults of 1 The state x of the state space form shall be accessible from the outside e g to set initial conditions It has to be declared as output because it is computed inside the block and every public declaration in a block must have the parameter constant input or output prefix The dimension of vector x is zero when the denominator coefficient vector a has only one element In the protected section the needed dimensions are defined and especially the utility vector bb This vector has at least the same size as the denominator coefficient vector a and is constructed by con catenating a zero vector with vector b The basic goal is that a and bb have the same size in order that the two vectors can be used in the same operation Therefore the desired definition of bb is Real bb na vector zeros na nb 1 b Note that zeros na nb 1 b is a column matrix that is transformed to a vector with the vector cast operator If a is e g defined as 1 2 3 and p is defined as 4 then bo 0 0 4 A user might define vector b with more elements as vector a and then the above construction gives a trans lation error because the built in function zeros has a negative input argument which is not allowed and this error is difficult to understand by the user One might declare the dimension of vector
233. he actual dimensions and with the prefix redeclare The above approach allows to define pop down menus in a quite convenient way There is however one drawback The modifier to the record instance is a function call i e an expression This means that the record cannot be modified in the plot variable browser before simulation starts If for example the diameter in the example above is changed Dymola has to re translate the model Providing the input arguments to the function as parameters e g as in parameter Modelica SIunits Diameter D 0 04 Beam beam data crossSection BeamChoices circle diameter D does not help either currently in Dymola since Dymola does not allow to change D before simulation starts and therefore changing D requires again a re translation of the model 5 5 3 Records with array elements partially available Records may have arrays as elements For example in the Modelica_ LinearSystems library a free li brary available from the File Libraries menu in Dymola linear state space systems are available as x A x B u y C x D u 5 7 The data of such a system is basically provided as a record record StateSpace Real A Real B size A 1 104 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 Real C size A 1 Real D size C 1 size B 2 end StateSpace where the 4 matrices are elements of thi
234. he definition is present and column icon is a screenshot of the connector icon In Modelica array dimensions are declared with square brackets e g A 3 4 is a two dimensional array where the first dimension has size 3 and the second size 4 see also sec tion 5 1 Every physical connector is available with two different icons one with a filled and one with a non filled icon Otherwise the Modelica definition is identical In the table above only the filled icon connectors are shown For 3 dim mechanical systems the position vector r 3 from the origin of the world frame to the ori gin of the connector frame the transformation matrix T 3 3 from the world to the connector frame as well as the cut force vector 3 and the cut torque vector t 3 are utilized The transformation matrix T 3 3 contains a redundant set of variables In Chapter 15 it is explained how the information about the constraint equations of T is defined in Modelica overdetermined connector and how a tool can automati cally remove connection restrictions due to this redundancy based on this additional information For 1 dim thermo fluid flow systems a third type of connector variables is needed called stream vari ables This variable type is used to describe properties that are transported with a flow of matter Details are given in Chapter 16 The stream variables of the FluidPort are specific enthalpy h independent mass frac tio
235. he freely designable part R of the desired reference transfer function with the stable part of the plant inverse Example s 1 s 2 s 3 s 4 s 5 s 1 SEB A ea T ra aaa In this example the open loop controller F is constructed with the inverse of P but not using the unstable term s 1 and adding additional poles until the relative degree Dr is zero and therefore the controller is causal The gain 1 is additionally introduced in order that the steady state gain of the reference transfer func tion is P s 0 F s 0 1 The two time constants T and T of the controller can be selected arbitrarily with the only restriction that they must be positive T gt 0 T gt 0 in order that the controller is stable 19 1 3 Standard controller with one structural degree of freedom If the plant is unstable and or the controller shall compensate for unknown disturbances a feedback con troller is needed The standard controller configuration for this case is shown in Figure 19 3 d y Tyr Tyn t Tyd r e u y td ly y C m P PC PC 1 2 Hre IC Pe d 3 Desired C such that T 1 T 0 T 90 Figure 19 3 Controller C with one structural degree of freedom for a plant P P might be unstable The identifiers in Figure 19 3 have the following meaning e P s is the plant transfer function e C s is the feedback controller transfer function y is the output signal of
236. he shortest name by using an unqualified import statement import Modelica SIunits Name before must be a package parameter Mass m 2 Mass is searched in all imports Velocity v start 3 It is a matter of taste which of the variants to use In the Modelica Standard Library often the third variant with a user defined name is present We are now in the position to show the recommended implementation of the simple example from above model MovingMass3 Mass pulled with force recommended implement import SI Modelica SIunits parameter SI Mass m 2 Mass of block Draft Jan 21 2009 2 1 Simple Declarations and Equations 21 parameter SI Force f 6 Pulling force SI Position s Position of block SI Velocity v Speed of block annotation Diagram Rectangle extent lt other definitions gt equation v der s m der v f end MovingMass3 Besides the equations of a model also additional information can be stored together with the model within the annotation definition In the Dymola GUI this is performed by clicking on button text layer see Figure 2 4 below to input the declarations and equations in the text layer and by clicking on button dia gram layer see Figure 2 4 below to draw a picture of the system for documentation purposes Figure 2 4 Diagram and text layer of MovingMass3 model in Dymola MovingMass3 MovingMas
237. his prop erty O n operations are needed It is then possible to transform the system above to b L b b ea A 0 12 22 J L Z The result is a BLT form with a usually much smaller diagonal block in the left upper corner The lower right corner is matrix L which has only trivial non zero diagonal elements As a result the original algebraic loop in dim z dim z2 equations has been transformed symbolically during translation into an algebraic loop of dim z2 equations It the number of tearing also called iteration variables z is small this reduc tion in the size of the algebraic loop leads usually to a much more efficient computation of the unknowns 12 4 Singular Systems Pantelides Dummy Derivative not yet available In the previous sections we have analyzed DAEs of the form 0 f x t x t y t t tEIR xER yelR fElIR 12 23 where x are variables that appear differentiated in the model equations and y are algebraic variables Algo rithms have been sketched and demonstrated by examples to transform this DAE into an ODE of the form X t _ gry bey f x t t 12 24 This transformation is possible if and only if the Jacobian with respect to the unknowns is regular at all time instants In this section systems are handled where this requirement is not fulfilled We will call them sin gular systems because the Jacobian is singular o
238. iable openValve is set as entry action The question is whether openValve will have value 0 or 1 after execution of the steps Stateflow changes this non deterministic behavior to a formally deterministic one by defining an execution sequence of the states that depends on the graphical position The light number on the right of the states shows in which order the states are executed In the upper part of Figure 18 5 this means that openValve 0 after execution of the parallel states In the lower part of the figure the second state fill2 is changed a little bit graphically and then open Valve 1 after filll and fill2 have been executed This is a dangerous situation because a slight changes in the placement of states might change the simulation result and b if the parallel execution of ac tions depends on the evaluation order errors are difficult to detect Similar problems occur in other state machine formalisms such as SFC Grafcet and Graphcharts Vari ables are changed according to an evaluation sequence of the simulator It seems not possible to provide an easy to grasp rule about evaluation order of actions that are executed in parallel Therefore either the simu lator just uses an internal evaluation order or non obvious rules are present as in Stateflow that do not solve the underlying problem In a StateGraph such a situation is detected by the translator resulting in an error since there are two eq
239. iable with v pre v after the event has been processed a new event instant is triggered pre v v is set and the variables are again computed accord ing to the previous paragraph This phase is called event iteration If an event is finalized with v pre v for all discrete variables v the integration is restarted and the differential and algebraic equations are again solved with a numerical integration method The above informal description can be formally described by the following procedure 1 Only equations of the form v expr or statements of the form v expr are allowed in a when clause Variable v on the left hand side of the equal sign can have a base type of Real Integer Boolean String or enumeration and or it can be a record record element array or array element reference of one of these base types 12The time instant at which a relation is changing its value is determined up to a precision determined by the integration method It is usually in the order of 10 s 130 Chapter 8 The Synchronous Principle Draft Jan 21 2009 In a first step a Modelica translator transforms a hierarchical Modelica model into a flat set of Modelica statements consisting of the equation and of all used components by e Expanding all class definitions flattening the inheritance tree and adding the equations and assignment statements of the expanded classes for every instance of the
240. iate branches For example the result of the output set assignment of the example above S l25n 4 0 f5 Z2 Z3 Z6 lt 0 felz2 0 f3 23 Zs 0 f z3 z 0 f z 2 can be transformed to the following set of nodes Draft Jan 21 2009 12 2 Sorting BLT 161 Figure 12 4 Output set assignment visualized as directed graph incomplete y lo f z lo f zl Jo f z O f z lo f z lo f z lo f zl Figure 12 5 Output set assignment visualized as directed graph complete lo s z where every equation f is a node together with its assigned variable The dependency of the equations from the respective variables is defined by appropriate branches For example a branch from node f z3 to node fi Z4 signals that variable z is needed to compute z4 The complete directed graph of this example is shown in the next figure The task is to determine the loops in this directed graph also called strong components For this the very efficient graph algorithm of Tarjan Tarjan 1972 is used that determines all loops in O n operations where n is the number of nodes i e the number of equations In the example only one loop is present as visualized in the figure above by a larger line width of the corresponding branches All variables in this loop can only be computed together and characterize the smallest algebr
241. ification Modelica 2007 defines how a Modelica model shall be mapped into a mathematical description as a mixed system of differential algebraic equations DAE and discrete equa tions with Real Integer and Boolean variables as unknowns This is also described in chapter 8 in this docu ment There are no general purpose solvers for such problems There are numerical DAE solvers which could be used to solve the continuous part However if a DAE solver is used directly to solve the original model equations the simulation will be very slow and initialization might be not possible for singular sys tems see section 12 4 It is therefore assumed that Modelica models are first symbolically transformed into a form that is better suited for numerical solvers In this chapter the transformation techniques are sketched that have been initially designed for the Dymola modeling language Elmqvist 1978 further developed in Omsim for the Omola language Pantelides 1988 Mattsson and S derlind 1993 and further improved in the commercial Modelica simulation environment Dymola Mattsson et al 2000 Dymola 2008 Dymola converts the differential algebraic system of equations symbolically to ordinary differential equations in state space form i e solves for the derivatives Efficient graph theoretical algorithms are used to determine which variables to solve for in each equation and to find minimal systems of equations to be solved simultaneously algebraic loo
242. ignal y is a discrete signal since it is assigned in a when clause at every sampling instant Therefore pre y is the value of y before the current event occurred i e it is the value of y from the previous sampling instant since y does not change its value during event in stants The implementation is then straightforward in the when clause Assume this discrete block is used as controller for a continuous plant For simplicity we collect the equa tions together in one model although in an actual model this would be a block and a model instance that are connected appropriately together Equations of continuous plant der xc f xc uc yc g xc Equations of discrete controller when sample 0 T then yd a pre yd b ud end when Equations that connect plant and controller together uc yd ud yc_ref yc A Modelica simulation environment will map the when clause to an instantaneous equation and will sort the equations resulting in Sorted equations with input yc ref xc pre yd yc g xc ud yc ref yc Cc sample 0 T yd if edge c then a pre yd b ud else pre yd uc yd der xc xc uc During continuous integration the plant equations are evaluated using yd pre yd i e the value of yd from the last event instant At a sampling instant the equations of the discrete controller are used to compute the input to the plant based on the value of the plant o
243. igure 4 6 Warning message of Dymola if not enough initial conditions are defined Translation of SpringMassSystem The DAE has 2 scalar unknowns and 2 scalar equations Warning The initial conditions are not fully specified Dymola has selected default initial conditions LogDefaultlnitialConditions true gives more information Additionally Dymola applies some suitable heuristics to construct a well defined initialization problem In many cases this is the problem that the user would like to solve However it might also be that the initializa tion problem constructed by Dymola is far away from the expectations of the modeler Furthermore the model is no longer portable between different Modelica tools because in the Modelica Language Specifica tion it is not defined in which way a tool should act if not enough initial conditions are provided It is very likely that different Modelica tools use different heuristics or different warning error messages strategies For these reasons it is highly recommended to always enable the option Log selected default initial conditions in the Simulation Setup Translation menu of Dymola see Figure 4 7 Draft Jan 21 2009 4 4 Too many or not enough initial conditions 67 Figure 4 7 Dymola menu to enable Log selected default initial conditions in Simulation Setup Translation Experiment Setup Optput Debug Compiler Realtime Mo
244. imple drive train consisting of motor and load inertia as well as of a gear box sine al torque inertial inertia2 i N Ta iL gt po 9 p tau KA T ratip 6 J 0 1 J 2 freqHz 2 In the right part of the figure the Modelica drive train is shown consisting of a motor inertia inertial a load inertia inertia2 a gear box idealGear and an applied external torque torque sine The load inertia is driven with the external torque and drives via the idealized gear box with a transmission ratio of 6 the load inertia Therefore the speed of the load inertia is 6 times slower as the speed of the motor iner tia In the left part of the figure a sketch of the 3 dim view of this system is shown The drive train is fixed on ground How to model drive trains that are moving in space is discussed in Chapter 14 Drive train components are connected by shafts and the shaft ends are the drive train interfaces In the Rotational library they are called Flanges The definition of these flanges is shown in Table 3 4 Table 3 4 Interface of drive train components of the Rotational library connector Flange b Modelica SIunits Angle phi rad flow Modelica SIunits Torque tau N m end Flange b There is a coordinate system along the axis of rotation which is fixed in the ground the red coordinate sys tem in the figure above Furthermore a coordinate system is attached rigidly with every shaft
245. in another generic modeling system The reason is that a differential algebraic equation system DAE of the form 0 f x x y t 11 1 where x are variables that appear differentiated and y are pure algebraic variables must be fulfilled all the time also at an event instant Assume that this DAE is fulfilled at the current event instant and that one or more elements of x and or y are changed discontinuously then 1 Other variables must be changed discontinuously too in order that the DAE is still fulfilled For example if there is an equation 0 yl y2 and y2 is changed discontinuously then y1 must be changed discontinuously too Ifm lt nx variables are changed discontinuously at the same event instant there is usually no unique solution to compute all the other variables so that the DAE is still fulfilled 2 If the variable that is changed discontinuously appears also differentiated in the DAE e g v is changed discontinuously and der v appears in the equations then a Dirac impulse occurs which means that other continuous variables need also to be changed discontinuously due to this Dirac pulse In section 11 1 the current unsatisfactory solution in Modelica to this problem is presented together with some examples In section 11 2 an experimental feature is described that was developed by Dynasim in the EU project REALSIM and that is considerably better as the current Modelica solution but there a
246. ince all three transitions fire af ter 1 second they are all candidates for the active branch If two or more transitions would fire at the same time instant a priority selection is made The alternative and parallel components have a vector of connec tors Every branch has to be connected to exactly one entry of the connector vector The entry with the Low est index has the highest priority Parallel Alternative and Step components have vectors of connectors The dimensions of these vectors are set in the corresponding parameter menu For example in a Parallel component Figure 18 9 Defining the number of branches in a Parallel component Parallel1 in ExecutionPaths2 21x General Add modifiers l Component Name Parallel1 Comment Model Path Modelica StateGraph Parallel Comment Parallel splitting of execution path use component between two transitions Parameters nBranches 2 Number of parallel branches that are executed in parallel OK Info Cancel Currently in Dymola the following menu pops up when a branch is connected to a vector of components in order to define the vector index to which the component shall be connected Figure 18 10 Menu to define the vector elements that are connected connect Parallel1 split step1 inPort 2 x Connect to only parts of the connectors by giving indicies below Make the connection connect Parallel split jz El step1 inPor 0 Bj i WV La
247. ing a complete array to the corresponding variable As an example the following function has a vector of complex numbers as input argument c represented as a two column matrix where the first column are the real and the second column are the imaginary parts of the complex numbers and returns the real values r that do not have a complex part therefore the dimension of r depends on the element values of c but not on the dimension of c function realNumbersl Return real numbers of a complex vector input Real c 2 Complex numbers col 1 real col 2 imag output Real r The real numbers of c protected Integer nr 0 Dimension of r Integer j 1 algorithm Determine the size of vector r for i in 1 size c 1 loop if c i 2 0 0 then nr nr 1 end if end for Generate vector r and copy the numbers r zeros nr for i in 1 size c 1 loop if c i 2 0 0 then Pi Y erp ely J j end if end for end realNumbers1 Note before assigning an element to array r this array needs to be defined by assigning an array to it here by assigning a vector of zeros with r zeros nr It is an error if a value is assigned to an el ement of an array if this element does not yet exist Alternatively a function can be provided to determine the dimension of a vector and this function is Draft Jan 21 2009 5 4 Functions 95 used in the declaration of this vector With this
248. ing of Nonminimum Phase Plants and Application to Flight Control AIAA 2002 4760 AIAA Guidance Navigation and Control Conference and Exhibit Monterey California Slotine J E and W Li 1991 Applied Nonlinear Control Prentice Hall Englewood Cliffs N J Steinhauser R G Looye and O Brieger 2004 Design and Evaluation of a Dynamic Inversion Control Law for X 31A Proc 6th ONERA DLR Aerospace Symposium Berlin June 22 23 pp 25 33 Th mmel M G Looye M Kurze M Otter J Bals 2005 Nonlinear Inverse Models for Control 4th Proceedings of the 4th International Modelica Conference Hamburg Germany ed G Schmitz pp 267 279 http www modelica org events Conference2005 online_proceedings Session3 Session3c3 pdf Th mmel M M Otter and J Bals 2001 Control of Robots with Elastic Joints based on Automatic Generation of Inverse Dynamics Models IEEE RSJ Conference on Intelligent Robots and Systems Oct 29 Nov 3rd Hawaii U S A Tietze U and C Schenk 2002 Halbleiter Schaltungstechnik Springer Verlag 12 Auflage pp 815 852 Walther N 2002 Praxisgerechte Modelica Bibliothek fiir Abtastregler Diplomarbeit HTWK Leipzig Fachbereich Elektro und Informationstechnik su pervised by Prof Miiller HTWK and Prof Martin Otter DLR 12 Nov 2002 Wichmann B A and I D Hill 1982 Algorithm AS 183 An efficient and portable pseudo random number generator Applied Statistics 31 pp 188 190 See also Wich
249. ing results in an index change of the DAE i e if the number of states is changing A typical example is Coulomb friction where this situation is present even in the simplest case To concentrate on the essentials first the simplified friction element in the next figure is discussed lt gt The friction force f acts between two surfaces see right part of the figure and is a linear function of the relative velocity v between the friction surfaces when the surfaces are sliding relative to each other When the relative velocity becomes zero the two surfaces are stuck to each other and the friction force is no longer a function of v The element starts sliding again if the friction force becomes larger than the maximum static friction force f0 This element can also be described as a parameterized curve as indicated in the left part of the figure leading to the following equations Simplified Coulomb friction element forward s gt 1 140 Chapter 10 Variable Structure Systems Draft Jan 21 2009 backward s lt 1 v if forward then s 1 else if backward then s 1 else 0 f if forward then f0 f1 s 1 else if backward then f 0 f1 s 1 else f0 s This model completely describes the simplified friction element in a declarative way Unfortunately current ly it is not known how to transform such an element description automatically in a form which can be simu lated Let us analyze the difficulti
250. ing the simulation speed considerably because the integrator has to be re initialized after every sample instant This is especially the case for small sample periods In some appli cations sampled data systems do not perform any action most of the time For example an ABS braking sys tem only acts if the brake is pressed and the car starts to slide Still in a Modelica simulation model the inte gration will be halted at every sampling instant Manuel Remelhe from University of Dortmund has proposed a scheme that considerably enhances simulation speed in such a case The idea is to trigger a state event when the controller should act This state event just sets a flag to trig ger a time event at the next sample time Only then the sampling actually occurs As a result the permanent sampling is removed from the code and replaced by a sampling that only occurs after the exact matching condition is triggered with a state event In Modelica this situation can be expressed simply as sampleTrigger condition and sample for example sampleTrigger v gt 500 and sample 0 0 001 This means that time events of the samp1e operator should only be triggered if v gt 500 Dymola does not yet perform this optimization and triggers time events every 0 001 s even ifv lt 500 but of course sampleTrigger is only true at the specified time instants In order to get a simulation speed up this situation has to be currently i
251. ional not expr not bl Unary negation and bl and b2 Logical and or bl or b2 Logical or expr expr expr 1 5 start step stop Array range or vector construction if expr then expr if b then 3 else x Conditional else expr ident expr x 2 26 Named argument or modifier 6699 Equality and assignment are not expression operators since they are allowed only in equations and in assignment statements respectively All binary expression operators are left associative The unary minus and plus in Modelica is slightly different than in say MATLAB since the following expressions are illegal in Modelica whereas in MATLAB these are valid expressions 2 2 4 in MATLAB is illegal in Modelica 2 2 in MATLAB is illegal in Modelica 4 2 2 in MATLAB is illegal in Modelica 2 2 4 in MATLAB is illegal in Modelica 21 2 Mathematical Functions Table 21 6 Mathematical functions All functions have Real or Integer input arguments have Real output arguments are vectorizable and none triggers an event Operator Description abs x Is expanded into noEvent if x gt 0 then x else x asin x Inverse sine 1 lt x lt 1 acos x Inverse cosine 1 lt x lt 1 atan x Inverse tangent Returns result in the range 7 2 n 2 atan2 x y Four quadrant inverse tangent x and y not both zero tan x y Ret
252. ircuit model R1 i Rl i R1 yv R1 v Rl u R2 i R2 i R2 y R2 y7 R2 u dC Ci Cir Cv C v2 dL i hi L L L dt a Vi V2 Uu S i Sis Sv S v2 Siu Draft Jan 21 2009 12 1 Transformation to State Space Form 155 ei eee Since we have 27 unknowns and 27 equations it is possible to solve the equations for the unknowns In the following sections the algorithms are sketched to achieve the following result input variables x C u L i t all parameters R R onlyStates true return only states false return all variables output variables dx dt dC u dt dL i dt y all other unknown variables if not onlyStates algorithm R2 u R2R Li Rly SA sin S w t Lu R1yv R2 u Rlu Rlv Ca Ci RlaulRIR dL i LulLL di dC u di Ci C C if not onlyStates then compute all other 20 variables S i Ci bi Gi Si Ci end if From the 27 equations a subset of 7 equations is sufficient to compute the state derivatives X in a recursive evaluation order Whenever this model function is called these 7 equations have to be evaluated However the other 20 equations are not needed for the integration algorithm Therefore they are only evaluated when variables shall be stored on file at a communication grid Dymola performs such type of equation partitioning in order to improve efficiency One might wonder why the 7 e
253. iteration variables the number of operations is reduced significantly if the number of iteration variables can be reduced as in the example above The main difficulty is now to determine symbolically during translation the tearing or iteration variables here zs and the corresponding residue equations here rs so that the number of iteration variables is as small as possible In Carpanzano and Girelli 1997 it is shown that this requires in general to try out all pos sible combinations and that the number of operations grows exponentially with the system size NP com plete problem Therefore it is impossible that such a nice optimal algorithm as for the BLT form exists and only heuristic algorithms are available Another severe difficulty is that the rank of the system shall not be changed by the tearing transformation For example a system of 3 equations in 3 unknowns might be de scribed by z sin w t ZZ sin z 12 18 2 0 2 2 By BLT transformation it is determined that z is computed from the first equation and that z and z have to be computed simultaneously from the last two equations With the tearing approach one could compute z from the second equation and then solve 1 non linear equation in z Draft Jan 21 2009 12 3 Solving Algebraic Equations Tearing 163 known Z input Zz iteration variable output r residue z i sin z z algorithm a 32 F3 Z227 23
254. ithmetic operations are interpreted as series and parallel connection of StateSpace sys tems connects systems in parallel connects systems in series Function invariantZeros computes the invariant zeros For single input single output systems these are the zeros of the transfer func tion Function plotEigenValues computes and plots the eigenvalues of the system 17 2 2 Record TansferFunction This record defines the transfer function between the input signal u and the output signal y by the coefficients of the numerator and denominator polynomials n s and d s respectively 17 2 Draft Jan 21 2009 17 2 Linear System Descriptions 179 The order of the numerator polynomial can be larger as the order of the denominator polynomial in such a case the transfer function can not be transformed to a StateSpace system but other operations are possi ble For example the transfer function 2 s 3 2 u 17 3 T MS 5846 17 3 is defined in the following way import TF LinearSystems TransferFunction import Modelica Utilities Streams TF tf n 2 3 d 4 5 6 print y TF String tf u The last statement prints the following string to the output window y 2 s 3 4 s 2 5 5 6 u Besides arithmetic operations on transfer functions and constructing them optionally also from polynomials and from zeros and poles the zeros and poles can be computed and a Bode plot can be construc
255. ization if discrete block sampleFactor Ts sampleClock sampleTime sampleFactor initType gt Type of initialization free initial steady state x_start Initial integrator state if InitialState else guess OK Info Cancel Finally with parameter initType the type of initialization can be defined by using either the setting from the sampleClock component default or a local definition The default of the sampleClock component initialization is steady state This means that continuous blocks are initialized so that the state derivatives are zero and discrete blocks are initialized so that a re evaluation of the discrete equations gives the same discrete states provided the input did not change Since the equations of the blocks of the Sampled library are linear the default setting leads to linear systems of equations during initialization that can be usually solved very reliably The steady state initialization for a control system or a filter has the significant advan tage that unnecessary settling times after simulation start are avoided 17 3 2 Discretization Methods The core of the Sampled library is function DiscreteStateSpace fromStateSpace that transforms a linear time invariant differential equation system in state space form Draft Jan 21 2009 17 3 Continuous Discrete Control Blocks 185 t A x t B u s y t C x t D u t 128 into a linear time invariant difference equation
256. ks Interfaces import SI Modelica SIunits IF RealOutput y parameter SI Time T discrete SI Time tNext start T fixed true U equation when time gt pre tNext then tNext time T end when y if time lt tNext T 2 then 1 else 0 end Pulse Parameter T is the period of the pulse The discrete Real variable tNext defines the time instance of the next rising edge of the pulse Due to the start attribute pre tNext T before simulation starts Variable time is a predefined variable that is present in all models and defines the actual simulation time When the condition time gt pre tNext of the when clause becomes true or more precisely at the time instant when the condition changes from false to true the body of the when clause becomes ac tive Since the equations in a when clause are only active at one particular time instant they are called in stantaneous equations In the above when clause the time instant of the next rising edge of the pulse is com puted once the when condition becomes true Outside of the when clause the output variable y is easily calculated based on the time instant tNext of the next rising edge and the desired shape of the pulse sig nal It is essential to distinguish precisely when to use pre tNext and when to use tNext In the condi tion the pre value has to be used because at the event instant when the condition becomes true the va
257. l Lumped HeatTransfer library the two most important elements are the thermal conductor and the heat capacitor see Table 3 7 Table 3 7 Lumped heat transfer elements in different description forms G C are constants 0 z Q ow 1 F O pow 2 Q aw Ti Ta Qhow 2 O awi GATT ThermalConductor gt A S tow 1 S pow 2 0 TiS aow 1 tT rS fow 2 S sews T G T T IT yet c Q fow C dt HeatCapacitor Q 7 Sie et S r w T dt flow 50 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 The equations of the ThermalConductor and of the HeatCapacitor are both linear in T and in Ojo but are non linear in T and in Spo which means that the latter are more difficult to solve numerically Furthermore the common boundary condition of perfect insulation can be more easily expressed as Qy 0 instead of T Spow 0 where the latter is again a non linear relationship Finally most engineers have a better under standing of Ojo as of Show 3 4 Connectors for Block Diagrams Blocks library Block diagrams consist of a connection of blocks where the interface of every block is defined by input and output signals In principal it would be possible to define these interfaces with the already introduced lan guage elements e g Do not use not a good design for block diagram interfaces connector ReallInputl connector RealOutputl Real signal Real signal end ReallInput1 end RealO
258. l may have also other singularities at particular operating points or regions that pre vent an inversion e g due to divisions by zero singular linear or singular non linear systems The reason is that the corresponding inverse model has no or infinitely many solutions in particular points or in particular regions of the state space Again one remedy is to change the plant model before the inversion e g by ne glecting dynamic elements or by approximating components with functions that are less problematic to in vert 19 8 3 Actuator limits Every control system is inherently limited by constraints in the actuator or other parts of the plant and there fore the question arises how to cope with these restrictions When inverting a plant model such constraints have to be removed before the inversion Otherwise no unique solution of the inverse exists anymore be cause there are infinitely many solutions when an actuator is in one of its limits As a result usually only the trivial action is possible to add appropriate limiters to the outputs of inverse models This will only help for short time violations of the constraints because the control system is effectively switched off when the actua tors are in their limits The most effective way to cope with actuator constraints in any control system is to adapt the desired control signals such as y in Figure 19 9 In the most general case this means to solve a trajectory opti mization proble
259. l variables Connected potential vari ables are identical The visual appearance of a connector is defined in the diagram and the icon layer see Figure 2 6 Figure 2 6 Visual appearance of connector Pin in icon and diagram layer r O iojxi e diagram layer lt A iojxi mands Window Help x Sin ieee renee ts Window Help 15 x MA QS W Ray i B M ERILL gt Packages fl Packages 3 J Modelica a Modelica 1 Users Guide 9 Users Guide gg Blocks Blocks ZJ Constants EJ Constants 5 fe Electrical fej Electrical Analog a JAnalog J Examples CJ Examples Basic 0 Basic Z Ideal FJ ideal M Intertaces CJ interfaces Pin z Br i l a EER a El eljl r S Modeling Modeing 4 The picture in the icon layer here a filled square see left part of figure above is shown at the place where the connector is placed in the icon of a component The picture in the diagram layer see right part of figure above is used when dragging the connector in the diagram layer of a component i e the internal part of a component It is usually best to have a smaller picture here in order that the default size is reasonable after Draft Jan 21 2009 2 2 Basic Models 23 cco dragging it otherwise it is too large Furthermore the meta tag sname this is displayed in the figure above as name should be included as a text string in order to
260. la to be able to get rid of the drawbacks of current state machine handling in Modelica 18 1 Steps and Transitions The basic elements of StateGraphs are steps and transitions as shown in the figure to the left Steps represent the possible states a StateGraph can have If a step is active the Boolean variable active of the step is true If it is deactivated active false At the initial time all ordinary steps are deactivated The InitialStep objects are steps that are activated at the initial time They are characterized by a double box see figure at the left Transitions are used to change the state of a StateGraph When the step connect ed to the input of a transition is active and when the transition condition becomes true then the transition fires This means that the step connected to the input to the transition is deactivated and the step connected to the output of the transition is activated The tran sition condition is defined via the parameter menu of the transition object Clicking on object transition1 in the figure to the left results in the parameter menu shown in next figure Aa S PWU LUOISUBA gt et oD o O ho Draft Jan 21 2009 18 1 Steps and Transitions 191 Figure 18 1 Parameter menu of Transition component transition in StateGraphDemos FirstExamp 2x General Add modifiers Component Icon Name transition Transition Comment Model Path Modelica StateGr
261. laring replaceable models etc will always lead to a model where the total number of unknowns and equations are equal Besides this strong guarantee it is possible to precisely pin point which submodels have too many equations or lack equations in case of error Before Modelica 3 0 it was possible to connect components from a library and the resulting system could have either too many or not enough equations Worse the tool could not pin point the source of the er ror Dymola improved this situation some years ago by introducing a heuristic to detect the component that leads to a mis match of equations and unknowns This heuristic works well in many cases Still it is not pos sible to trust the result In fact there are cases where this heuristic points erroneously to a correct compo nent Since Dymola 7 0 the Modelica 3 0 restrictions are checked if the Modelica Standard Library version 3 0 and later is loaded since this library requires Modelica 3 0 For example this means that an error mes sage is triggered for components that are not locally balanced If a previous version of the Modelica Standard Library is loaded instead e g version 2 2 2 Dymola works slightly different for backward compatibility If the model is correct according to Modelica language version 2 but erroneous according to version 3 only a warning message is printed In order that you are able to find an error in your model easily it is highly advis able to first c
262. lica has deterministic concurrency This implies that the number of active equations and the number of unknown variables in the active equations at every time instant are identical This feature is sometimes also called single assignment rule meaning that a variable is defined by one equation this phrase is not completely correct because a variable might be determined from an algebraic system of equa tions Modelica does not have one of the implementation schemes of synchronous languages sketched above event driven or sample drive model because Modelica handles continuous and discrete equations and then such an event model is not possible because time is advanced according to the numerical integration of the continuous system and the triggered events and not solely by a clock or by events If needed it would be possible to introduce in Modelica a new type of restricted model class say controller that corresponds to a data flow oriented synchronous language like SattLine or Lustre The controller class shall have all the properties of a block class and additionally the restriction that inside the controller the operators der and sample and the predefined variable time cannot be used The definition of the event iteration is not yet satisfactory in Modelica because there is no guarantee that event iteration converges Practically a Modelica tool will terminate a simulation with an error if the
263. lica mo Alternatively a package can be stored in several directories and files In this case every package hierarchy is mapped on a corresponding directory hierarchy For exam ple package Modelica could also be stored in a Windows file system as Figure 2 14 Hierarchical storage of Modelica package in the file system x Name Fl Modelica af A MultiBody 9 Blocks 9 package ma Electrical Rotational mo E Math Translational mo O Mechanics Here Modelica Mechanics and MultiBody are stored as directories with the same name Pack age Modelica Mechanics Rotational is stored in file Rotational mo within directory Modelica Mechanics In every directory a file with the name package mo must be present This file con tains the definition of the corresponding package class For example file package mo in directory Multi Body might be defined as 34 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 File Modelica Mechanics MultiBody package mo within Modelica Mechanics Full name Modelica Mechanics MultiBody package MultiBody Library to model 3 dim mechanical systems annotation Documentation info lt html gt Overview of package MultiBody lt html gt end MultiBody The first line in a file consists of the within clause consisting of the full name of the package in which the subli
264. ling of Hybrid Systems Proceedings ESS 93 European Simulation Symposium Delft The Netherlands pp xxxi xli Elmqvist H S E Mattsson and M Otter 2001 Object Oriented and Hybrid Modeling in Modelica Journal Europeen des systemes automatises 35 1 2001 pp 1 a X Franke R M Rode M and K Kr ger 2003 On line Optimization of Drum Boiler Startup 3rd Int Modelica Conference Link ping Nov 3 4 pp 287 296 Download http www Modelica org Conference2003 papers shtml Follinger O 1994 Regelungstechnik Hiitthig Verlag 8 Auflage F llinger O 1998 Nichtlineare Regelungen I Oldenbourg Verlag 8 Auflage Gautier T P le Guernic and O Maffeis 1994 For a New Real Time Methodology Publication Interne No 870 Institut de Recherche en Informatique et Systemes Aleatoires Campus de Beaulieu 35042 Rennes Cedex France ftp ftp inria fr INRIA publication publi pdf RR RR 2364 pdf Halbwachs N P Caspi P Raymond and D Pilaud 1991 The synchronous data flow programming language LUSTRE Proceedings of IEEE vol 79 pp 1305 1321 http www esterel technologies com files LUSTRE synchronous programming language pdf Draft Jan 21 2009 Chapter 22 Literature 237 Halbwachs N 1993 Synchronous Programming of Reactive Systems Springer Verlag http www esterel technologies com files synchronous programming of reactive systems tutorial and references pdf Holman J 2001 Heat Transfer 9
265. lization still fails the first action is to inspect the actual values of all guess values of the iteration variables This is performed by setting the command Advanced LogStartValuesForIterationVariables true in the input field of the command window at the bottom of the simulation window and translating the mod el Dymola reports the iteration variables and their start values in the message window For example when translating the example model Modelica Mechanics MultiBody Examples Loops Fourbar1 and the above option has been set then the following output is presented in the message window 72 Chapter 4 Initialization Draft Jan 21 2009 Figure 4 13 Iteration variables and their start values in the message window E Messages Dymolz 15 xj Syntax Error Translation Dialog gt Start values for iteration variables j3 phi start 0 j4 phi start 0 j5 phi start 0 rev phi start 0 rev1 phi start 0 Finished experiment StopTime 5 Therefore this model has the iteration variables 33 phi j4 phi j5 phi rev phi revl phi and as guess value for every iteration variable zero is used The list of start values for the iteration variables should be carefully inspected It suffices that one start value is not meaningful in order that initialization fails 4 5 2 Difficult nonlinear equations There are nonlinear algebraic equation systems that are difficult to solve Such systems can occur
266. ller and the event iteration is fin ished i e the integration is restarted 128 Chapter 8 The Synchronous Principle Draft Jan 21 2009 Chapter 8 The Synchronous Principle In the previous chapters we have discussed the basic language elements of Modelica to model discrete and combined continuous discrete systems In this chapter the basic underlying principle of Modelica is summa rized and compared with other synchronous languages Synchronous languages Halbwachs 1993 Benveniste et al 2003 such as SattLine Elmqvist 1992 Lustre Halbwachs et al 1991 Signal Gautier et al 1994 or Esterel 1999 are used to model discrete con trollers to yield safe implementations for real time systems and to verify important properties of the discrete controller before executing it These languages are used in industrial applications since quite a long time For example SattLine has a large installation base at ABB in the process industry Lustre is the basis of the com mercial SCADE tool used e g to generate certified real time programs for aircraft controllers Synchronous languages have one of the implementation models shown in the next table Benveniste et al 2003 that can be categorized as event driven left and sample driven right Table 8 1 Implementation models of synchronous languages event driven sample drive Initialize memory Initialize memory for each input event loop for each clock tick loop compute outputs r
267. llers The subject of this chapter is the systematic design of controllers for non linear systems based on inversion of the non linear plant model This chapter is based on the article Th mmel et al 2005 Traditional design techniques require the nonlinear plant model to be linearized around a stationary operating point Afterwards linear methods may be applied to synthesize a controller for this operating point In order to make this con troller work over the full operating range of the plant robust design techniques and or gain scheduling are applied The first approach may considerably reduce achievable performance if the plant dynamics vary strongly over the operating range whereas the latter may involve designing many controllers at a grid of op erating points and finding an interpolation scheme in between them In linear design inversion of plant dynamics is sometimes used to compensate for coupled input output responses or as an easy way to impose specific dynamic behavior of the closed loop system Enns et al 1994 In a nonlinear context the application of model inversion additionally provides compensation of non linear dynamic behavior of the plant This is exploited in design techniques such as feedback linearization Slotine and Li 1991 The design approach proposed in this chapter starts from any controller structure that is based on a linear inverse model of the plant This model is replaced with a nonlinear inverse one resu
268. lly better implementation alternatives 6 1 Relation triggered events The Modelica language has no construct to explicitly trigger an event Instead events are always implicitly triggered if a relation such as v1 gt v2 changes its value Since relations are the only means in Modelica and also in programming languages to change the value of a variable discontinuously e g by an if clause every potential discontinuous change in a Modelica model or block triggers an event and halts the numerical integration This basic mechanism is explained in more detail on the basis of the following simple block that models a two point switch Figure 6 1 Two point switch block to demonstrate relation triggered events during even iteration model TwoPoint import YA twoPoint f Modelica Blocks Interfaces u y Interfaces RealInput u gt Interfaces RealOutput y equation NI ru y if u gt O then 1 else 1 A end TwoPoint Assume that u is negative and becomes continuously larger until it is positive At the beginning y 1 since u is negative When the integrator performs a step so that u is positive an iteration is started to detect the zero crossing of u within a small time interval During this iteration y remains still on the else branch i e y independent of the value of u Once the zero crossing time instant is found precisely enough usually close to machine
269. ls These structures are attractive since it is possible to cope directly with operating point dependencies The difficult part to construct an inverse model can be performed automatically even for complex systems The plant is modeled with Modelica inputs and outputs are exchanged and a Modelica simulation environment such as Dymola generates automatically the appropriate C code for the inverse plant model including real time in tegration algorithms The generated code can be easily embedded into Simulink from Mathworks using Dy molas Simulink export option Via Mathworks Realtime Workshop the code can be finally downloaded to different target processors The presented controller structures can be used in all types of areas such as control of robots vehicles aircrafts satellites ships motors air conditioning systems The most important requirement is that an appro priate plant model is available Then the inverse modeling approach is in principle fully automatic although the practical application is usually more difficult The essential issues have been discussed in section 19 8 and also possible remedies Appendix 226 Chapter 20 Contributors to Modelica Draft Jan 21 2009 Chapter 20 Contributors to Modelica Many persons have contributed to the Modelica language definition and to the Modelica Standard Library Find below a list of all contributors Draft Jan 21 2009 Chapter 21 Modelica Built In Functions and Operator
270. ls that have identical input arguments are only evaluated once and the result values of the first call are used at all other places where this function is called with the same input arguments Such type of code simplification is not possible if functions with internal memory would be present Note common subexpression elimination makes external C code with internal memory unreliable be cause a tool might just replace several identical calls to the same C function by just one call and then the expected behavior is gone The only reliable way to use external C code with internal memory in Modeli ca are therefore ExternalObjects see section 5 6 Draft Jan 21 2009 5 4 Functions 97 Although Modelica functions cannot have internal memory it is possible to emulate them by having the in ternal memory as input argument and the modified internal memory as output argument This is demonstrat ed with the next example a pseudo random number generator In a programming language like C a ran dom number generator is typically utilized in the following way Wrong Modelica code to use a random number generator randomInit seed 23 87 187 x random x 0 67206 y random y 0 36786 The internal memory of this random number generator consists of an Integer vector with 3 elements With randomInit this internal memory is initialized Every call of random produces now another number and updates the internal
271. lse and true high false i e y remains on the lower branch e pre high false and u gt 1 high true and true or false and true high true i e y changes from the lower to the upper branch e pre high true and u gt 1 high false and true ortrue and true high true i e y remains on the upper branch e pre high true and 1 lt u lt 1 high false and false or true and true high true i e y remains on the upper branch e pre high true and u lt 1 high false and false or true and false high false i e y changes from the upper to the lower branch Once high is computed it is trivial to calculate the output y with the Real equation 112 Chapter 6 Discontinuous Equations Draft Jan 21 2009 y if high then 1 else 1 The declaration of high defines the attributes start true and fixed true and we will now discuss the precise meaning of this declaration For non discrete Real variables the start attribute de fines the value of a variable after initialization provided fixed true This means initialization has to be carried out so that the variable has the required start value after initialization Often this requires solving a non linear algebraic system of equations For Boolean Integer String discrete Real and enumeration variables v this is different The start attribute defines the value of pre v be fore initi
272. lseandy 1 Since high pre high the initialization is finished Since the equations of the Hysteresis block are not obvious at the current stage one might think to imple ment the block without using a Boolean equation Wrong Modelica model y if u gt 1 or pre y gt 0 5 and u gt 1 then 1 else 1 However this is not allowed in Modelica because outside of a when clause the pre operator can only be applied on discrete variables and y is a non discrete Real variable One reason for this restriction is that otherwise the following type of equations could be defined Draft Jan 21 2009 6 2 Discrete equations and the pre operator 113 Wrong Modelica model y noEvent if u gt 1 then u else pre y 2 and then it is not clear how pre y should be computed Due to the noEvent operator no event is generated when u gt 1 changes its value The problem is that y changes its value discontinuously at this time instant and the value of y before this change is not defined because the switching time instant is not determined 114 Chapter 7 Instantaneous Equations Draft Jan 21 2009 Chapter 7 Instantaneous Equations 7 1 When clauses and mapping to instantaneous equations At an event instance additional equations can be activated with the when clause This is demonstrated in the next example by means of a pulse generator block y model Pulse import IF Modelica Bloc
273. lting in a controller that is valid for the full operating range of the plant In case the plant model is available in Modelica it will be demonstrated that inversion can be performed automatically exploiting symbolic algorithms and code gener ation features of a Modelica simulation environment as discussed in Chapter 12 This often allows for an au tomated design process that directly results in nonlinear controllers that work in all operating conditions of the plant avoiding the need for gain scheduling 19 1 Inversion of Linear SISO Models The goal is to use a nonlinear plant model in a controller in order that the nonlinearities of the plant are di rectly taken care of in the control system Some basic properties to use inverse models in controllers for lin ear single input single output plants are recapitulated as needed for the rest of the chapter This section is based on the fundamental article of Kreisselmeier 1999 A single input single output plant shall be described by a rational transfer function of the form y P s u u 19 1 where u s is the plant input or actuator signal y s is the plant output P s is the plant transfer function consisting of the numerator polynomial mp s and the denominator polynomial d s The two polynomials np s and de s shall have no zero in common All following properties hold both for continuous and discrete linear systems i e the independent variable can either be the derivative
274. lue of tNext from the last event has to be used and in the when clause the value of tNext for the next event is calculated Variables of type Real that appear at the left hand side of an equal sign in a when clause are defined to be discrete Real variables independently whether they are declared as Real or as discrete Real The prefix discrete in a declaration is therefore just a safe guard to explicitly state that the modeler expects that this variable is computed in a when clause and therefore can change its value only at event instants If this is not the case e g due to a typing error a translation error occurs Several built in operators are provided in Modelica for the handling of discrete equations The most im portant ones are listed with a short explanation in the following table Table 7 1 Build in operators for discrete equations initial Returns true during the initialization phase and false otherwise terminal Returns true at the end of a successful analysis sample start period Returns true and triggers time events at time instants start i period i 0 1 During continuous integration the operator returns always false Arguments start and period have to be parameter expressions Draft Jan 21 2009 7 1 When clauses and mapping to instantaneous equations 115 pre v Value of variable v before the current event instant If pre v is used outside of a when
275. lue yo 1 The appropriate model can be defined with Mod elica in the following way model FirstOrderl initialize via the start attribute parameter Real k gain parameter Real T time constant RealInput u input signal RealOutput y start 1 fixed true output signal equation T der y y k u end FirstOrderl This model defines the following two equations for the initialization phase y 1 T der y y k u Under the assumption that the initial value of the input variable u is computed somewhere else the two equa tions above have the solution y 1 der y k u y T and this is the solution of the initialization problem Dragging the above model FirstOrder1 in a model as firstOrder1 and double clicking on the model icon in Dymola opens the following parameter menu that is automatically constructed from the definition above Figure 4 1 Automatically constructed parameter menu with start and fixed attribute for y 5 DEN 2x General Add modifiers Component Icon Name firstOrder1 FirstOrdert Comment Model Path FirstOrder1 Comment Parameters k gt gain T time constant Initialization y start output signal OK Info Cancel As can be seen not only parameters but also start values are automatically introduced into the parameter gt To be precise for parameters the default value of fixed
276. m i e to determine actuator signals u t such that the plant outputs y have a desired be havior e g reaching the desired position in minimum time or with minimum energy without violating the plant constraints The result is used as y t A typical example can be found in Franke et al 2003 Note if the plant is unstable and the inverse plant model is stable it might be considerably simpler to solve the tra jectory optimization problem with the inverse plant instead with the plant model Usually trajectory opti mization problems are difficult to solve and therefore highly simplified plant models are used Take for example the crane model The basic requirement is to move the crab from position s a to s b in a short time The plant model is simplified by fixing the angle to g 0 resulting in equation 19 36 Based on this equation the actuator limit lt fmax can be directly transformed into a limit of the acceleration ls lt F mast Merab Mioa max 19 37 Together with limits on the maximum speed due to the maximum speed of the motor RA lt Sima and the requirement to move in minimum time from a to b it is straightforward to construct the analytic solution of the desired movement s a t This solution is e g available via the block Modelica Blocks Sources KinematicPTP Note the plant model used for the trajectory optimization problem and for the inverse plant model in the feedforward path according to Figur
277. mann B A and I D Hill 1984 Correction to Algorithm AS 183 Applied Statistics 33 pp 123 McLeod A I 1985 A remark on Algorithm AS 183 Applied Statistics 34 pp 198 200
278. marize the structure of the inverse model equations is different at these two points and at 0 and 180 the DAE index is 5 for g 0 and g 180 and the DAE index is 3 otherwise Since the divi sion by zero occurs when computing p the plant model should be changed to compute in a different way This can be accomplished by taking the inertia of the load into consideration previously it was neglected With a non zero inertia the transfer function from f to sz of the linearized plant has 2 conjugate complex ze ros on the imaginary axis and a relative degree of 2 Again by introducing damping in the revolute joint these two zeros are moved to the negative real axis It turns out that the inverse model is very insensitive with respect to the newly introduced load inertia To summarize for the crane example the inverse plant models from s to f and from sz to f can be con structed by inverting a modified plant that has a load inertia and additionally damping in the revolute joint A simpler alternative is also available Before inversion the angle is fixed to 0 and therefore s s2 and the plant to be inverted is described by the following equation 77 lt is the mass of the crab and mjoaa is the mass of the load Map t Moa 8 f 19 36 which can be easily inverted This example demonstrates that it might be necessary to slightly modify the original plant model in order that the inverse model of the plant can
279. me instant t At an event in stant y t is the value of y after the last event iteration at time instant t The pre operator can be applied if the following three conditions are fulfilled si multaneously a variable y is a subtype of a simple type Real Integer Boolean String b y is a discrete time expression c the operator is not applied in a function class Before initialization starts pre y y start edge b Is expanded into b and not pre b for Boolean variable b The same restrictions as for the pre operator apply e g not to be used in function class es change v Is expanded into v lt gt pre v The same restrictions as for the pre op erator apply reinit x expr In the body of a when clause reinitializes x with expr at an event instant x is a Real variable resp an array of Real variables in which case vectorization ap plies that must be selected as a state resp states at least when the enclosing when clause becomes active expr needs to be type compatible with x The reinit operator can only be applied once for the same variable resp array of variables It can only be applied in the body of a when clause 21 6 Array Operators Table 21 11 Operations and functions for arrays Operator Description Array dimension and size functions 234 Chapter 21 Modelica Built In Functions and Operators Draft Jan
280. memory in order to get this behavior As previously explained it is not pos sible to implement such a behavior with a Modelica function It would be possible to implement this behav ior with external C functions However Dymola would transform the above code fragment into the follow ing form Common subexpression elimination of the above code results in randomInit seed 23 87 187 x random x 0 67206 y X y 0 67206 Therefore every call to random would produce exactly the same output The reason is that Modelica functions generate the same output arguments if identical input arguments are provided It is therefore fine that a tool replaces all occurrences of a function call with identical input arguments by just one function call In order to achieve the expected behavior the random number generator must be implemented in the following way where the internal memory is both an input and an output argument of the Modelica func tion function random Pseudo random number generator input Integer seedIn 3 Seed from last call output Real x Random number between 0 and 1 output Integer seedOut 3 Modified seed for next call algorithm seedOut 1 rem 171 seedIn 1 30269 seedOut 2 rem 172 seedIn 2 30307 seedOut 3 rem 1 70 seedIn 3 30323 Zero is a poor seed therefore substitute 1 for i in 1 3 loop if seedOut i 0 then seedOut i 1 end
281. model e Replacing all connect equations by the corresponding equations of the connection set e Mapping all algorithm sections to equation sets e Mapping all when clauses to equation sets As a result of this transformation process a set of equations is obtained consisting of differential algebraic and discrete equations of the following form where v X x y t m pre m p m f v relation v 8 1 0 f v relation v 8 2 and where Table 8 3 Meaning of variables in the operational model of Modelica p Modelica variables declared as parameter or constant i e variables without any time dependency t Modelica variable time the independent real variable x t Modelica variables of type Real appearing differentiated m te Modelica variables of type discrete Real Boolean Integer enumera tion which are unknown These variables change their value only at event in stants te pre m are the values of m immediately before the current event oc curred y t Modelica variables of type Real which do not fall into any other category algebraic variables relation v A relation containing variables v e g v1 gt v2 or v3 gt 0 and the relation is not within anoEvent operator definition For simplicity the special cases of the reinit operator as well as of higher index DAEs are not con tained in the equations above and are not discussed below The generated set of equatio
282. more practical to define the network from left to right since it is easier to read the labels on the icons The example from above has then the following layout Figure 18 2 First StateGraph example drawn from left to right initialStep transition1 step transition2 18 2 Conditions and Actions With the block TransitionWithSignal the firing condition can be provided as Boolean input signal in stead as entry in the menu of the transition with block Transition see example in the next figure 192 Chapter 18 Hierarchical State Machines StateGraph Library Draft Jan 21 2009 Figure 18 3 Example of a StateGraph with Step WithSignal and TransitionWithSignal components transition transition2 initialStep step active V Component step is an instance of StepWithSignal that is a usual step where the active flag is avail able as Boolean output signal To this output component Timer from library Modelica Blocks Logical is connected It measures the time from the time instant where the Boolean input i e the active flag of the step became true up to the current time instant The timer is connected to a compar ison component The output is true once the timer reaches 1 second This signal is used as condition in put of the transition As a result transition2 fires once step step has been active for 1 second Of course any other Modelica block with a Boolean output signal can be connected
283. mplemented in Dymola as if condition then sampleTrigger sample 0 T else sampleTrigger false end if Here no unnecessary time events are generated Since it is inconvenient to write these statements at all the needed places one would encapsulate it in a block But then it would be convenient to call this block as a function as discussed below 7 4 5 Calling a Modelica block as a function As shown by four examples Modelica is suitable to model various kinds of time synchronization mecha nisms The described blocks can be easily used in a block diagram However in many cases the modeler The proposed implementation is from Hans Olsson Draft Jan 21 2009 7 4 Time synchronization multi rate delays range limited sampling 125 would like to apply these very basic definitions as a function call such as wrong Modelica code slowSampling Counter fastSampling samplingFactor y InertialDelay u delayTime pressButtons B1 and B2 and not OnDelay Bl or B2 0 5 This is not possible because the functionality was implemented as a block and therefore requires to first in stantiate the block and then connect input and output signals correct Modelica code Counter counter samplingFactor samplingFactor InertialDelay inertialDelay delayTime delayTime OnDelay onDelay delayTime 0 5 equation counter u fastSampling counter y slowSampling inertialDel
284. mportant data for an electrical motor Model Motors Motor is a physical model of the motor using the motorData record The motorData record declaration has the prefix parameter in order that the record appears in the parameter menu of a Motor instance New motor data records are defined by inheritance from the MotorData record and providing concrete values for the parameters Inheritance is used here via the short class definition E g Mo torTypel is completely equivalent to the following definition record MotorTypel extends MotorData inertia 0 0015 end MotorTypel The defined Motors library is utilized to build up a Robot model In the Robot model several motors are instantiated and the motor data is given as modifier Note it is not possible to directly provide Mo torTypel as modifier because MotorTypel is a class and not an instance and only instances can be provided as modifiers Instead the record constructor function has to be used to generate on the fly an in stance of MotorTypel If no arguments are given the defaults from the Motors library are used However 102 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 it is also possible to overwrite the defaults of the Motors library by providing actual values to the input ar guments like changing inertia from 0 001 to 0 0012 for motor2 above Functions that return a record can nicely be utilized to provide different pr
285. n cation the function div s a b Scalar Real power of a variable or scalar Integer power of a square matrix i a b Element wise multiplication division and exponentiation of scalars and arrays Equal operator of an equation element wise on arrays Assignment operator in the command window of Dymola Assignment operator element wise on arrays 228 Chapter 21 Modelica Built In Functions and Operators Table 21 2 Relational Operators operate on Real Integer Boolean String enumeration scalars For Boolean false lt true for String comparison based on lexicographical order where 1 lt A lt a For enumeration comparison according to definition order Operator Example Description a s True if equal for strings true if identical characters For Real only allowed inside functions lt gt a lt gt b True if not equal for strings true if not identical characters For Real only allowed inside functions lt a lt b True ifa less than b lt a lt b True ifa less or equal b gt a gt b True ifa greater than b gt a gt b True ifa greater or equal b Table 21 3 Boolean Operators operate on Boolean scalars or element wise on Boolean arrays Operator Example Description and a and b logical and true false and true true true fal
286. n trolBus in the variable list of the plot browser the list on the right side of the figure appears which shows the hierarchical structure of the bus Without special action the three buses above are all visible in the icon layer because connectors are al ways visible in the diagram and in the icon layer Usually this is not desired for sub buses because only the top most bus connector shall be visible in the icon and connections shall be made from the outside only to this bus This can be achieved by right clicking on a sub bus and selecting in the pop down menu Attributes A menu opens where attributes of variables and component instances can be defined When selecting Protected then the corresponding connector is no longer visible in the icon layer and it is not possible to connect to it from the outside Since this is a bit tedious to perform for every used sub bus a better alterna tive is to define the sub bus in the following way expandable connector ControlSubBus annotation defaultComponentPrefixes protected end ControlSubBus As a result whenever an instance of this connector is dragged the protected prefix is automatically in troduced and therefore the connector is not visible in the icon layer Dragging introduces the following code protected ControlSubBus controlSubBus When connecting two buses the connecting line has usually not the desired d
287. n 21 2009 3 3 Connectors for Heat Transfer HeatTransfer library 49 Figure 3 6 Simulation results of simple electrical circuit with heat transfer temperature in windings and core speed of motor inertia winding T core T 45 0 100 200 300 400 500 motorlnertia w 2000 1000 0 rpm 1000 2000 62 0 62 4 62 8 63 2 63 6 64 0 The heat flow connector of the HeatTransfer library has the following definition Table 3 6 Interface of heat transfer components of the HeatTransfer library connector HeatPort a E Modelica SIunits Temperature T K flow Modelica SIunits HeatFlowRate Q flow W end HeatPort a The temperature T in the connection point is the potential variable of this connector The resultant heat flow rate Q flow flowing from the outside into the component is the flow variable This means that at a connec tion point the boundary condition of identical temperatures as well as the energy balance sum of the heat flow rates is fulfilled An alternative to the above interface variables would be to use entropy flow rate Show instead of heat flow rate Qno The reason is that temperature and entropy flow rate are sometimes used as interface variables in thermodynamics and bond graph literature since their product is energy flow rate Ejow T Spow However this would not be a good choice here For the lumped heat method as used in the Modelica Ther ma
288. n equation To improve this situation in Modelica version 2 0 so called short class definitions have been intro duced to avoid a superfluous naming hierarchy if a connector consists only of one variable Input output sig nal interfaces can now be conveniently defined as connector RealInput input Real connector RealOutput output Real In fact these definitions are present in sublibrary Modelica Blocks Interfaces and are used in the components of the Modelica Blocks library These definitions imply that a connector instance is both a connector and a variable i e has a value The additional advantage is that arrays of such connectors can be easily defined for example RealInput u A vector of real input signals The example above can now be implemented as 52 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 firstOrder block FirstOrder Modelica Blocks Interfaces RealInput u k Modelica Blocks Interfaces RealOutput y fom parameter Modelica Slunits Time T Time constant equation T der y y u T y y U end FirstOrder In order to still improve the handling of block diagrams a new specialized class block is provided in Modelica and is used in the FirstOrder component above A block is a model with the restriction that each public variable must have one of the prefixes constant parameter input or output By us ing the block keyword
289. n is evaluated and assigned to the left hand side variable In such a situation the same variable can be used on the left and right hand side Function print from the Modelica Standard Library prints a string to the command win dow Here the string consists of the actual value of x This function produces the following output in the command window of Dymola when called interactively 6720613541740214 3678597646977231 6041085613242592 1667374195729034 4629589578171096 9289893342375364 6531921316126386 5926293930235833 5294703487946859 7684225743864563 xx xm mM MM KM KM Il ll O OOOO GOO OO x In a model it is a bit more tricky to call this function and there are different variants A random number gen erator is usually used to generate noise for the measurement signals of a plant model A typical usage is therefore a sampled data system where the random function is called at every sample time model testRandom2 parameter Integer seed start 3 23 87 187 Integer seed 3 start seed_start each fixed true Real x equation when sample 0 0 2 then x seed random pre seed Modelica Utilities Streams print time String time format 1f x String x format 6 end when end testRandom2 The details of a when clause and of the built in operators sample and pre are discussed in Chapter 7 In short the body of the when clause
290. n that has been constructed with the Modelica Mechanics MultiBody library Table 4 2 Steady state initialization of spring mass system model SpringMassSystem steady state initialization import SI Modelica SIunits parameter SI Mass m mass of mass point parameter SI Position sO unstretched spring length parameter SI Acceleration g 9 81 gravity acceleration parameter SI TranslationalSpringConstant c SI Position s distance of mass from attachment point SI Velocity v der s speed of mass equation der s v m der v m g c s s0 initial equation der s 0 der v 0 end SpringMassSystem During the initialization phase this model is described by the 4 initialization equations Draft Jan 21 2009 4 3 Initialization with initial equations 65 der s v m der v m g c s s0 der s 0 der v 0 Solving this system of 4 equations for the 4 unknowns results in the following sequence of assignment state ments to compute the initial values of these variables s s0 m g c vi 0 der s 0 der v 0 It is also possible to include different initial conditions in the same model by using an if clause in the initial equation section Example model FirstOrder4 initialize with y0 or in steady state parameter Real k gain parameter Real T time constant parameter Boolean steadyState true initialization type pa
291. n the left side of the equal signs are not changed and keep their value from a previous event instant Note discrete equations are still equations and not assignment statements e g they are sort ed with the rest of the other equations details will be given below The discussed features are demonstrated by means of some examples Although only the restricted form of equations v expr is allowed in a when clause it is still pos sible to define implicit algebraic equations in a when clause Implicit algebraic equation in when clause parameter Real A 1 2 3 4 parameter Real b 3 4 Real x 2 equation when sample 0 1 then solve 0 A x b time for x x x A x b time end when The trick is to add the unknown x on both sides of an implicit equation 0 f x in order to not change the implicit equation but to just state that the unknown should be a discrete variable It is also possible to have an algebraic loop between discrete and continuous equations Again one has to be careful how to formulate the discrete part Wrong Modelica code 2 x 3 y time when sample 0 1 then O x 4 y end when 7 The following example is from Hans Olsson 116 Chapter 7 Instantaneous Equations Draft Jan 21 2009 The above code fragment is not allowed in Modelica because in the when clause no variable is referenced at the left side of the equal sign As a result it
292. n to the ground The ball starts at height h 1 m When it reaches the ground at h 0 m an impact occurs At the impact point the velocity of the ball changes discontinuously according to Newtons impact hypothesis v tt e e v t e e 0 11 2 where e is the impact coefficient 0 lt e lt 1 model BouncingBalll no proper description of bouncing ball import SI Modelica SIunits parameter Real e 0 7 parameter S1I Acceleration g 9 81 SI Height h start 1 fixed true SI Velocity v equation der h v der v g when h lt 0 then reinit v e pre v end when end BouncingBalll The impact is simply modeled by a when clause that triggers an event when the height becomes less than zero Then the velocity v of the ball is re initialized with e pre v where pre v is the velocity of the ball just before the event h lt 0 occurred i e the velocity with which the ball touches the ground The simulation result of the above model is shown in the next figure Figure 11 1 Simulations results of BouncingBalll the solution is not satisfactory h 1 2 1 05 0 85 0 65 0 45 0 25 0 0 0 2 4 0 4 l l l i l l l l l 0 0 0 4 0 8 1 2 1 6 2 0 2 4 28 After about 2 6 s the model does no longer behave as expected because the ball falls through the ground The reason is that h is smaller as zero when the event occurs When the re bou
293. nalyze them manually as shown above In such a case one should assume that Dymola is right and that the initializa tion problem does not have a solution and should at once try to re formulate the initialization problem 74 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 Chapter 5 Advanced Modelica Language Constructs In this chapter more advanced Modelica language constructs are discussed 5 1 Arrays Modelica supports arrays with any number of dimensions In this section the most important properties of arrays are introduced on the basis of examples 5 1 1 Array declaration and construction Arrays are defined by declaring an array name with the dimensions appended in square brackets for exam ple Real v 3 Vector with 3 elements Real M 3 4 Matrix with 3 rows and 4 columns Real A 2 3 4 5 4 dimensional array Alternatively the dimensions can be defined with the type for example Real 3 vl v2 Two vectors with 3 elements each Real 3 4 M Matrix with 3 rows and 4 columns Unknown dimensions are declared with for example parameter Real denominator dimension is not yet known parameter Real M size of matrix M is not yet known As will be explained in section 12 5 a Modelica environment that shall be able to process the Modelica Stan dard Library has to preprocess the model equations symbolically during translation This is only possible if at tr
294. nceptually included in model Inductor that is the model is completely equivalent to model Inductor Modelica Electrical Analog Interfaces PositivePin pin p Modelica Electrical Analog Interfaces NegativePin pin_n Modelica SIunits Voltage u Voltage drop between pin p and pin n Modelica SIunits Voltage i Current flowing from pin_p to pin_n parameter Modelica SIunits Inductance L equation 0 pin_p i pin tei u pin_p v pin ney i pin peiz L der i u end Inductor With the OnePort model also other electrical components can be easily defined such as model Resistor extends OnePort parameter Modelica SIunits Resistance R equation u R i end Resistor model Capacitor extends OnePort parameter Modelica SIunits Capacitance C equation C der u i end Capacitor As with all declarations in Modelica there is no particular order required and the extends keyword can be placed where ever appropriate For example if the partial model contains parameter declarations one has to decide whether these should appear in the parameter menu before or after the parameter declarations of the Draft Jan 21 2009 2 3 Inheritance 27 model class using the partial model and place then the extends clause correspondingly Several extends clauses in a declaration are possible multiple inheritance is supported The mean ing is that all decla
295. nd velocity e pre v is not large enough the ball will not fly above h 0 and since it will then remain permanently below h 0 the when clause does not trigger a new event Draft Jan 21 2009 11 1 The reinit operator 145 The reason for this behavior is a wrong model because the constraint of the ground is not properly de scribed A reasonable solution requires describing the surfaces and the material properties of the two parts that collide From this information the impact equations should be automatically derived and handled proper ly This is currently far away from what can be achieved in Modelica and only certain simple forms of hard impacts e g between two parts with simple surface descriptions can be defined in a reasonable way In the simple case of the bouncing ball one solution is the following model BouncingBall2 import SI Modelica SIunits parameter Real e 0 7 parameter SI Acceleration g 9 81 SI Height h start 1 fixed true SI Velocity v Boolean flying equation der h v der v if flying then g else 0 flying not h lt 0 and v lt 0 when h lt 0 then reinit v e pre v end when end BouncingBall2 The basic idea is always to switch between different model structures when two colliding parts remain on each other here when the ball remains on the ground In the above model the two model structures are de fined with the Boolean variable
296. nerators AIMC1 A aaie Modelica Thermal pipe Simple thermo fluid pipe flow especially for ma x chine cooling systems with water or air fluid Con pump fs tains pipes pumps valves sensors sources etc ee ee BE Furthermore lumped heat transfer components are gz present such as heat capacitor thermal conductor convection body radiation etc reservoir level_start 2 2m pipe valve Modelica_ Fluid 1 dim thermo fluid flow in networks of pipes A unique feature is that the component equations and the media models are decoupled Contains pipes pressure loss components valves pumps sensors sources etc will be included in the next main ver sion 3 1 of the Modelica Standard Library Enthalpy kJkg 4000 4 Pressure bar E g density as a function of pressure and enthalpy Modelica Media Large media library for single and multiple substance fluids with one and multiple phases e 1241 high precision gas models mixtures be tween these gas models e Simple and high precision water models IAP WS IF97 e Dry and moist air models e Table based incompressible media 4A library in Modelica is defined with the language keyword package It is common in the Modelica community to use the words library and package interchangeably Draft Jan 21 2009 1 2 Modelica Standard Library 15 Example system Name of sublibrary Mo
297. ng because differentiating a discontinuous expression leads to Dirac impulses that are not supported by Dymola in this general form This check of Dymola is avoided by providing the smooth operator around the noEvent expression Technically this statement is wrong because the expression above is not smooth A better solution is to real ly provide a smooth transition from the 1 u to the 1 eps case and provide the correct differentiation order in the smooth operator 6 2 Discrete equations and the pre operator Equations are characterized according to the type of variable that are defined by the equation Therefore Modelica has Boolean Integer String and enumeration equations besides Real equations e Continuous equations An equation is called continuous equation when the equation defines a relationship between non discrete Real variables In general a continuous equation can have the form expri expr2 where expr and expr2 are expressions that evaluate to a Real expression e Discrete equations An equation is called discrete equation when it defines a discrete variable v that is a Boolean Integer String discrete Real or enumeration variable It is required that a discrete equation has the form v expr i e the discrete variable v to be calculated has to be present on the left hand side of the equal sign and the expression on the right hand side of the equal sign eval
298. nics end Modelica In model Inertia two connectors from another package are accessed The Interfaces Flange a connector is determined by searching the first part of the name here Interfaces outside of Iner tia In this case the name is already present in the next hierarchical level The rest of the name is then searched downwards starting from Modelica Mechanics Rotational Interfaces An alternative is to provide the full name of the class as if it would be accessed from the outside such as Modelica Mechanics Rotational Interfaces Flange a above Note that the first part of the full name here Modelica is still searched from the inside to the outside until it is found on the highest level The remaining part of the name Mechanics Rotation al Interfaces Flange_a is then looked up inside package Modelica Due to this search rule the name of a root level package such as Modelica should not be used as name of a subpackage Classes are stored in the file system to keep them persistent Modelica has a simple mapping rule to map Modelica names to names in the file system in order that a Modelica translator can automatically detect and load packages that are referred in a model In the simplest case the content of a package is stored in one file and the name of the file is the package name with the extension mo For example package Modelica would be stored in a file called Mode
299. nitial event new values of the discrete variables m are determined accord ing to the following iteration procedure Draft Jan 21 2009 Chapter 8 The Synchronous Principle 131 known variables x t p unknown variables dx dt y m pre m pre m value of m before event occurred loop solve 8 1 and 8 2 for the unknowns with pre m fixed if m pre m then break pre m m end loop Solving 8 1 and 8 2 for the unknowns is non trivial because this set of equations contains not only Real but also Boolean Integer and enumeration unknowns Usually in a first step these equations are sort ed In many cases the Boolean Integer and enumeration unknowns can be just computed by a for ward evaluation sequence In some cases there remain systems of equations e g for ideal diodes Coulomb friction elements and specialized algorithms have to be used to solve them In other cases algebraic loops between Boolean Integer and enumeration equations are present This is treated as an error Modelica is called a synchronous language because it has the most important properties of discrete synchronous languages First Modelica has instantaneous communication since at an event instant all equa tions have to be fulfilled concurrently If needed it is possible to model the computing time by explicitly de laying the assignment of variables Second unknown variables are uniquely computed from a system of equations i e Mode
300. nnecting the output of the speedSensor component with the controlBus controlBus iniia speedSensor lim 4 2 After drawing the connection line the menu from Figure 3 10 pops up to define the variable on the bus for this connection In the left column the definition of connector speedSensor w is shown The right column is used to define the variable on the cont rolBus to which speedSensor w is connected 54 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 Figure 3 10 Step 2 A menu pops up to define the variable on the controlBus connect speedSensor w controlBus p 2 xi Connect to component of outside connector by selecting sub components below Name Unit Description Name Unt Description speedSensor E controlBus pw Absolute angular velocity of flange lt Add variable gt Connection connect speedSensor w controlBus E JV Label connection to bus connector 3 Clicking on lt Add variable gt opens an input field in which the name of the variable on the bus can be defined alternatively clicking on the sign at the left side of lt Add variable gt shows a list of vari ables that have been defined in other classes where connections to a ControlBus connector have been made Clicking on one of them selects the signal for the connection Figure 3 11 Step 3 Defining variable speed on the controlBus connect speedSensor w controlBus 2 x
301. ns are Modelica functions because function results may be discontinuous and it is not possible to trigger events in a function It is planned to fix this defect of Modelica in a future language version Furthermore external functions are outside of the control of a Modelica translator and can therefore lead to any type of misbehavior of an integrator There are additionally restrictions in the language to prevent the implementation of models that have a memory during the continuous integration phases since this would violate the integrator prerequisite in an even more drastic way as discontinuities are doing it Therefore it is for example impossible to determine the integrator step size from a Modelica model of course in an external function with a memory such types of properties can be determined Most important properties of Modelica with respect to discrete systems e A variable keeps its value until it is explicitly changed It is always possible to access the value of a variable both during continuous integration and at event instants e Itis only possible to access the left or the right limit of a variable at the actual time instant but not at previous time instants to avoid memories during the continuous integration phases During continuous integration the two values are identical At an event instant t a variable name such as v is defined to be the right limit v t and pre v is defined to be the left limit
302. ns can be seen in Figure 17 3 a screenshot of the input menu of this function Draft Jan 21 2009 17 2 Linear System Descriptions 181 Figure 17 3 Parameter menu of function ZerosAndPoles filter filter Description Generate the data record of a ZerosAndPoles transfer function from a filter description Inputs analogFilter El j Analog filter characteristics g CriticalDamping Bessel Butterworth Chebyshev filterType zje Type of filter LowPass HighPass order Order of filter f_cut Hz Cut off frequency default is w_cut 1 rad s Ain ar Gain amplitude of frequency response at zero g frequency A_ripple oo os dB Pass band ripple for Chebyshev filter otherwise not used o me true if amplitude of low pass filter at f_cut is nO UES i 1 sqrt 2 otherwise unmodified filter Info Copy Call Execute Close Besides type of filter and whether it is a low or high pass filter the order of the filter and the cut off frequen cy can be defined With option normalized filters can be defined in normalized default and non normal ized form In the normalized form the amplitude of the filter transfer function at the cutoff frequency is 3 dB 0 7071 When trying out different filter types with different orders the result of a comparison makes only sense if the filter is normalized Note when comparing the filters of this function with other software systems the settin
303. ns is used for simulation and other analysis activities Simulation means that an initial value problem is solved i e initial values have to be provided for the states x The equations de fine a DAE Differential Algebraic Equations which may have discontinuities a variable structure and or which are controlled by a discrete event system Such types of systems are called hybrid DAEs Simulation is performed in the following way 1 The DAE 8 2 is solved by a numerical integration method under the assumption that the discrete vari ables m pre m and relation v are kept constant pre m m due to the event handling Therefore 8 2 is a continuous function of continuous variables and the most basic requirement of numerical inte grators is fulfilled 2 During integration the relations relation v from 8 1 and 8 2 are monitored If one of the relations changes its value an event is triggered i e the exact time instant of the change is determined and the in tegration is halted 3 At an event instant 8 1 and 8 2 are a mixed set of algebraic equations which is solved according to the iteration procedure given below 4 After an event is processed the integration is restarted with Note that the values of m all discrete Real Boolean Integer and enumeration variables are only changed at an event instant and that these variables remain constant during continuous integration At every event instant including the i
304. ns of multi substance fluids Xi and trace substances C Besides physical connectors also a set of signal connectors are provided especially for block diagrams and for hierarchical state machines Type signal bus characterizes an empty connector that has the addi tional prefix expandable This connector type is used to define a hierarchical collection of named sig nals where the full connector definition containing all signal definitions is defined implicitly by the set of variables occurring in all connections of this type This connector is used to communicate a large amount of signals in a convenient way between components Details are given in section 3 5 Draft Jan 21 2009 3 1 Connector Design 45 In the following sections some of the connector definitions above are explained in more detail Furthermore connectors for 3 dim mechanical systems are discussed in Chapter 15 and connectors for 1 dim thermo flu id flow are discussed in Chapter 16 3 2 Connectors for Drive Trains Rotational library Drive trains are mechanical systems where only the rotation around a fixed axis is under consideration This abstraction allows to model many technical systems with electrical drives and mechanical transmission ele ments in a very simple and efficient way Drive trains are modeled with the Modelica Mechanics Ro tational library A simple drive train build with the Rotational library is shown in Figure 3 2 Figure 3 2 S
305. nt iteration would take place During the event iteration it might be that u becomes zero or so small that 1 u results in an overflow leading to a run time error The noEvent operator can only be applied in equations that involve Real variables It is not allowed to apply it for Integer Boolean String enumeration or discrete Real expressions since otherwise it would be possible to define very drastic changes of a model during continuous integration that cannot be handled by a reliable numerical integration method For example during continuous integration the number of equations could be changed in combination with when clauses see next section or the integrator step size could be computed and used to make the model evaluation dependent on the actual integrator step size It is sometimes useful to combine the smooth and the noEvent operators y smooth 100 noEvent if u gt eps then 1 u else 1 eps The reason is that the equation above might be differentiated and the guard to prevent a zero division must be present in a noEvent in order that it is guaranteed that no event is generated remember it is tool de pendent whether the smooth operator generates or does not generate events If Dymola differentiates an expression where the noEvent operator is applied it checks at run time whether the equation that was differentiated is continuous Otherwise the simulation result could be wro
306. o cases a s s0 forward sliding at v 0 m a E eae Fok 10 3 SS Solving for s results in 142 Chapter 10 Variable Structure Systems Draft Jan 21 2009 1 S Spt f rma fr z Rmax m 1 So Final Feat F a S Ries for m o 10 4 IV So On the other hand if during the iteration it is temporarily assumed that the model is sliding backward i e a s So we have m a Fona ie a S S 10 5 Solving for s results in 1 S So I m S ore fr f kmar 1 10 6 gt So 2 Toat max F 2 So J rma for MO This means that a contradiction is present because in the first case the result is that forward sliding is true and in the second case the result is that backward sliding is true This means that the mixed system of equations has two solutions The exact formulation of the friction characteristic at zero velocity should be uncritical a if s gt fing then s f max elseif s gt f Pe then s f oe else 0 fe if S fma then fia elseif s f ma then f ing else s 70 9 lt lt Draft Jan 21 2009 Chapter 11 Re initialization of Continuous Variables 143 Chapter 11 Re initialization of Continuous Variables At an event instant non discrete Real variables might change discontinuously e g due to an impact To de fine this with a modeling language is a very difficult issue and we are not aware of a really satisfactory solu tion neither in Modelica nor
307. o design connectors is presented and some im portant connectors are discussed in detail 3 1 Connector Design The goal is to provide a set of useful connector definitions to describe physical interfaces such as electrical pins mechanical flanges hydraulic or pneumatic ports heat transfer through a wall As a guide line we start with the following rule General Rule for Interface Design The abstracted interface of a component should contain all independent variables that are necessary to describe the desired effects in the real interface Although this rule is quite obvious it is surprisingly difficult to apply it in an actual situation This rule is re fined in a more concrete way based on the following requirements for port based component coupling 1 Independent Interface Variables A connector should contain an independent set of variables that describes the desired physical effects in the interface Usually mean values over the interface area are used e g a resultant cut torque Variables ci in a connector are not independent if at least one variable cj can be computed by alge braic equations by differentiation and or by integration from the other variables ck in the connector In Modelica it is possible to have a redundant set of variables in a connector In such a case there are unnec essary restrictions how components can be connected together The basic reason for this is rule 1 of bal anced models see page 3
308. oblem in two sub problems 12 2 1 Step 1 Output set assignment In a first phase called output set assignment the question is answered for which variable an equation is solved for Since this algorithm is also the basis to handle singular systems in section 12 4 we will discuss it in some detail The basic idea shall be first sketched on the basis of a simple example original step 1 step 2 step 5 0 f 25 24 0 f 2 Bj E O f 2324 O f 23 Z4 O f2 21 Z6 Z7 0 f2 Z1 26 27 0 f z 26 27 0 f 21 2627 0 f z Z5 0 f3 23 Z5 O f 23 Z5 O f 25 Zs 0 f z Z gt 0 f z Z gt 0 f z 2 me 0 f z Z2 O f 2 23 Z6 0 f 5 Z5 23 26 O f 2 Z3 Z6 0 f 2525125 0 f z 0 f z 0 f6 z2 0 f z 0 7 z3 z7 0 f1 Z3 Z7 0 f1 Z3 Z7 0 f z3 27 In the first column called original we have 7 equations f in 7 unknowns zi The goal is to uniquely assign one unknown to one equation with the intention that the assigned variable is computed from the correspond ing equation The important point is that no variable can be assigned twice because it can only be once com puted from an equation In step 1 we start with the first equation and assign arbitrarily one of the un knowns Here we assign zz In step 2 we continue with the second equation and assign one of the un knowns of this equation that has not yet been assigned before We select arbitrarily z In this way the algo rithm continues
309. oblems will occur when applying it in a real control system In such cases one might remove dynamic elements from the plant and try it again One might even use a stationary plant model for the inversion In the following sections different controller structures will be discussed in more detail that follow the general approach outlined above 19 3 Inverse Model in Feedforward Path of Controller The controller with two structural degrees of freedom from Figure 19 4 on page 205 is generalized according to the approach sketched in the previous section This structure has been applied in Th mmel et al 2001 to the control of robots and has been successfully validated with hardware experiments In flight control the model following approach see for example Adams and Banda 1993 is a special case of this structure whereby the reference model is known as the command block providing state references for the inverse Draft Jan 21 2009 19 3 Inverse Model in Feedforward Path of Controller 211 model as well as the feedback controller Figure 19 9 Controller with two structural degrees of freedom and an inverse plant model in the feedforward path plant might be unstable however the inverse plant must be stable Sree ene iene ee Set oe Set ey ae ee ee ee ee ok ee en eet ek ye ye oe ee N u t Y 9 reference inverse yO ___ p gt model plant model f
310. od To get a guaranteed behavior the time synchronization has to be explicitly programmed with a counter as shown in the following code fragment parameter Modelica SIunits Time basicSamplingPeriod parameter Integer sampleFactor 5 Boolean fastSampling slowSampling Integer ticks initial equation pre ticks sampleFactor 1 equation fastSampling sample 0 basicSamplingPeriod when fastSampling then ticks if pre ticks lt sampleFactor then pre ticks 1 else 1 end when slowSampling fastSampling and ticks gt sampleFactor when initial fastSampling then equations for fast sampling controller end when when initial slowSampling then equations for slow sampling controller end when Integer variable ticks is used to count the number of occurrences of the fastSampling events At every event instant this counter is incremented by 1 until the desired sampleFactor is reached Then slowSampling is set to true at this time instant and the counter is reset to 1 at the next fastSam pling event During initialization the controller equations are activated in order to e g initialize in steady state 7 4 2 Inertial delay Another often occurring situation is the modeling of an inertial delay especially to model digital electrical circuits The purpose is to delay the input by a given time delay but only if the input holds its value for at least the delay time
311. ode to C and use the C libraries provided with f2c that emulate the FORTRAN run time environ ment This is the most portable and less cumbersome way to call FORTRAN 77 functions within Modelica in any Modelica tool In the Modelica Standard Library this approach is used for the more advanced nu merical functions that operate on matrices Modelica Math Matrices Here the standard linear algebra FOR TRAN 77 library LAPACK provides sophisticated algorithms to e g solve linear systems of equations or calculate eigen values In Dymola a C version of LAPACK is used that has been transformed from FOR TRAN 77 to C with the f2c tool After transforming the FORTRAN 77 code to C the further usage is of course identical to native C code However there is one remaining issue Arrays of two or more dimensions have a different storage lay out in FORTRAN column wise storage as in C row wise storage and f2c does not change this layout For notational convenience functions means both functions and subroutines Draft Jan 21 2009 5 6 External Functions partially available 105 scheme For example the 3x2 array A 11 12 A 21 22 5 8 31 32 is stored in the following way in contiguous memory in the two programming languages Table 5 3 Storage layout of matrix A in different programming languages C A 11 12 21 22 31 32 FORTRAN A 11 21 31 12 22 32 Dymola us
312. of equations in the different branches of an if clause might be different and therefore the if clause above is correct It is a bit involved to provide different useful options for the initialization of the block Above only the default initialization is defined that initializes the block in steady state This initialization is also correct if the dimension of vector x is zero because then the initialization equation is not present The above model works usually fine Let us simulate the transfer functions 1 10000 0 000l st1 5410000 Ga y The two transfer functions are mathematically equivalent and should give the same simulation result When simulating the step response of these systems using the default options of Dymola integrator DASSL relative tolerance le 4 different simulation results are obtained as shown in Figure 5 3 below note the step has to start at time gt start time because the transfer function initializes in steady state furthermore dif ferent simulation runs have to be performed for the two cases since otherwise the step size of y1 is also used for y2 and the problem disappears 80 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 Figure 5 1 The same transfer function gives different results for a relative tolerance of le 4 eo y1 1 0 0001 s 1 y2 10000 s 10000 u rel tol 0 0001 0 84 0 44 0 04 T r T
313. of the components are used e g only a test model to test the engine model above then dummy values have to be provided for not used bus signals in order that every declared variable can be computed For large signal sets this becomes very inconvenient e Different model levels and or model variants for a component might be present e g in Figure 3 8 component chassis might be a simple 1 dim model or a complicated 3 dim model and they might provide different signals to the bus or might need different signals from the bus The effect is that whenever simplified models are used then again dummy values have to be provided for not used bus signals Worse if a new variant is introduced that requires more signals from the bus then existing models have to be adapted by introducing dummy values for the newly needed bus variables Draft Jan 21 2009 3 5 Connectors for Signal Buses 53 e Ifthe component models are developed by different people or organizations and or are even in different libraries then one has to organize the definition of the single contro1Bus connector used by all components This is especially cumbersome if new signals are introduced or deleted since all libraries must be updated at the same time Due to the problems sketched above expandable connectors have been introduced in Modelica 2 2 This connector type does not have a pre defined set of declarations but allows to automatically introduce new va
314. ollowing table 94 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 Table 5 1 Functions in the Modelica Standard Library Modelica Math Mathematical functions e g sin cos Vectors Functions operating on vectors e g isEqual length sort Matrices Functions operating on matrices e g norm solve eigen Values Media Functions computing media properties Utilities Files Functions to work with files and directories e g remove Streams Functions to read from file and write to file e g print Strings Functions operating on strings e g sort scanReal System Functions for environment e g getWorkDirectory Some of these functions will be discussed in more detail in the following chapters 5 4 4 Functions with dynamically sized arrays Arrays inside functions can be dynamically sized as function of input arguments For example in function polynomialMultiply on page 88 the dimension of output vector c is defined as function of the dimension of input vectors a and b respectively using the size operator input Real a bl output Real c size a 1 size b 1 1 This is the standard way to dimension arrays in functions It is also possible to dimension arrays in a more flexible way after the array has been declared and or to resize arrays dynamically This is performed by assign
315. olution if start values for the iteration variables are provided that are sufficiently close to the solution In Modelica it is possible to formulate the above initialization problems by providing the missing infor mation in the models The two basic approaches are discussed in the next sections 4 2 Initialization with attributes With the attribute start an initial value of an arbitrary variable can be defined The start value must be a literal number or an expression that contains directly or indirectly only references of parameters and con stants With the additional Boolean attribute fixed it is defined whether the corresponding start value is the initial value of the corresponding variable fixed true or whether the start value is just used as a 62 Chapter 4 Initialization Draft Jan 21 2009 guess value fixed false i e as first value of the variable before the initialization problem is solved The default value of fixed is false i e it has to be explicitly set to true if an initial value for a variable shall be defined As an example we analyze a simple first order system with the transfer function a Tes l Y s U s 4 8 Transforming this transfer function into a differential equation results in T y y k u 4 9 Since this differential equation has one variable that appears differentiated y we need to provide one initial condition The goal is here to provide an initial va
316. omponents Modelica interfaces are called connectors Variables associated with a connector define the possible interaction with other components e Directed or undirected connection lines between connectors represent the actual physical connection e g an electrical wire a rigid mechanical connection heat transfer pipe fluid flow or a signal connection e A component such as the motor or the gear box in the figure above is either described by another composition diagram see the electrical circuit diagram of the motor above or is a basic component that is described by equations in the Modelica language see the Resistor model above e When double clicking on a component a parameter menu opens and displays the parameter data that can be modified for this component For example in Figure 1 2 below the parameter menu of the gear box Draft Jan 21 2009 1 1 Composition Diagrams 13 Figure 1 2 Parameter menu of gearbox component B gear in DriveSystern 2x General Add modifiers Component Icon Name gear Gow Comment T Model Path Gear Comment Realistic model of a gearbox Parameters ratio 10 transmission ratio flange_a phi flange_b phi eta 0 9 Gear efficiency friction_pos 0 1 10 2 w tau positive sliding friction characteristic w gt 0 peak 1 1 peak friction_pos 1 2 maximum friction torque at zero velocity c 1e5 gt N m rad Gear elasticity spring constant d 100 N m s rad relative g
317. op c i 0 0 for j in max 1 it l nb min i na loop cli cse t a Ab lrt rS end for Draft Jan 21 2009 5 4 Functions 89 end for end polynomialMultiply The dimensions of vectors a and b are arbitrary and are defined when calling the function Output vector c is the coefficient vector of the product of the polynomials represented by the coefficients vectors a and b The dimension of this vector must be defined before terminating the function In the example above the actual dimension of c is directly given when c is declared as function of the dimensions of a and b This is the usual way to dimension output arrays Afterwards in the algorithm section the output array is calculated Functions are terminated when the last statement of the respective function is reached Alternatively a function can be terminated after every desired statement by using the language keyword return This is demonstrated in the following example yet another variant to test whether two matrices A and B are identi cal function isEqual Determine whether matrices A and B are equal input Real A input Real B size A 1 size A 2 B has same sizes as A input Real eps min 0 0 0 Two numbers rl r2 are identical if abs rl r2 lt eps output Boolean identical algorithm identical true for i in 1l size A 1 loop for j in 1 size A 2 loop if abs A i j B i j gt eps then id
318. options when translating the inverse model Simulation Setup Realtime Inline integration method Only fixed step integrators can be used for a real time application Via simula tions the appropriate step size of the integrator has to be determined 19 8 5 Robustness As already mentioned the use of inverse model equations gives rise to robustness issues since any mismatch Draft Jan 21 2009 19 8 Difficulties with Inverse Models 223 between the inverted model equations and the actual plant will leave part of the nonlinearities and couplings uncompensated The usual approach is to provide robustness to model uncertainty via the linear feedback controller Figure 19 9 19 10 19 11 or the disturbance observer Figure 19 12 This can be done by appli cation of a robust control synthesis technique Adams and Banda 1993 or by robust parameter tuning in a classical structure e g using multi model techniques and enforcing sufficient stability margins Looye 2001 Tolerances on parameters in the model also appear in the inverse model equations In Looye 2001 it has been shown that these parameters may be very effectively used as additional tuning parameters in multi objective optimization The result is a model that is basically inverted at a location in the parameter space that provides the highest level of robustness 19 8 6 Summary Several control structures have been discussed that are based on non linear inverse plant mode
319. or x of the overall system Here the original plant states c T are used as state vector x Transforming the equations to state space form 19 12 results in the following sequence of assignment statements to com pute the derivative 7 of the state vector and of the output T as function of c T y ck ei ayar yta x c lom ga x s cux w ct i ee a 19 28 y e ar an 2 E E E E Ec kye T A T ay T an y an b For notational clarity the time derivatives of variables that are treated as purely algebraic variables dum my derivative method are denoted with an apostrophe such as x Equations 19 28 are a set of differential equations in state space form Given the desired concentrations Caes it is possible by numerical integration to compute the desired cooling temperature T ug in Figure 19 9 and the desired substance temperature T Yma in Figure 19 9 The latter is compared with the measured substance temperature forming the control er ror e as input to the feedback controller Even for this rather simple system the derivation of the nonlinear feedforward controller is not so easy Such a manual derivation becomes impractical if the plant model consists of hundreds or of thousands of equations as it is usual in complex Modelica models It is now demonstrated how to derive this nonlinear feedforward controller in an automatic way with Modelica and Dymola 19This is a CriticalDam
320. ore the current flowing in to pin_p is flowing out of pin_n The second equation states that the time derivative of the current flowing into the component via pin_p is proportional to the voltage drop over the two pins As a last action a suitable icon is drawn in the Icon layer of the inductor model see figure on the right Again the meta tag sname displayed as name in the figure is introduced in order that the instance name of the inductor is displayed when dragging it Addi tionally the actual value of the inductance is displayed by adding the meta tag L L displayed as L L in the figure This tag is displayed as a string in the icon L means that the actual value of the parameter with the name L shall be displayed When dragging the just defined model into another model the inductor icon is shown Double clicking on the icon opens the parameter menu This menu is automatically constructed from the parameter declara tions in the inductor model and shows the parameter name its unit displayUnit and description text as well as an input field in which the actual value of the inductance can be defined Figure 2 8 Automatically generated parameter menu from Inductor definition 2x General Add modifiers Component Icon Name inductor ero Comment e Model Path Inductor Comment Parameters E 0 001 H Inductance OK Info Cancel Note that the unit H is displayed in th
321. orrect all components where such warnings are reported before using them Otherwise the strong guarantee about balanced models does not hold because one or more used components are not lo cally balanced The basic rules for balanced models are rather simple and are introduced together with examples to demonstrate the issues Rule 1 Restrictions on Connectors The number of flow variables in a connector must be equal to the number of non flow variables that do not have an input output parameter constant prefix 38 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 The following connector is correct according to this rule connector Pin correct Modelica 3 0 connector Real u flow Real i end Pin because the connector has one flow i and one non flow u variable The following connector is not al lowed in Modelica 3 0 but was fine in previous Modelica versions connector Flange wrong Modelica 3 0 connector Real angle Real speed flow Real torque end Pin because this connectors has one flow torque and two _non flow angle speed variables If the number of flow and non flow variables has to be different for a particular application the modeler has to use the input output prefix see also section 3 4 Example connector FluidPortA correct Modelica 3 0 connector Modelica SIunits Pressure p flow Modelica SIunits MassFlowRa
322. ote that the plant model is horizontally flipped and connected to the inner input output connectors of the Inverse BlockConstraints block If the filter order is too low the DAE is not causal and Dymola prints an error message of the following form deducing this information from the result of the Pantelides algorithm Error The model requires derivatives of some inputs as listed below Order of input derivative 4 ul 2 u2 3 u3 Error Failed to reduce the DAE index In the second column the Modelica names of the input signals are listed that need to be differentiated accord ing to the differentiation order of the first column The numbers in the first column are therefore the mini mum order of the corresponding filters For example a low pass filter with a minimum order of 4 has to be used to filter the input signal u1 in order that no derivatives of u1 are needed for the inverse model Note in linear and nonlinear controller theory the Order of input derivative numbers shown in Dymolas error mes sage are the negative relative degrees of the respective inputs For example u1 has a relative degree of 4 If the inversion is to be based on a time derivative of the measured output a sufficient number of inte grators might be added instead of increasing the filter order There is always a filter order and or a number of integrators for which the system will translate The higher the filter order the more pr
323. oth in the continuous and in the discrete case In Modelica notation the difference equation above is implemented as when initial sample Ts Ts then new x d A xd B u RC Sed sk D Fag pre new x d xd BA un x X aK ll end when Since no next value operator is available in Modelica an auxiliary variable new _x_d stores the value of x q for the next sampling instant The relationship between new _x_ d and x qd is defined via equa tion x _d pre new_x d The body of the when clause is active during initialization and at the next sample instant t 7 Note the when equation is not active after the initialization at 0 due to sample Ts Ts instead of sam ple 0 Ts since the state x _d of the initialization has to be used also at t 0 Additional equations are 66 099 added for the initialization to uniquely compute vectors x and x q at the initial time initial equation if init InitialState then x x start elseif init SteadyState then x d new x d end if If option InitialState is set the discrete state vector x _d is computed such that the continuous state vector x x start i e the continuous state vector is identical to the desired start value of the continu ous state If option SteadyState is set the discrete controller is initialized in steady state default setting in sampleClock This means that the output y
324. ow certain classes of finite state machines can be defined in Modelica In general only formalisms can be imple mented conveniently in Modelica that are deterministic and have a black box module behavior This means e g that most Statechart types cannot be implemented in a convenient way because they may have transi tions over several hierarchies and the hierarchical layers are not hidden in a black box but are visible on the highest layer In the Modelica StateGraph library is described that can be directly utilized in a large class of discrete event applications It is a Modelica based hierarchical state machine with a similar modeling power as Statecharts In this chapter the basic method is explained how to implement state machines This mechanism was utilized to implement the StateGraph library but it can also be used to implement other types of deterministic au tomata such as prioritized Petri nets 9 1 Implementation of Finite State Machines There are several ways to implement state machines in Modelica If components shall be implemented in a library it is usually desired to have an object oriented implementation where the equations use only infor mation from the connected elements and nothing else The basic mechanism is explained by means of the following figure Figure 9 1 General method to implement a state machine in Modelica inCond1 outCond1 inCond2 outCond2 1 outCond
325. owing limiter equation y if u gt 1 then 1 else if u lt 1 then 1 else u As usual the first branch y 1 is selected ifu gt 1 otherwise the second branch y 1 is selected provided u lt 1 and if also this does not hold the else branch is selected y u Every if expression must have an else branch in order that an equation is always well defined Alternatively conditional expressions can be defined with if clauses such as if u gt 1 then yl elseif u lt 1 then elseif not else if as for if expressions yu l else y u end if This if clause has the same effect to compute y as the previous if expression Every if clause must have an 28 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 else clause and every branch must have exactly the same number of equations Again the reason is that in every situation the number of equations must be the same in order that the number of unknowns and the number of equations is always identical In fact since Modelica is an equation based system every language construct is finally mapped to equa tions in order that the symbolic transformation algorithms see Chapter 12 can be applied Typically a Mod elica translator and especially Dymola transforms an if clause in to a set of equations with if expressions For example the if clause below has two equations in every branch if condition then exprla exprlb 0
326. package shall be included as sub library in another package the desired package has to be selected from the pull down list Insert in package optional Draft Jan 21 2009 2 6 Model Libraries 35 Figure 2 15 Definition of a new package with Dymola Dymola Dynamic Modeling Laboratory Diagram loj x File Edit Simulation Plot Animation Commands Window Help 218 x fe Model e 4 gt elemee2 i00 c Open Ctrl O Connector P rete T Libraries Record fad Demos Block E Save CHS Function 2x Save As Package Name of new package Save All Duplicate Class ServoSystem Save Total are Vein x Description Clear All Q Search r Partial C cE Extends optional cS P DD You are adding servosystem as E a new top level package S Insert in package optional We just want to verify that you did PEERS A not forget to give the name of the r enclosing package m Save contents of package in one file al E If a sub library shall be created or a model or connector should be created in a library it is usually most con venient to first select with the right mouse button the desired package here ServoSystem Then the menu entry Edit New Class in Package New Model has to be selected Figure 2 16 Definition of a new model inside a package with Dymola S File Edit Simulation Plot Peicreate new Model B a
327. ping filter without normalization for simplicity 216 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 Figure 19 13 Modelica model of mixing unit with constant cooling temperature T const mixingUnit c k 308 5 In Figure 19 13 a Modelica model of the mixing unit is shown The constant input is the cooling temperature the outputs are the concentration c and the temperature 7 of the substance This model contains just the plant equations 19 22 Simulation results of this model are shown in Figure 19 14 Figure 19 14 Simulation results of mixing unit for c to 0 237 mol l T t 323 9 K T t 308 5 K mixingUnit c 0 25 _ 0 24 m E 0 23 0 22 0 1000 2000 3000 4000 mixingUnit T 325 324 323 0 1000 2000 3000 4000 As can be seen the system is unstable at this operating point In Figure 19 15 the inverse model of the mix ing unit is constructed by connecting the input c_des via a filter to the c output of the mixing unit i e the concentration c is treated as known input signal Figure 19 15 Inverse model of mixing unit with pre filter fitter c_des f 1 When this system is translated without the filter Dymola reports that the second derivative of c_des is needed In a second step the filter is included with order 2 and Dymola translates without an error Af terwards the inverse model is connected with the plant model according to Figure 19 8
328. ports the differentiability order order 0 continuous order 1 con tinuous and first derivative is continuous order 2 continuous first and second derivatives are continuous etc Since u 2 u 3 0 for u 0 and der u 2 2 u der u 3 3 u 2 0 foru 0 ar 110 Chapter 6 Discontinuous Equations Draft Jan 21 2009 gument order 1 in the example above The actual behavior of the Modelica simulation environment is not defined for the smooth operator The usual interpretation is to not generate an event if order 2 0 If a statement containing the smooth operator is differentiated during the symbolic preprocessing the order argument is decremented for every differentiation If order 1 the expression is discon tinuous and an event has to be generated If such a statement is differentiated again a Modelica environment will usually report a translation error because the derivative of a discontinuous expression results in a Dirac impulse that is difficult to handle In some exceptional cases it is necessary to force the Modelica simulation environment to never gener ate an event This is performed with the noEvent built in operator Example y noEvent if u gt eps then 1 u else 1 eps This if expression is taken literally as in programming languages i e no event is triggered Without the noEvent operator this example might fail because if u is larger as eps and then becomes smaller as eps an eve
329. pr Real elementary relations within expr are taken literally i e no state or time event is triggered This operator must be used if the definition range of a variable is left and this would trigger an error In the following example noEvent is used to avoid a division by zero y noEvent if x gt eps then 1 x else 1 eps smooth p expr If p gt 0 then smooth p expr returns expr and states that expr is p times continuously differentiable i e expr is continuous in all real variables appearing in the expression and all partial derivatives with respect to all appear ing real variables exist and are continuous up to order p The only allowed types for expr in smooth are Real expressions arrays of al lowed expressions and records containing only components of allowed expres sions Examples yl smooth 1 y2 smooth 1 if x gt 0 then x 2 else x 3 noEvent if x lt 0 then 0 else sqrt x x noEvent is necessary sample start interval Returns true and triggers time events at time instants start i interval i 0 1 During continuous integration the operator returns always false The starting time start and the sample interval interval need to be parameter expres sions and need to be a subtype of Real or Integer Example when sample 0 0 1 then executed every 0 1 s exactly once end when pre y d Returns the left limit y t of variable y t at a ti
330. pre Operator cccceccccessccessceesseessecesseeeeseecsseeessceeseeceeneessteeenteeenaees 110 Chapter 7 Instantaneous Equations ccccscsccccssccscsssccscssccsscsesssssceccssccessssccsssssccsessssssssessssseesees 114 5 Draft Jan 21 2009 7 1 When clauses and mapping to instantaneous CqUuatiONS ccscceessceesseeeseeeeseeeesesneeeeeeeeseeeeeeeees 114 7 2 Sampled data systems and imitialiZation cesccescceseeeeceseceeeceseceseceeceaeceneceaeceeeceseceaeeeaeenseeeeas 117 7 3 Prioritizing event aACtlOMSs ies ec ss ecsiccriirsesendesaudasticsacdaeessasbsncacsedaastadand caseataeeaneccnedsccaniiedaceaensaiaees 119 7 4 Time synchronization multi rate delays range limited sampling cc cccssesseeeteeeeeeeennees 121 7 5 Algebraic loops between continuous and instantaneous CquatiONS ccccceeeseeeteeesteeeeteeeeteeeees 125 Chapter 8 The Synchronous Principle ccssssccsssssscssscsccsscscesscccescsscsscssssssecscscscecccceeseesseessees 128 Chapter 9 Finite State Machines cccccsscccssscsssscssscsssscsssccsssscsssesscsssscsssssssssssssssssssssssesssssseseoenes 132 9 1 Implementation of Finite State Machines cccccecsscessscceseeseeeseeeeeseeeeeeseeeeeesneeeeesseneeeeeees 132 O72 Event Werati Om eiio ates adh al Medel REG Reside healed vind eee 134 Chapter 10 Variable Structure Systems cccccccssscssssscssscsssscsssscsssscsssscscsssssssssssscsssessesssss
331. pre v values of the variables v are given values from the previous event Ifpre vi vi for at least one discrete variable vi after the equations have been solved then pre vi vi is set and the system of equations is solved again This event iteration is terminated once pre vi vi for all discrete variables vi A solver will usually terminate with an error if a maximum number of it erations is reached It is an unresolved problem in Modelica to either detect during translation that event iter ation will not converge indicating an erroneous model or to modify the semantics of Modelica so that such a situation cannot occur As an example the initialization of the hysteresis block from above is analyzed eu gt I Since pre high true the following equations are solved during initialization high false and u gt 1 or true and u gt 1 y if high then 1 else 1 This results in high trueandy 1 Since high pre high the initialization is finished eu lt it Since pre high true the following equations are solved during initialization high false and u gt 1 or true and u gt 1 y if high then 1 else 1 This results in high falseand y 1 Since high pre high the initialization equations are again evaluated using pre high high false i e solving the following equations high true and u gt 1 or false and u gt 1 y if high then 1 else 1 This results in high fa
332. precision the integration is halted an event is triggered the i clause is changed from the else to the then branch and the integration is restarted This means that the above if expression is not equivalent to an if expression of a programming language such as Fortran C or Java because the if expression remains on the branch that it had previously until the event iteration is finalized indepen dently of the actual value of the i condition The advantage of this approach is that in most cases discontinuous changes are handled automatically in a numeric reliable and efficient way If a relation has the form time gt discrete expression or time lt discrete expres sion it is possible to compute the time instant of the event in advance In such a case no event iteration takes place and the integrator adapts its step size so that it steps exactly to the event point i e the simulation is more efficient Example y if time lt 1 then 0 else 1 The latter case is called time event whereas an event is called state event when an event iteration has to take place In some cases it is unnecessary that a relation triggers an event because the corresponding expression is smooth The modeler can report such a property with the smooth order expr built in operator of Modelica Example y smooth 1 if u gt 0 then u 2 else u 3 The first argument to smooth re
333. ps The equations are then if possible solved symbolically or code for efficient numeric solution is generated Other Modelica simulation environments might work differently although the basic algorithms such as BLT section 12 2 and the Pantelides algorithm section 12 4 will most likely be used 12 1 Transformation to State Space Form In Chapter 8 it is sketched how a Modelica model is mapped to a set of differential algebraic and discrete equations We will demonstrate this mapping on the basis of a simple electrical circuit and show how this ba sic set of equations is further processed In a first step a small library of electrical components is modeled using the connector definition of an electrical pin as explained in section 2 2 The result is shown in the next table Draft Jan 21 2009 12 1 Transformation to State Space Form 151 Table 12 1 Equations of basic electric circuit components Resistor i i 0 iti D x 2 o R o u VTV Y _ Y u Ri u Capacitor i C i 0S iiti Ea a w vsv Ms _ V c 7 u dt Inductor i L 0 iti 1 2 LAN u vv v v di 1 gt 2 1 L u u dt Voltage source i u i 0 iti v v u Asin w t Ground i v 0 gyv cor 99 In order to simplify the notation in this example a connector variable such as pin_p i is defined as i Otherwise the same equations are used as for the electrical components in the Mod
334. pushbuttons are operated again The implementation with a StateGraph is shown in the next figure under the assumption that the onDe lay block operator would be available see page 123 Figure 18 6 StateGraph to model the safe operation of two pushbuttons 51 transition3 wait transition2 not B1 or not B2 not B1 and not B2 MTV B2 buttonsReleased transition buttonsPressed pressButtons B1 and B2 and not onDelay B1 or B2 0 5 As can be seen e property 1 to be proved A cycle can only be initiated by pressing the two pushbuttons simultaneously Draft Jan 21 2009 18 2 Conditions and Actions 195 with 0 5s is used as condition of transistion1 e property 2 A cycle is interrupted by releasing one or both buttons to stop the output is used as condition of transition2 and e property 3 The output signal can only be re initiated after both inputs have been released and the pushbuttons are operated again is used as condition of transition3 At transition3 also an explicit wait time of 1 s is added to enforce that the machine is really coming to a halt before the buttons can be pressed again This program is so simple and obvious that verification of the properties 1 2 3 seems not to be neces sary 18 3 Parallel and Alternative Execution Parallel activities can be defined by component StateGraph Parallel and alternative
335. quations are not inserted into each other in order to arrive at only 2 equa tions for the state derivatives The reason is that the number of operations usually increases dramatically when expressions are substituted and therefore the efficiency of the model evaluation is considerably re duced This can be already seen at the simple example above There are 2 multiplications and one sin func tion evaluation needed in order to compute variable R v This variable is present at two places It would therefore be possible to remove the assignment statement to compute R v and replace this variable at every occurrence by its definition equation However this would mean that the number of operations increases be cause then 4 multiplications and 2 sin function evaluations are necessary The symbolic transformation algorithms to be discussed are suited for very large systems of equations E g in Dymola it is possible to process more than 100000 equations in a few seconds on a PC One reason is that the effort for the symbolic transformation grows usually linearly with the number of equations In Dymola the default integrator to numerically solve the ODE form of the symbolically processed equations is DASSL DASSL is an integrator that is designed to solve DAEs This can be easily accom plished by just re writing the ODE into the DAE form 0 x t f x t t 12 8 The numerical solution of 12 8 with DASSL is usually very reliable and efficient due
336. r Real T time constant RealInput u input signal RealOutput y start 1 fixed true output signal Real der y start 0 fixed false der y derivative of output equation der y der y T der y y k u end FirstOrder2 When using the default setting of this model the initial equation y 1 is used during initialization When clicking on the icon of an instance of this model the following parameter menu opens Figure 4 2 Parameter menu with start and fixed attributes for y and der y eral Add modifiers Component Icon Name firstOrder1 peter Comment a Model Path FirstOrder1 Comment Parameters k gain T time constant Initialization y start M output signal der_y start der y derivative of output ok Info Cancel When marking the check box of der_y start and removing the mark for y start then the model ini tializes in steady state Since the feature to include the start value definition automatically into the parameter menu is simple to 64 Chapter 4 Initialization Draft Jan 21 2009 Figure 4 3 Steady state initialization with der_y fixed attribute y start E output signal der_y start der y derivative of output define and the user interface is very clear for someone using such a model this is the recommended way to define initialization in Modelica models Definin
337. r elements have to be dragged Every Modelica element may have an optional description text enclosed in apostrophes such as Mass of block Since this description text is provided before the that terminates an element the text is uniquely associated with the corresponding element Description texts are used by a Modelica tool for vari ous purposes e g as description text in a parameter menu or as label in a plot window It is also possible to use comments in C Java notation i e either lt until end of line gt Draft Jan 21 2009 2 1 Simple Declarations and Equations 17 or lt text over several lines gt Comments are just ignored during parsing of the model The equation keyword separates the declaration section from the equation section In the equation section equations between the declared variables are present Note these are not assignment statements as in procedural languages such as C C or FORTRAN Instead the general form expressionl expression2 defines that the expressions on the left and right hand side of the equal sign are mathematically identical Ev ery equation has to be terminated with a semicolon The built in der operator of Modelica de fines the time derivative of the enclosed variable e g der v dv dt Modelica has only an operator for the first time derivative In order to define derivatives of higher order auxiliary variables have to be in
338. r minimum phase plants We will now considerably enhance the control system by a second structural degree of freedom In a first step only minimum phase plants are taken into account to simplify the discussion see Figure 19 4 Figure 19 4 Controller F R C with two structural degrees of freedom R C for a minimum phase plant P P might be unstable y Tyrt Tyn ded dy PC 1 R i d TEPC TEPC u y P Desired R C such that Pie T a Restriction R stable and Dr Z Dp In this case the controller consists of 3 parts the feedback controller C the pre filter R and the feedforward controller F RP which is computed from the pre filter and the inverse plant model From the transfer functions in the right part of Figure 19 4 it can be seen that all of them have a different dependency on the controller parts Especially the reference transfer function is identical to the pre filter R and the measurement noise and disturbance transfer functions depend only on C but not on R This means that the reference trans fer function can be designed completely independent from the measurement noise and disturbance transfer functions Due to this important property the controller structure above has two structural degrees of free dom The 3 controller parts are computed from the design transfer functions C and R and the given plant transfer function P The design of R is simple and st
339. r other tools provide diagnostics here As an example consider the following model Real x start 1 y start 2 algorithm x 2 y yi 2 amp During translation such a code fragment is transformed to the following evaluation sequence input xX Start y start output ep Sy algorithm x x start y y start KX 1 2 y y 2 X With the given start values x start 1 y start 2 the result of this algorithm section is therefore x 4 y 6 Since the result depends on y start the user probably made a mistake and a rewriting of the algorithm section would be useful A model may have any number of equation and algorithm sections e g model Test equation equation section 1 algorithm algorithm section 1 algorithm algorithm section 2 equation equation section 2 algorithm algorithm section 3 end Test Draft Jan 21 2009 5 3 Algorithm Sections 87 The semantics is that every algorithm section is mapped to a set of equations and all equations are sorted together Practically this means that all statements within an algorithm section always remain together However different algorithm sections may be sorted relative to each other This means that in the Test example above the algorithm sections might have a different evaluation order with respect to each other after sorting e g algorithm section 3 is evaluated say before algorithm section 2 which is evalu
340. r the unknowns x and y This is not as difficult as it looks like because balance equations in physics are linear in the derivatives X and all connection equations are simple linear equations This means that a large portion of f is usually linear in the unknown variables and not non linear which will considerably simpli fies the transformation to ODE form In order that a unique solution of the non linear equation exists the Jacobian J of this equation with re spect to the unknowns must be regular For the moment we will only treat models where this basic require ment is fulfilled J a ar is regular everywhere 12 7 In such a case all potential states x of the DAE are states of the ODE form Models with a singular Jaco bian are treated in section 12 4 Before transforming to ODE form the states of the ODE form must be identified from the set of variables used in a Modelica model because these variables are treated as known during the transformation The states are simply all variables on which the der operator is applied in the Modelica model In the simple electrical circuit example above the variables C u and L i are used with the der op erator and therefore these variables are assumed to be known Additionally all constants and parameters such as R J R or R2 R are known variables All other unknown variables are summarized below Table 12 3 The 27 unknowns of the SimpleC
341. raft Jan 21 2009 Chapter 18 Hierarchical State Machines StateGraph Library In many applications a user would like to define a discrete algorithm graphically as a finite state machine e g as a Petri net a Sequential Function Chart SFC or a Statechart In this chapter it is described how cer tain classes of finite state machines can be defined in Modelica In general only formalisms can be imple mented conveniently in Modelica that are deterministic and have a black box module behavior This means e g that Statecharts cannot be implemented in a convenient way because Statecharts may have tran sitions over several hierarchies and the hierarchical layers are not hidden in a black box but are visible on the highest layer In the Modelica Standard Library there is a sublibrary called Modelica StateGraph that is used to model hierarchical finite state machines It is based on SFCs but with the extensions from JGraphcharts Arzen et al 2002 so that the modeling power is similar to Statecharts This library should be used to model finite state machines in Modelica if there are no specific reasons to do it differently In this section the components of the StateGraph library are introduced by examples to show how it can be used in applications Currently a new state machine library called ModeGraph is under development that shall be much bet ter as the StateGraph library by including new features in Modelica and Dymo
342. ragments only once and reference them correspondingly In Modelica this is performed with the extends clause For example several electrical base components consist of 66199 two electrical pins and the voltage drop u between the two pins is needed to formulate the material law 26 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 of the component partial model OnePort odelica Electrical Analog Interfaces PositivePin pin _p odelica Electrical Analog Interfaces NegativePin pin_n odelica SIunits Voltage u Voltage drop between pin p and pin n odelica SIunits Voltage i Current flowing from pin p to pin_n equation 0 pin pri pin_n i u pin _p v pin_n v i pin_p i end OnePort Additionally model OnePort defines that the in flowing current is identical to the out flowing current The model is declared as partial meaning that it is incomplete and that it cannot be dragged into an other model For example the following usage of OnePort is not allowed OnePort onePort wrong Modelica code fragement since OnePort is partial partial models can only be used via inheritance with the language construct extends model Inductor extends OnePort parameter Modelica SIunits Inductance L equation L der i u end Inductor The meaning is that the whole definition of OnePort is co
343. raightforward since R is the desired reference transfer function It has only the slight restriction that it must be stable and that the relative degree cannot be smaller as the relative degree of the plant Details how R is designed in a suitable way are given below It is important to understand the construction above in order to be able to generalize it to nonlinear plant models in section 19 3 In the next figure the feedback controller C and the disturbance and measurement noise signals are removed Figure 19 5 Construction of two degree of freedom controller for minimum phase plant 4 F R P gt R p tro Output y is computed by the open loop controller from Figure 19 1 Therefore restrictions on R apply as de 206 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 fined in Figure 19 1 The reference transfer function is independent of the plant 1 y P P R r R r 19 9 e Rr y Rr Rr 0 A control error e is constructed by adding the reference transfer function also as pre filter and subtracting it from the measurement y Due to this construction the control error e is identical to zero if disturbance and measurement noise are zero and the plant inverse cancels the plant identically In this idealized situation the feedback controller C is not needed because its input is zero Therefore C is only used to act against di
344. rameter Real y0 0 y yO at initialization ifsteadyState false annotation Dialog enable not steadyState RealInput u input signal RealOutput y output signal equation T der y y k u initial equation if steadyState then der y 0 else y y0 end if end FirstOrder4 Via parameter steadyState the type of initialization is defined Depending on this condition a value for pa rameter y0 is or is not needed In order to make this more clear in the graphical user interface the annotation Dialog enable not steadyState is included for y0 This is a standard Modelica annotation and defines that the input field for parameter yO is only enabled when the enable attribute is true In other words it is not possible to define a value for yO in the parameter menu if parameter steadyState has val ue true The parameter menu of the above model has the following layout in Dymola Figure 4 4 Steady state initialization defined with Boolean parameter Erster tin Unnamed ax General Add modifiers Component Name firstOrder4 Comment Model Path FirstOrder4 Comment Parameters k te gain IE time constant steadyState true sp initialization type yo oO y y0 at initialization if steadyState false OK Info Cancel 66 Chapter 4 Initialization Draft Jan 21 2009 4 4 Too many or not enough initial conditions It is quite a lot of work to defin
345. rations and equations of the model classes referred in extends clauses are conceptually in serted If two different model classes have the same declarations and both models are inherited the declara tions must be syntactically identical and only one of them is inserted For example model C in model A parameter Real v end A model B parameter Real v end B model C extends A extends B end C is conceptually equivalent to model C parameter Real v end C If model B would define a default value for v v 1 whereas this would not be the case in model A then model C would be no longer a valid Modelica model and the Modelica translator will complain 2 4 Conditional Equations A Modelica model can utilize similar control structures as in a programming language like C Due to the equation based nature and since Modelica is designed to define simulation problems there are some differ ences The most important control structures if expressions and if clauses are discussed below They are used to describe conditional equations More details are given in Chapter 6 page 108 Loops in equation sections are discussed in section 5 2 whereas control structures in algorithm sections are discussed in sec tion 5 3 An expression in Modelica e g in an equation or in a declaration may contain an if expression i e an if clause that is written directly in the expression e g as shown for the foll
346. rd Library once some missing features in Modelica and in Dymola are available such as operator overloading and Dymola spe cific build in functions are replaced by equivalent standardized functions e g for plotting and for lineariza tion Library ModelicaLinearSystems provides different representations of linear time invariant differential and difference equation systems as well as typical operations on these system descriptions In the right fig ure a screen shot of the first hierarchical level of the library is shown StateSpace TransferFunction ZerosAndPoles and DiscreteStateSpace are records that provide basic data structures for linear control sys tems Every record contains a set of utility functions that operate on the corresponding data structure Espe cially operations are provided to transform the respective data structures in to each other Sublibrary Math contains the basic data structures Complex and Polynomial that are utilized from the linear system descriptions Sublibrary Sampled contains a library of input output blocks to conveniently model and simulate sam pled data systems where it is convenient to quickly switch between a continuous and a discrete block repre sentation 17 1 Basic Mathematical Data Structures Sublibrary Math provides the basic data structures Complex and Polynomial that are utilized and needed from the linear systems data structures to be discussed in the next section In the Modelica As
347. re there might be several state ments to compute the same variable which is not possible in an equation section This is demonstrated in the Limiter example above where first an assignment to y is performed and then a second assignment to y is present to reassign a new value if u is outside of the defined limits Another example with nested for loops is shown in the next code fragment to determine whether two arrays are identical up to a given precision Determine whether matrices A and B are equal input Real A input Real B size A 1 size A 2 B has same sizes as A parameter Real eps min 0 0 0 Two numbers rl r2 are identical if abs rl r2 lt eps output Boolean identical algorithm identical true for i in l size A 1 loop for j in 1 size A 2 loop if abs A i j B i j gt eps then identical false break end if end for Draft Jan 21 2009 5 3 Algorithm Sections 85 if not identical then break end if end for With the two nested or loops all elements of matrices A and B are tested pairwise whether they are identi cal If one pair of elements is found that are not identical the flag identical is set to false and the for loops are at once terminated without checking all the remaining elements The termination is performed with the language keyword break which terminates the inner most for or while loop The break key
348. re still un resolved issues and further research is needed 11 1 The reinit operator In Modelica it is possible to change non discrete Real state variables discontinuously at an event instant with the reinit built in operator that is defined as reinit x expr In the body of a when clause reinitializes x with expr at an event in stant x is a Real variable resp an array of Real variables in which case vectorization applies that must be selected as a state resp states at least when the enclosing when clause becomes active expr needs to be type compatible with x The reinit operator can only be applied once for the same variable resp array of variables It can only be applied in the body of a when clause The reinit operator is processed by assuming that the previously known state variable x refer enced in the reinit operator is treated temporarily as unknown with the equation x expr Af ter sorting BLT partitioning the reinit clause is present so that all references to x are after the reinit equation and all variables referenced in expr are computed before this equation Note this gives some restrictions what can be referenced in expr e g no variable that depends explicitly or im plicitly on x Additionally code must be introduced to inform the integrator that the corresponding state has changed at an event instant Since state variables x are a
349. reasons why functions with internal memory are not allowed in Modelica 1 Functions are potentially called directly or indirectly in an equation section which is mapped to the right hand side f of a differential equation system x f x t Numerical integration algorithms might fail if f is computed via a hidden internal memory that accesses values of variables from previous integra tion steps As usual Modelica uses the safe approach and guarantees that such a potential misuse is not possible with Modelica functions and therefore has this restriction with external C functions such type of misuse cannot be prevented 2 The symbolic transformation algorithms see Chapter 12 cannot be applied on functions that have an in ternal memory since equation sorting requires that all variables that are relevant for the sorting process are taken into account and an internal memory of a function would consist of hidden variables 3 Modelica tools especially Dymola enhance efficiency by common subexpression elimination For ex ample the following code fragment y sin phi x 1 cos phi x 2 cos phi x 1 sin phi x 2 is translated by Dymola into the following form during code generation wl sin phi w2 cos phi y wl x 1 w2 x 2 w2 x 1 wl x 2 This means that the function calls of sin and cos are evaluated only once and not twice In general function cal
350. riables on the connector when connecting to it An example is shown in Table 3 8 Table 3 8 Example of bus definition with an expandable connector expandable connector ControlBus no declarations copy the content of Modelica Icons SignalBus here end ControlBus Icon provided in Modelica Icons SignalBus The connector usually does not contain any variable declarations and has only the prefix expandable If variable declarations are present they are handled exactly in the same way as in other connectors However there is the restriction that an expandable connector may not have flow variables The actual content of an expandable connector is built up by connecting input and or output signals to it Example model TestBus Modelica Blocks Continuous FirstOrder firstOrder ControlBus controlBus equation connect firstOrder y controlBus filteredSpeed end TestBus Due to the connect statement variable filteredSpeed is implicitly introduced on the con trolBus This means that the new variable controlBus filteredSpeed is implicitly declared Only signals declared with the input or output prefix can be connected to an expandable connector Dymola supports expandable connectors in a nice way in the diagram layer Example 1 The output signal of a speedSensor component is connected to an instance of connector ControlBus Figure 3 9 Step 1 Co
351. rogram and to utilize the re sult of a FE calculation in the Modelica model A simple example is shown in Figure 3 5 Figure 3 5 DC motor model with heat transfer to the environment 8 3 amp erd 3 Y S S winding core x as ambientTemperature 2000 windingToCore 20000 YY fA convection T 20 a inductor ig Lo Born a heatingResistor L 0 0061 sine ar 5 a motorlnertia D gt lt mo a i S emf m m J 0 0025 freqHz 5 a ground It consists of an uncontrolled DC motor where the voltage source is driven by a sine wave The heat transfer in the winding resistor is modeled by using a resistor with a heat port Modelica Electrical Ana log Basic HeatingResistor The heat generated in the resistor port i port_a v port _b v is transfered to the heat capacitance of the winding named winding is further transfered via conductance to the heat capacitance of the core named core and is finally transfered to the environ ment via convection The heat flow between these lumped elements is characterized by the red connection lines in the figure above Typical simulation results are shown in Figure 3 6 The temperatures of the winding and core reach a steady state value after some time Since the excitation is a sine wave of the source voltage the speed of the motor inertia is also a sine wave Draft Ja
352. rrespond ing potential variables are set equal and the sum of corresponding flow variables sums up to zero Plug A B A phase v B phase v equation A ground v B ground v l connect A B 0 A phase i B phase i me 0 A ground i B ground i Assume that a resistor shall be connected to a pin in a Plug connector One solution is to directly connect from the pin of the resistor to the plug see Figure 3 15 Figure 3 15 Connecting the pin of a resistor to a Plug connector plug resistor HO R 100 Then the menu of Figure 3 16 pops up and shows in two columns the definitions of the connected connec tors In the right column a sub connector in the plug must be selected that has the same definition as the connector in the left column of the menu In this case sub connector phase is selected 58 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 Figure 3 16 Menu to select the connection in to a Plug connector connect resistor p plug 2 x Connect to component of outside connector by selecting sub components below Name Unit Description Name Unit Description resistor plug W p Positive pin potential p v gt n v for posi Positive pin of an electric compon i Y y Potential at the pin po Potential at the pin ij A Current flowing into the pin i A Current flowing into the pin ground Negative pin of an electric compo by y Potential at
353. rtialDelay y 44 24 0 i T T T T 0 1 2 3 4 5 6 7 4 3 On Off delay Modelica has a delay operator to model time delays of Real Integer and Boolean signals Be sides the InertialDelay also other delays need to be modeled e g onDelay A rising edge of the Boolean input gives a delayed output A falling edge of the input is immediately given to the output This can be defined as block OnDelay odelica Blocks Interfaces BooleanInput u Modelica Blocks Interfaces BooleanOutput y parameter Modelica SIunits Time delayTime protected Boolean delaySignal start false fixed true discrete Modelica SIunits Time tNext start 0 fixed true algorithm when u then delaySignal true tNext time delayTime elsewhen not u then delaySignal false tNext tim T end when equation 124 Chapter 7 Instantaneous Equations Draft Jan 21 2009 y delaySignal and time gt tNext end OnDelay The implementation is similar to the InertialDelay but with some changes Simulation results of a typi cal example of the OnDelay block with a delay time of 0 1 s is shown in the next figure Figure 7 4 Simulation results of OnDelay block with a delay time of 0 1 s onDelay u onDelay y 1 5 1 0 0 5 0 0 0 0 0 5 1 0 7 4 4 Range limited sampling Sampled data systems are often reduc
354. rules it is now possible to understand the basic philosophy of balanced models For ev ery unknown variable in a model or block exactly one equation must be provided in the class with the fol lowing exceptions 1 Ifa variable has an input prefix the equation for this variable must be provided when using this component therefore no equation has to be provided in the model or block class where this variable is defined 2 Ifa connector has N flow variables then it must have also N non flow variables without a prefix according to rule 1 Therefore such a connector introduces 2N unknowns Exactly N equations must be provided in the component class where the connector instance is defined and N equations must be provided when using the component Note that the latter requirement is automatically fulfilled when connecting connectors together If M connec tors are connected together via connect equations and every connector has N flow and N non flow variables then the Modelica connecting semantics provides M 1 N equality equations and N zero sum equations i e the missing M N equations for M connectors For example if 3 Capacitors are connected to gether Capacitor Cl C2 C3 equation connect Cl p C2 p connect Cl p C3 p the following 3 equations are generated Cl p v C2 p v Clp y C3 put 0S Clop de C2 pI t C3 ps3 and these are the three missing equations for the connectors C1 p C2
355. s action language Deterministic behavior tA is guaranteed UZLL import Modelica Math Matrices Modelica Math Matrices Modelica Utilities A 1 2 3 Functions operating on matrices e g to solve linear 3 4 5 systems and to compute eigen and singular values 2 1 4 b 10 22 12 x Matrices solve A b Matrices eigenValues A Also functions are provided to operate on strings streams and files Sublibraries Modelica Fluid and Modelica LinearSystems mentioned above will be included in the Modelica Standard Library in one of the next releases Beta releases are already shipped with Dymola and are available via the File Libraries menu in Dymola 16 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 Chapter 2 Basic Modelica Language Constructs 2 1 Simple Declarations and Equations Some basic language elements of Modelica are introduced on the basis of the very simple model of a mass that is pulled with a constant force see Figure 2 1 Figure 2 1 Mass that is pulled with a constant force m 2kg It would be very easy to model this system using components of the Modelica Mechanics Translational li brary However modeling this system directly with Modelica language elements can be done with the fol lowing model code model MovingMass1l Mass pulled with a constant force parameter Real m 2 Mass of block parameter Real f 6 Pullin
356. s 227 Chapter 21 Modelica Built In Functions and Operators This chapter contains a summary of all built in operators and built in functions of Modelica 3 0 e Elementary operators on page 227 e Mathematical functions on page 229 e Conversion functions on page 230 e Derivative and special purpose operators on page 232 e Event related operators on page 232 e Array operators on page 233 This information is also available from the following package which is provided in Dymola s package browser ModelicaReferenc The tables in this chapter are partly a copy of the corresponding tables from the Modelica Language Specifi Operators cation 3 0 and or from the ModelicaReference package 21 1 Elementary Operators Table 21 1 Arithmetic Operators operate on Real Integer scalars and arrays Operator Example Description t H a b Addition and subtraction element wise on arrays a t b ar Ap Multiplication scalar array element wise multiplication vector vector vector product element wise multiplication and summing the result result scalar matrix matrix matrix product vector matrix row matrix matrix result vector matrix vector matrix column matrix result vector a fb Division of two scalars or division of an array by a scalar is defined element wise The result is always of type Real even if both argu ments are of type Integer In order to get Integer division with tru
357. s an associated coordinate system which is rigidly at tached to the component A typical component is shown in Figure 3 3 46 Chapter 3 Component Coupling by Ports Draft Jan 21 2009 Figure 3 3 Coordinate system and vectors of an inertia component i The x axis of the coordinate system fixed to the inertia is denoted as unit vector All vectors are ex pressed with this unit vector This gives the following mathematical description Table 3 5 Mathematical and Modelica description of inertia component Mathematical description Modelica definition we pe gt an gt hi flange a phi Je t T e p oe T q r phi flange b phi l w der phi w P J der w flange a tau flange_b tau JW T T Since all vectors are expressed with the same unit vector the vector equations can be replaced by equations of the vector coordinates as shown in the lower left part of the table above This is the basis of the Modelica implementation shown in the right part of the table When two components are connected together conceptually the coordinate systems fixed to every com ponent are fixed to each other i e the unit vectors Z in the connected connectors are identical Since the unit vectors are identical it is allowed to operate on the coordinates of the vectors especially to sum up the cut torques The question arises why the absolute angle is used in the connector and not the
358. s constructed by exchanging the meaning of variables A subset of the in put vector Uin With dimension Min is treated no longer as known but as unknown and nin previously un known variables from the vectors x and or y are treated as known inputs The result is still a DAE which can be handled with the same methods as any other DAE Examples are given in the following sections This technique of constructing non linear inverse models has been first applied in Mugica and Cellier 1994 Otter and Cellier 1996 For linear systems a complete theory for inverse models can be given as sketched in section 19 1 For nonlinear systems such a complete theory does not exist and it is very unlikely that it will ever be developed For example one critical issue which seems unsolvable is the useful definition of stability that can be checked by a tool The current state in the theory of nonlinear control systems is e g summarized in Slo tine and Li 1991 Isidori 1995 Isidori 2006 The practical approach taken in this chapter is not well known outside of the Modelica community because the very powerful symbolic algorithms to transform singular DAEs as presented in section 12 4 seem to be not widely known There are several meaningful definitions of stability for non linear systems e g Ljapunov stability Bounded Input Bounded Output stability Input to State Stability Since it is not important for the de scribed approac
359. s record The data is not provided as parameter since Sa 5 6 External Functions partially available The Modelica language has a basic mechanism to call functions of other programming languages within a Modelica model or a Modelica function Details are standardized for C and FORTRAN 77 functions Dymola supports the calling of external C FORTRAN 77 and Java functions The details for call ing Java functions in Modelica are not discussed here see instead Chapter 8 3 2 in the document Dymola User Manual Volume 2 pdf in the installation directory of Dymola under Dymola Documentation 5 6 1 Calling C functions in Modelica not yet available 5 6 2 Calling C functions in Modelica not yet available 5 6 3 Calling FORTRAN 77 functions in Modelica Although FORTRAN 77 is a stone age programming language it is still a major language to implement nu merical algorithms A huge collection of useful and free FORTRAN 77 libraries is available from http www netlib org One goal is that these libraries can be utilized in Modelica Calling FORTRAN 77 functions directly in Modelica requires to have a FORTRAN 77 compiler that generates compatible code to the C compiler that is used by Dymola for translating Modelica models to object code If such a compiler is available FORTRAN 77 object libraries can be directly linked to the C code generated by Dymola The problem is that a portable m
360. s the same number of dimensions or if there is a special built in rule for the respective operation Exam ples with cast operators Real v 3 Vector with 3 elements Real M 3 1 Column matrix with 3 elements equation v M error since different number of dimensions v vector M fine since matrix M is transformed to a vector M matrix v fine since matrix v generates a column matrix M v fine since v generates a column matrix With the cast operator vector A a vector with all elements of A is generated provided there is at most one dimension size of A gt 1 For a scalar s vector s is a vector with s as element With the cast operator matrix A a scalar A is transformed to a matrix with 1 row and 1 column a vec tor A is transformed to a column matrix and an array with more than two dimensions is transformed to a ma trix using the first two dimensions all other dimensions must have size one When a scalar or vector is used inside the matrix constructor it is interpreted as a column matrix With the cast operator scalar A an array is transformed to a scalar provided it consists only of one el ement The matrix constructor can be used to construct a matrix from other scalars vectors matrices or ar rays Example Real v1 2 1l 2 The definitions on the left Real v2 3 4 5 6 side construct array M2 as Real M1 3 2
361. s to on s gt 0 u vl v2 u u_knee if on then 0 else s i0 if on then s else 0 R1 i0 v0 vil i2 v2 R2 il 10 12 der v2 i1 C The first 5 equations build a system of equations in the 5 unknowns on s u v1 and i0 The remaining as signment statements are used to compute the state derivative der v2 During continuous integration the Boolean variables i e on are fixed and the Boolean equations are not evaluated In this situation the first equation is not touched and the next 4 equations form a linear system of equations in the 4 unknown vari ables s u v1 and i0 which can be solved by Gaussian elimination An event occurs if one of the relations here s gt 0 changes its value At an event instant the first 5 equations are a mixed system of discrete and continuous equations which can not be solved by say Gaussian elimination since there are both Real and Boolean unknowns However appropriate algorithms can be constructed 3 Make an assumption about the values of the relations in the system of equations 4 Compute the discrete variables 5 Compute the continuous variables by Gaussian elimination discrete variables are fixed 6 Compute the relations based on the solution of 2 and 3 If the relation values agree with the assumptions in 1 the iteration is finished and the mixed set of equations is solved Otherwise new assumptions on the relations are necessary and the
362. s to be restarted to reliably handle the discontinuous change of the actuator signal which reduces again the simula tion efficiency For some design phases a continuous approximation of a controller might be sufficient e g when tun ing the controller parameters by parameter variation or multi criteria optimization By reducing the simula tion time with a factor of say 20 30 will then significantly reduce the optimization time It is also necessary to validate or fine tune the controllers with the most detailed models available Then effects such as sampling AD DA converter quantization resolver quantization and noise computing time of the control algorithms signal communication times as well as additional filters have to be taken into ac count Practical experience at DLR shows that it is difficult to maintain the consistency between a continuous and a digital representation of a control system model For this reason in a master thesis project Walther 2002 a Modelica library was developed that allows to easily switch between a continuous and a discrete rep resentation of a controller This library has been in use at DLR for some years Based on the gained experi ence in using this library as well as new features in Modelica and in the Modelica simulation environment Dymola Dynasim 2008 the library was considerably restructured and completely newly implemented Be sides Walter 2002 the books of Astr m and Wittenmark 1997 and
363. s3 Modelica Text eS text layer lol x Elle Edit Simulation Plot Animation Commands Window Help SARS Nm oy Am Hh Z 3X1 model MovingMass3 import SI Modelica SIunits parameter SI Mass m 2 mass of block parameter SI Force 6 force to pull block SI Position s position of block SI Velocity v velocity of block a equation v der s m der v f end MovingMass3 Line 1 Modeling y a Movingrtasss Bt proram diagram layer 5 x B Eile Edit Simulation Plot Animation Commands Window Help 18 x CECELIA FANA gfo z E H Modelica Reference a C Modelica l Unnamed MovingMess3 7 id Te Sle gt nw Modeling V Simulation The information about the diagram layer is stored in annotation Diagram Simulating this model with Dymola leads to the following results where the speed v grows linearly and the position s quadratically By default Dymola shows the variable name and its displayUnit or unit as label for the respective curve in the plot 22 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 Figure 2 5 Simulation results of model MovingMass3 s m v m s 4 2 0 0 0 0 5 1 0 2 2 Basic Models Connections specify interactions between components and are represented graphically as lines between con nectors A connector should contain all quantities needed to describe the interaction For exampl
364. se or a or b logical or not not a logical not Table 21 4 Other operators Operator Example Description Real A 2 2 Array dimension in declaration or A 1 2 3 array index or A 1 2 3 4 matrix constructor separates rows separates columns A 1 2 3 4 Array constructor every adds one dimension i 7 string value String literal is used inside a string for string value abc def Concatenation of string scalars or string arrays element wise Table 21 5 Operator precedence in expressions operators with higher precedence are evaluated first e g has higher precedence as Draft Jan 21 2009 Operator Example Description ee 2 3 5 Parenthesis arrayName arr index Postfix array index operator name name a b c Postfix access operator funcName function arguments sin 4 36 Postfix function call expr 2 3 Array construction concatenation expr 5 6 Draft Jan 21 2009 21 1 Elementary Operators Operator Example Description expr expr 2 3 7 8 A 2 3 Exponentiation Ke je afp aT 2435 273 Multiplicative and 1 2 3 4 2 375 6 array element wise multiplicative expr expr atb a b a a Additive and tg e 1 2 3 4 2 3 5 6 array element wise additive lt lt gt gt lt gt a lt b ax lt b a gt b Relat
365. sociation there is currently a proposal to introduce operator overloading into the Modelica language The data structures in the Modelica_LinearSystems library are organized so that their us age becomes convenient once operator overloading is available The basic approach will be explained by means of the record Math Complex Its content is displayed on the right side The record contains the basic definition of a complex number consisting of a real re and an imaginary im part record Complex Real re real part Real im imaginary part function definitions end Complex Additionally a set of functions is present that operates on this record Especially function construc tor constructs an instance of this record and all functions xx provide basic operations such as addi tion and multiplication Note the apostrophe is part of the function name In the operator overloading pro posal predefined function names are defined such as that will be automatically be invoked when a Com plex number is used in an arithmetic expression The Complex number record can be utilized in current Modelica environments such as Dymola Dynasim 2008 in the following way Draft Jan 21 2009 17 1 Basic Mathematical Data Structures 177 import Modelica Utilities Streams import Modelica_ LinearSystems Math Complex Complex cl Complex re 2 im 3 2 33 Complex c2 Complex 3 4 NS 3 2a algorithm
366. soeseoesoeesoeesoeesoessossoeesoeessooeeesseoo 75 VE ATRAVS o a A inate A A E E AT 75 Dd FOr LOOPS a ssncicseteliss cieiseheselasieei teener a R EEEE EEEE REEE EEA ENEE N EE E EE 83 5 3 Algorithm Sections aenn 25 ech na A EE R A A N ie 85 54 FUncton S scccesssseassccgunetowstagadeaseasvois seosngeaate cng dees santasivedounceedeaba divas sah Gonsdecduadaadeedsabedenstevasaguaueagoedinedde 88 DD ROCOTAS ts scdsenesnae cuvaeseceerscusredgeuedsceeniutielieoustd niaacnd se A ibestiasneisee E EEE 100 5 6 External Functions partially available cccccecsccesseesseeeesneeeeeeeeseeeeseeenseeeseeeseeeeeesseeneeeseeees 105 5 7 Modelica as Scripting Language not yet available cccccecsscesseccesseeeseeeeseeesseeeeeesseseeeeeneees 106 5 8 Component Arrays not yet available cccccecsscessseessecesneeseneeceeeesneeesseeeeeeeeesenaeeeseneenteeeeeees 106 5 9 Component Coupling by Fields not yet available ccccceescessseeesneceeneeeeeeceaeeesseeeeteeesseeeeseees 106 5 10 Replaceable Models not yet available cccccscesssecssceeseneeesteeceseeeseeesseeeseeeesenteeeeessesneeeeeeees 106 Part B Modeling of Discontinuous and Variable Structure System 107 Chapter 6 Discontinuous Kquations cccsccccscsssccsscssccsscsccsccsecsscccesssccsesssscsesssscsssssscesssccseeeeseees 108 6 1 Relation triggered events iarr ai eee E EEE EEE vas sends EE E caso saananessadeeddceds 109 6 2 Discrete equations and the
367. solution of the initialization problem ex ists 2 If there are several choices Dymola gives the highest priority for a selection if a start value is set for the corresponding x variable This is a good heuristic because modelers often forget to define fixed true after setting a start value If there are not enough start values where the fixed attribute is not set or is set with fixed false Dymola uses an arbitrary selection for the remaining subset of x and initializes these variables with a start value of zero 3 If there are more subset variables of x available that have a start value but not with fixed true priorities are used for the selection that depend on the model level where the start variables are set Start values set on a high level of a model have a higher priority over start values set on a lower level of a model The heuristics of Dymola works quite well if zero start values are good initial conditions for a subset of x This is for example the case for drive trains where the absolute angle and the absolute angular velocity of drive train elements are used as variables x The reason is that zero angles and zero speed for all shafts of a drive train are meaningful initial conditions the drive train is fully at rest This works also quite well for electrical circuits because voltage drops or currents flowing through elec trical components are usually utili
368. ssel filter should be selected if a filter is mainly used to make a non linear inverse model realizable 17 2 4 Record DiscreteStateSpace This record defines a linear time invariant difference equation system in state space form At the time of writ ing this record contains only the core function fromStateSpace to transform a StateSpace descrip tion in a DiscreteStateSpace form The details of this record and of this function are discussed in section 17 3 Continuous Discrete Control Blocks The core of the Modelica_LinearSystems library is sublibrary Sampled to model continuous and discrete multi rate control systems Experience shows that the combination of physical plant models that are con trolled by digital controllers slow down the simulation speed significantly if the sampling rates of the digital controllers are small compared to the step size that could be used for the continuous plant For example at DLR detailed calibrated models of robot systems are available A stiff solver with variable step size inte grates this system with step sizes in the order of 10 20 ms When replacing the continuous approximation of the controllers by the actual digital controller implementation the simulation time is increased by a factor of 20 30 The reason is that the digital controllers have a sample time in the order of 1 ms and therefore the step size of the integrator is limited by 1 ms Additionally at every sample point the integrator ha
369. ssion so that the when equation is only active when the condition changes from false to true and in all other cases the value of the discrete variable from the previous event is used Furthermore it can be seen that the when equations are not active during initialization but become only active when the simulation starts After the mapping equations are present that are handled in the same way as the continuous equations The only new issue is that the set of all equations are solved under the assumption that all pre variables besides the continuous states are known As usual a first step during the solution procedure is to sort the equations To distinguish this special type of when mapped equations from the continuous equations they are called instantaneous equations because the then branch is only active at an event instant Note the above mapping can also be performed manually in a Modelica model for Boolean Inte ger String and enumeration variables This means that when clauses are not really necessary for these variable types although the model will be easier to read and to understand if a when clause is used On the other hand the above mapping cannot be performed manually in a Modelica model for Real variables The reason is that the pre operator can only be applied on Real variables that are assigned within a when clause As previously explained this restriction is present to prevent con
370. ssscssssssssscsssscssssssssscssssssssessssscssssscsssssesessenoees 169 13 1 Integration Algorithms nisnin E ER EE AR EEs 169 13 2 Method Order not yet available ccccccessccsssecssceeeseecesneeseeeceseeeeseeeeseeseeeeseeesseeessaeesesaeeeeees 169 13 3 Stability Region not yet available ccccccsscessseeseeessnsceeeeseeeeseeesseeeesseceseeesseeeseaeesseeseaeees 169 13 4 Inline Integration not yet available ccccccesceessseesseeeeseeesseeeeseeesseeceeaeeceseeseseeeneeesseeessteeeeess 169 13 5 Event Handling and Chattering not yet available ccccccccssseeesseeeneeeeneeeeneeeeseeeeeeeeeteeeeeesees 169 13 6 Practical Considerations not yet available cccccsccecsseceseceseeeeeeeeeeeeseeseseeeeseeesteeeetteeeeeeees 169 Part D Libraries for Physical Systems ccccssscssssscssssssssssscsccscecesesscsseel JL Chapter 14 Drive Trains Rotational library sssccsssssssscseecsscceecsscccessssccesssccsesssscscsssssceeessees 172 Chapter 15 Mechanical Systems MultiBody library sssssssccsssssssssccsccsssssssceeeesesssssesssecseees 173 Chapter 16 Thermo Fluid Systems Media Fluid libraries scsssccssssssssscsssscsssecsssssceecesssssees 174 Part E Libraries for Control System cccccccscsssscccsssscssssssssssssccscsssccsscoeeeel 15 Chapter 17 Continuous and Digital Control Systems LinearSystems library csssssssssoeee 176 17 1 Basic
371. sssees 42 Ball Connector Des ii zresiss csesescvcssaagaes cdivcevsa a R E E RS TEE Eaa 42 3 2 Connectors for Drive Trains Rotational library cccceccceescceeseceeseeeeseeeeeeesseeenseeeessnneeeeenseeaaes 46 3 3 Connectors for Heat Transfer HeatTransfer library ccccecsscssceeesceeeseeceseeeeeneeessesteeeeessennaes 48 3 4 Connectors for Block Diagrams Blocks library cccccescccessceesseeeneeeeeeeeseeceeeesseeesseeenseessenaes 51 3 5 Connectors for Signal BUSES iiis sccsedetanivieecadsdeaes aN nba beta AE EE NE EN E E N 53 3 6 Hierarchical ConmectOrs ssri iiras arranoa se eea EEA N KEENE NEEE AEAEE RON EEE ERES 57 Chapter 4 mitialization cccccsssccsssccssscsssecsssccsssscscssssssssscssscssssesssscsssscssssssescsesssssssssessssseesessccesea 61 4 1 Mathematically defining the initialization problem ecceeseeseeeceeeeeeeeeeeeeeeeeeeeeceeeeeteneeeeseaeeees 61 4 2 Initialization with attributes eniron i naei E caseadeodcceeshdasventedveaseetessesesthaacesens 62 4 3 Initialization with initial equations ccccccesscesseccsseessneeeeeeceeeeeneeesseeeeeeenseeeeeeseeneeeeceseneeeeenes 65 4 4 Too many or not enough initial CONdITIONS ecceeseceteeeseeesteeeeseeceeeceseeessceeneseeseaeeeseeeeeeeeeees 67 ASS initialization fails as ccecvsedescenccsneestesdedccudeneds ste adhe e EE aE EEE AE EEEE IE E Eiai 70 Chapter 5 Advanced Modelica Language Constructs e seesseeseeesoeesoes
372. ssumed to be known during equation sorting they can be changed at an event instant in the way described above and the DAE is still fulfilled For other variables this property does not hold and therefore only this restricted version of the reinit operator is supported in Modelica From a user point of view this approach is problematic because it is not known in advance whether a 699 potential state variable x is used as real state from the simulation system However the 144 Chapter 11 Re initialization of Continuous Variables Draft Jan 21 2009 reinit operator can only be applied on the real states that are selected from the modeling system during code generation Therefore the only way how this can work is that the Modelica translator tries to use variables x appearing in a reinit clause as real states So this is equivalent to set attribute stateSelect StateSelect always on variable x If this is not possible a translation error will occur The reinit operator is a very dangerous construct and a modeler can easily construct models that are physically not correct but the Modelica translator cannot detect this error Therefore this operator should be removed from the Modelica language once a better alternative is available The usage of the reinit operator is demonstrated in the next example that describes a ball a mass point that is vertically falling dow
373. ssumption that the input u 1 To get a feeling for the situation let us compute the steady state value of the 4th order transfer function a a a as 1 I ay ay ay ay a as 0 x 0 0 o xX o u gt x 0 5 4 0 1 0 0 0 0 0 0 1 0 0 0 The steady state value is computed by setting the derivatives to zero The result implies that we should use a state transformation 1 _ CE i x 5 5 leading to the slightly changed state space form 43 4g Gs as a a a a a x 1 1 0 0 oO o 0 1 0 0 0 5 6 0 0 1 0 0 1 b b Te b y b a b a b a b a lt x y a ay a a as a Setting again the state derivatives to zero results in a steady state value of x u Changing the Modelica 82 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 model above according to this scaling leads to a model where the discussed problem is again resolved and the simulation result is correct block TransferFunctionWithScaling extends Modelica Blocks Interfaces SISO parameter Real b 1 Numerator coefficients parameter Real a 1 Denominator coefficients output Real x size a 1 1 State of transfer function protected parameter Integer na size a 1 parameter Integer nb size b 1 parameter Integer nx na l parameter Real x scale if abs alend gt 1000 Modelica Constants eps Modelica Math Vectors length a then
374. strated by means of the following non minimum phase system stable zeros unstable zeros ___ sti s 2 s s 3 s 4 s 5 stable poles ll Pa nA uo lt oe ENS x lt unstable poles 19 1 1 Open loop controllers for stable minimum phase plants Our first goal is to design an open loop controller F s for a causal stable and minimum phase plant P s This can be achieved with the simple controller structure of Figure 19 1 r Al u y K yY X gt F R P F P gt gt R gt R arbitrary with e R stable and e Dr2 Dp Figure 19 1 Open loop controller F for a stable minimum phase plant P where r is the reference signal i e the desired value for output y The open loop controller F is connected in series to plant P and is designed by inverting the plant and multiplying it with an arbitrary transfer function R which has the only restriction that controller F must be stable and causal in order to be realizable and in or der that the overall system is stable Since P is minimum phase by assumption plant P has only stable zeros and therefore P p l D D 19 4 that is the inverse plant model is stable and its relative degree is Dp In order that F is stable and causal and since Np dp re 19 5 R must be stable and Dr Drt Dp Dp Dp 0 Therefore the relative degree of R must be identical or larger as the relative degree of pl
375. strategy the function above could be implemented in the fol lowing way function numberOfReals Return the number of real numbers input Real c 2 Complex numbers col 1 real col 2 imag output Integer nr 0 Number of Reals in c algorithm for i in 1 size c 1 loop if c i 2 0 0 then nr lt nre 1 end if end for end numberOfReals function realNumbers2 Return real numbers of a complex vector input Real c 2 Complex numbers col 1 real col 2 imag output Real r numberOfReals c The real numbers of c protected Integer j 1 algorithm for i in 1 size c 1 loop if c i 2 0 0 then Ei e eli lly j ely end if end for end realNumbers2 The difference to the previous implementation is that the dimension of vector r is computed with the func tion call numberOfReals c and the returned result is directly used to declare the dimension of r Functions realNumbers1 and realNumbers2 can be called directly or indirectly in the inter active environment of Dymola without any problems Calling these functions directly in a model is not possible because the dimensions of any array in a mod el must be known during translation in order that symbolic processing is possible and the two functions re turn arrays where the dimensions are only known after the actual values of the input vector c is known Calling these functions indirectly in a model should be
376. structions of the form Wrong Modelica model v noEvent if u gt 1 then u else pre v 2 because the value of pre v is not well defined if no event is generated when v changes its value dis continuously 8 This is a conceptual mapping rule because it is a quality of implementation whether a Modelica environment implements exactly this mapping rule or whether it uses a different implementation scheme that results in the same semantics Draft Jan 21 2009 7 2 Sampled data systems and initialization 117 7 2 Sampled data systems and initialization The implementation of sampled data systems is basically straightforward with the currently discussed Mod elica language elements For example the simple first order discrete system y ti a y ti T b u ti ti 0 T 2 T can be implemented in the following way block DiscreteFirstOrderBlockVersionl Modelica Blocks Interfaces RealInput u Modelica Blocks Interfaces RealOutput y parameter Modelica SIunits Time T Sample time parameter Real a b equation when sample 0 T then y a pre y b u end when end DiscreteFirstOrderBlockVersionl Due to the sample operator time events are generated at every sampling point i T The input signal u can be a continuous or discrete variable In any case the value of u at a sampling instant is utilized as dis crete input to the discrete system The output s
377. structure of Figure 19 6 Therefore this is the most gener al linear controller structure for single input single output systems There are other controller structures with a feedforward path However they are not necessarily as general as the controller above For example in a first step a feedback controller could be used to stabilize the plant and then an open loop controller could be designed according to Figure 19 2 for the stabilized plant system to provide a desired reference transfer function For minimum phase plants this would result in 1 P C EPE Dy gt Dpt De 19 10 and therefore the class of realizable reference transfer functions T F is both smaller and depends on the controller C whereas the reference transfer functions 7 R in Figure 19 4 do not depend on C For exam ple if C is strictly proper i e Dc gt 0 the class of realizable reference transfer functions is obviously small er as in the controller structure in Figure 19 4 since Dr gt Dp instead of Dr gt Dp It is unlikely that the controller structure from Figure 19 6 can be generalized to nonlinear plant models because this would require that a nonlinear model can be split up in a part that has a stable inverse and the remaining part This might be possible for special plant models but not in general For this reason the generalizations to nonlinear systems as presented in the following sections are based on Figure 19 4 Draft Jan
378. stur bances measurement noise and imprecise plant model and can be optimized for these tasks without taking into account the desired reference transfer function 19 1 5 Controller with two structural degrees of freedom for non minimum phase plants The above approach can also be generalized to any type of SISO plant as shown in Figure 19 6 Figure 19 6 Controller F R C with two structural degrees of freedom R C for a plant P P might be unstable and or non minimum phase ee y ITyrt Tyn t Tyd A CSR n Rr TC n d P 2 1 PC 1 PC r u f l nt R e C Desired R C such that 7 T 1 T 0 T 0 Restriction R stable and D D deg n With the same argument as before but by using the more general open loop controller from Figure 19 2 in the feedforward path and the desired reference transfer function as pre filter it can be easily seen that again the control error e is identical to zero when disturbance d and measurement noise n are zero and the exact plant model is used in the controller Therefore controller C is again only used to act against disturbances measurement noise and imprecise plant model and can be optimized for these tasks without taking into ac count the desired reference transfer function In Kreiselmeier 1999 it is shown that every causal controller with inputs r y and output u that stabi lizes the plant can be represented by the controller
379. switching to forward mode note that even for exact calculation without numerical errors a waiting phase is necessary because v 0 when sliding starts Since der v gt 0 this will occur af ter a small time period This waiting procedure is most easily described by the state machine of the next figure Figure Switching structure of friction element Collecting all the pieces together finally results in the following equations of a simple friction element Draft Jan 21 2009 10 2 Coulomb Friction not yet complete 141 part of mixed system of equations StartFor pre mode Stuck and sa gt 1 startBack pre mode Stuck and sa lt 1 a der v For then sa 1 else Back then sa 1 else 0 For then f0 f1 v else Back then f0 f1 v else f0 sa 3 if pre mode Forward or star if pre mode Backward or star a f if pre mode Forward or star if pre mode Backward or star GCheGh Et ict state machine to determine configuration mode if pre mode Forward or startFor and v gt 0 then Forward else if pre mode Backward or startBack and v lt 0 then Backward else Stuck Note that the equations within the mixed system are evaluated based on the value of mode when the event occurred i e on pre mode After the new sliding or stuck mode is determined by the solution of a mixed set of continuous discrete
380. t entry If a used package is referenced in a model and it is not yet loaded in to Dymola then it is automatically loaded provided Dymola can find it in its library directory in the current directory or in one of the directo ries defined in the MODELICAPATH environment variable details see page 34 A package can be manual ly loaded in the following ways e File Open opens a dialog box to read a file If the complete package is stored in one file then this file name has to be selected If a package is stored in several files in a directory then file package mo of the root level package has to be selected For example the Modelica_Fluid library is loaded manual ly by selecting file Dymola Modelica Library Modelica_Fluid package mo in the File Open di alog box e When command import lt package name gt is given in the command window in the simulation tab then package lt package name gt is loaded if it is in the search path Dymola library directory cur rent directory and MODELICAPATH For example if the command import Modelica Fluid is given then package Modelica_Fluid is loaded in to Dymola and is shown in the package brows er e File Libraries allows to select a package from a list of libraries The selected package is loaded af ter searching it in the search path In Figure 2 17 the packages are shown that are by default available in this menu Figure 2 17
381. t because the zero dimensions effectively remove the equation Example Real A 3 0 B 3 0 C 0 0 Arrays have zero dimension parameter Real D 111 0 0 0 0 D is empty by default equation A B 2 A fine the equation is not present A C 2 A error arrays with different dimensions are added It is not allowed to access elements of an empty array because an empty array does not have elements Real x 0 empty vector equation x 0 x 0 2 error since an element x 0 does not exist Empty arrays are very useful to handle limiting cases without extra effort For example the connector of a Fluid library might be designed for single and multiple substance fluids connector FluidPort connector for single multiple substance fluids import SI Modelica SIunits parameter Integer ns number of substances flow SI MassFlowRate m flow mass flow rate SI AbsolutePressure P pressure stream SI SpecificEnthalpy h outflow stream Real Xi_outflow ns 1 Real 0 if ns 1 end FluidPort For single substance fluids ns 1 Therefore array Xi_outflow has dimension zero and is actually not present For multiple substance fluids the array is present The connector is for example used to model a fluid volume model Volume import SI Modelica SIunits FluidPort port SI Mass ms ns 1 substance masses Real 0 if ns 1 equation der ms port m_flo
382. t Modelica is very well suited for this approach be cause Modelica is designed to model complex systems and since Modelica tools like Dymola can generate nonlinear inverse models automatically A continuous Modelica model is primarily mapped to a DAE set of Differential Algebraic Equations of the form 0 f x t x t y t u t t tEIR xEIR yelR uelR f eR 19 11 where x t are variables that appear differentiated in the model y t are algebraic variables and u t are known inputs as function of time It is possible to transform system 19 11 to the following state space form at least numerically as sketched in section 12 4 f x u 19 12 where x and x form vector x such that the subset vector x is the state vector and contains the independent variables of x The new vector w contains higher order derivatives of X and of y that appear when differenti ating equations of f and that are treated as algebraic variables For the computation of f it might be necessary to solve linear and or non linear algebraic systems of equations The equations to be differentiated can be determined with the algorithm of Pantelides Pantelides 1988 The selection of the state variables x can be performed with the dummy derivative method of Mattsson and S derlind Mattson and S derlind 1991 Both algorithms are for example available in the Modelica simulation environment Dymola An inverse model of the DAE i
383. t example model InconsistentRedundantEquations parameter Boolean steadyStateInit false parameter Real all 2 a12 3 parameter Real a21 2 a22 3 parameter Real bl 1 parameter Real b2 3 Real x1 Real x2 initial equation if steadyStateInit then der x1 0 der x2 0 else xl 1 x2 2 end if equation der x1 al1 x1l al2 x2 bl Draft Jan 21 2009 4 5 Initialization fails 73 der x2 a21 xl a22 x2 b2 end InconsistentRedundantEquations This model can be simulated without problems if steadyStateInit false If steadyS tateInit true the model shall be initialized in steady state but fails during initialization with the er ror message Residual component 1 of system 1 is 1 000000e 000 Residual component 2 of system 1 is 1 000000e 000 LINEAR SYSTEM OF EQUATIONS IS SINGULAR AT TIME 0 Linear system of equations number 1 Variables which cannot be uniquely computed x1 0 307692 NOT ACCEPTING SINCE TOO LARGE RESDIUAL The message indicates that a linear system of equations is solved but this system does not have a solution The listed variables which give rise to the problem here x1 may already indicate the source of the prob lem If this is not the case one should have a look at the equations This is performed by selecting in the sim ulation tab Simulation Setup Translation Generate listing of tr
384. t is also possible to introduce sub buses especially to structure large sets of signals For sub buses the slightly different icon Modelica Icons SignalSubBus should be used An example from the Vehi cleInterfaces library is shown in Figure 3 13 Figure 3 13 Sub buses used in the VehicleInterfaces library left diagram layer right plot window E acontrolBus E sbrakesBus 3 obrakesControlBus schassisBus edriverBus sengineBus no SengineControlBus a transmissionBus lengineStart _ GengineStop E _ OfuelFlowOn ei throttle transmissionControlBus E transmissionBus _ CoutputSpeed _atransmissionControlBus _ OCeurrentGear lt OlockUpClutchDemand CtorqueDownRequest A sub bus is also an expandable connector When connecting two expandable connectors together the above menu also pops up If the transmissionBus shall be a sub bus inside the controlBus then in the menu the transmissionBus has to be defined in the controlBus input field of lt Add variable gt leading to the statement connect controlBus transmissionBus transmissionBus and this statement introduces the sub bus controlBus transmissionBus In Figure 3 13 above on the left side the two sub buses transmissionBus and transmissionControlBus are shown that are connected to the controlBus in the diagram layer When translating the model and selecting the co
385. t mixingUnit T PlantAith20PercentLargerParameters mixingUnit T 340 320 K 100 200 300 400 500 Oo Plant vith20PercentLargerParameters mixingUnit T_c NominalPlant mixingUnit T_c SS SE 100 200 300 400 500 320 K 280 19 8 Difficulties with Inverse Models When constructing inverse models for industrial systems it is often the case that the generated inverse mod els do not work as expected In this section the major reasons are discussed and it is explained how to cir cumvent such problems 220 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 19 8 1 Unstable inverse models Usually it is required that the inverse model is a stable system For example in the structure of Figure 19 9 the inverse model is in the feedforward path and if it would not be stable the overall system would be unsta ble as well For linear single input single output systems this situation is well known and can be easily ana lyzed For example take the following linear plant model s 1 a 2543 19 33 The inverse model together with a reference model is _ s 2 s 3 As can be seen the inverse model is unstable because the plant has an unstable zero In other words for lin ear systems the plant must be a minimum phase system in order that the inverse model is stable For a gener al DAE no stability proof exists Therefore many simulations have to be performed wi
386. t of regular DAEs The blocksets use domain specific algorithms to transform singular DAEs such as a mechanical system to state space form As a result reliable solutions of certain classes of singular DAEs with discontinuous variable changes can be expected It is however not possible to handle generic singular DAEs as e g occurring when building inverse systems see Chapter 19 e In the electronic field modeling is performed with HDLs Hardware Description Languages such as MAST www openmast org or VHDL AMS www eda org vhdl ams HDL simulators have usually numerical integration methods without symbolic or domain specific preprocessing to transform a singular DAE to a regular DAE and it is therefore questionable whether they can solve reliably singular DAEs with discontinuous variable changes This is usually not critical in the electronic field since singular systems 16t is of course possible to model mechanical and or fluid systems in another modeling environment e g in Modelica perform the transformation to a regular hybrid DAE in this other environment and use the result in a HDL simulator or 168 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 do not often occur here alternatively use co simulation Draft Jan 21 2009 Chapter 13 Integration Algorithms 169 Chapter 13 Integration Algorithms 13 1 13 2 13 3 13 4 13 5 13 6 Integration Algorithms Method Order not
387. tate that the electrical potentials at the three pins are identical and that the sum of the electrical currents is zero These are exactly Kirchhoff s voltage and current laws and therefore describe cor rectly this electrical circuit In section 3 1 on page 41 it will be explained in more detail how and why the two Modelica connection rules are able to describe physical component coupling phenomena In a similar way also other basic components can be constructed For example a rotational inertia is de scribed by the following model model Inertia 1l dim rotational inertia import SI Modelica SIunits import Modelica Mechanics Rotational Interfaces Interfaces Flange a flange a Interfaces Flange b flange b a e parameter SI Inertia J e SI AngularVelocity w speed of inertia equation flange a phi flange b phi w ae ae J 0 001 J der w flange a tau flange b tau end Inertia It consists of a shaft with two flanges flange a and flange b Since the two flanges are rigidly con nected the angles flange _a phi and flange _b phi are identical Furthermore the momentum balance along the axis of rotation states that the time derivative of the momentum i e d J w dt J W is propor tional to the sum of the cut torques 2 3 Inheritance It is often the case that different models share the same code fragment It is then convenient and less error prone to define these common code f
388. te m flow input Modelica SIunits SpecificEnthalpy h inflow output Modelica SIunits SpecificEnthalpy h_outflow end FluidPortA connector FluidPortB correct Modelica 3 0 connector Modelica SIunits Pressure p flow Modelica SIunits MassFlowRate m_ flow output Modelica SIunits SpecificEnthalpy h_inflow input Modelica SIunits SpecificEnthalpy h outflow end FluidPortB The two connectors are correct because both have one flow m_ flow and one non flow p variable and the remaining variables have an input or an output prefix Whenever a connector has variables with the input output prefix restrictions are present how connectors can be connected together because con nected components must follow the block diagram semantics e g two input variables cannot be connected together For the two connectors above this means that two instances of FluidPorta or two instances of FluidPortB cannot be connected together However an instance of FluidPortA can be connected to gether with an instance of FluidPortB FluidPortA Al A2 FluidPortB Bl B2 equation connect Al A2 error since two inputs are connected connect Bl B2 error since two inputs are connected connect Al Bl fine Based on the restrictions of connectors with rule 1 it is now possible to precisely define how many equa tions a component must have in order to be
389. ted as shown in the next figure Figure 17 2 Bode plot of two transfer functions generated with Modelica_LinearSystems TransferFunction Examples bodePlot2 Bode diagram tf1 tf2 1 o 3 01 c oa G 0 01 0 01 0 1 1 10 Frequency in Hz 50 Phase in deg dn O 100 150 0 01 0 1 1 10 Frequency in Hz The function call plotBode tfl legend tf1 automatically selects an appropriate frequency interval and uses the selected legend The useful frequency range is estimated such that the phase angle of the plot of one numerator or denominator zero is in the range p min lt phase angle lt Pun 17 4 where n is the number of numerator or denominator zeros Note the phase angle of one zero for a frequen cy of 0 up to infinity is in the range 180 Chapter 17 Continuous and Digital Control Systems LinearSystems library Draft Jan 21 2009 0 lt phase angle lt mn 17 5 Therefore the frequency range is estimated such that the essential part of the phase angle defined by min is present in the Bode plot default is 10 rad 17 2 3 Record ZerosAndPoles This record defines the transfer function between the input signal u and the output signal y by its zeros poles and a gain y k u 17 6 IT s p where the zeros and poles are defined by two Complex vectors of coefficients z and pj The elements of the two Complex vectors must either be real numbers or conjugate
390. ter 8 The Synchronous Principle 129 Table 8 2 Definitions needed to define the Modelica execution semantic discrete A discrete equation is either of the following two forms equation 1 An equation of the form v expr ora statement of the form v expr where v is of type Boolean Integer String or enumeration If v isa record or array reference it is conceptually mapped to a set of scalar variables equations 2 An equation or a statement in a when clause that is conceptually mapped to an equation or statement as defined in section 7 1 This type of discrete equation is also called instantaneous equation differential An equation or a statement where the der operator is applied at least on one equation variable reference algebraic An algebraic equation is an equation or a statement that is neither a differential equa equation tion nor a discrete equation discrete A variable of type Integer Boolean String enumeration or a variable of variable type Real that appears on the left hand side of an equal sign in a when clause continuous A subset of the variables on which the der operator is applied or that are de state fined with the StateSelect prefer or StateSelect always attribute The selection of the proper subset is the task of the Modelica translator pre v Value of variable v before the current event instant If pre v is used outside
391. th the inverse DAE to check whether it is stable in the desired operation region For certain classes of DAEs it might be possible to proof that the inverse model is stable An alternative is to linearize the plant model around several stationary operating points and check whether the transmission zeros are stable Of course none of these checks can guarantee that the inverse DAE is stable for simulations or stationary points that have not been analyzed If the inverse plant is unstable only approximate inverse plant models can be used for the design For linear single input single output systems this can be achieved by removing unstable zeros before inverting the plant E g in the example above the approximate inverse plant model would be aaa s 2 s 3 ae Tstiy gt Ce For a non linear plant one might choose other outputs of the plant as inputs to the inverse model since this might change the stability behavior of the inverse plant see for example Snell 2002 Alternatively the plant might be modified before inversion These advices are demonstrated by the crane example in Figure 19 21 Figure 19 21 Crane consisting of a horizontal moving crab and a load on a rope The crane consists of a horizontally moving crab and a rope on which the load is attached For simplicity the load is modeled as a mass point The crab is driven by the external force f The horizontal position of the ce 29 66 39 crab s
392. the declarations of the input variables of the function Therefore also user de fined functions can be conveniently called in this way 5 4 3 Modelica built in functions and functions from the Modelica Standard Library Modelica has a large set of built in functions and of built in operators with function syntax A summary of these functions is provided in the appendix on page 227 and in package ModelicaReference Opera tors Built in functions and operators are provided if they cannot be expressed as standard Modelica func tions e g because the number of input arguments is varying and or the input output arguments are of vary ing types and or the operator communicates with the simulation engine For example the built in function String transforms variables to a string representation The first input argument can be of type Real Integer or Boolean With a standard Modelica function this cannot be defined and therefore String has to be a built in function Although built in functions cannot be defined with Modelica the calling mechanism of such functions is identical to calling Modelica functions There are some built in func tions such as sin or cross that could be defined as Modelica function but are provided as built in function mostly for historical reasons Additionally about 550 Modelica functions are provided in the Modelica Standard Library A short overview is given in the f
393. the dummy derivative of v i e dder v is treated as an algebraic variable All the equations are integrated from t0 to t0 in order to get rid of the Dirac impulse This results in equations for the new initial values of all variables For the integration the following standard formula is used tte J v t 6 t dt v t 11 4 ty which holds under the assumption that v t is continuous in the integration interval The final result is input xl x2 vl pre v_rel v_rel dummy lte pre v_rel v1 pre vl m2 v rel dummy ml m2 v_ rel e pre v_rel v2 a v_rel vil x1 pre xl x2 pre x2 These are the correct equations for the impact of two mass points 149 Part C Simulation The following chapters provide an overview about the symbolic and numeric algorithms to perform a simula tion of a Modelica model as well as what to do when a simulation fails Algorithms are sketched in order that a Modelica modeler understands how a Modelica translator and simulator operates This knowledge is useful to understand error messages and to resolve problems with a Modelica translator or when a simulation fails It is not the intention to explain the algorithms in such detail that they can be used for an implementa tion in a Modelica tool 150 Chapter 12 Symbolic Transformation Algorithms Draft Jan 21 2009 Chapter 12 Symbolic Transformation Algorithms The Modelica Language Spec
394. the equation can be directly assigned because at least one of them is not yet assigned If this is possible the function returns successfully Otherwise it is tried systematically to perform a reassignment of the unknown variables For this process the Visited array is used to mark which variable was already tried to be reassigned in order to not repeat the same computations Then the assign function is called recursively to perform a reassignment If an assignment is not possible after trying all possible combi nations of reassignments the system of equations is structurally singular The worst case complexity of the algorithm is O n m where n are the number of equations and m is the sum of the variables appearing in all equations In the aie above n 7 m 15 This means that the number of operations to compute the complete assignment is at most k n m with a suitable con stant k The worst case complexity usually only appears for specially constructed examples It turns out that in most practical cases for Modelica models the complexity is about O n i e the number of operations grows linearly with the number of equations 12 2 2 Step 2 Loops of a directed graph algorithm of Tarjan With the result of the previous phase a directed graph can be constructed containing the functions with their assigned variables as nodes and the dependency from the other unknowns described by appropr
395. the variable units are compatible with the usage of the variables in equations For example if variables are used in an equation of the form m a then the units on the left hand side m a kg m s 2 must be identical to the units on the right hand side f N gt kg m s 2 when representing all units with the 7 SI base units kg m s A K mol cd If the equation would be written in the form m v f a warning message reports that the units on the left and right hand side are not compatible to each other and that either the equation or the unit definitions of the variables are erroneous If units are not defined Dymola deduces the units from the context and propagates this information For 20 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 example if a connect equation defines the equation a b and a has a unit whereas b does not have a unit defined then Dymola assumes for the dimension analysis that bp has the unit of a unit propagation The sames holds in expressions For example c a b and a has unit kg whereas c and b do not have a unit defined then c and b are assumed to have also unit kg In order to avoid that every modeler defines different sets of derived types in the Modelica Standard Library about 450 units from ISO 31 1992 General principles concerning quantities units and symbols and ISO 1000 1992
396. thering the user to provide nominal values Unfortunately it is always possible to construct a model where the default behavior of a li brary component will fail due to inappropriate nominal values since the library has not enough information about the simulation problem to be solved However it is possible to enlarge the number of models that re sult in a reliable simulation Basically one has to utilize information from the numerator and or denominator polynomial coefficients to set appropriate nominal values e g in the form output Real x size a 1 1 each nominal 1 a end The each prefix is explained in section 5 8 It defines that the provided value is used for all array ele ments In principal the above declaration would be a good solution Unfortunately Dymola has currently the significant restriction that the parameter vector a cannot be modified after translation any more if an ele ment of a is used to define an attribute such as nominal min or max This means that the model has to be retranslated whenever an element of a shall be changed Due to this drawback another approach is used which is based on the scaling of the state vector so that the critical state is usually in the order of 1 and no longer in the order of le 4 as in the example above For this the heuristics is used that the magnitude of x is in the order of its steady state value under the a
397. this results in the following equation inertial flange b phi angleSensor flange phi This formulation is unproblematic Especially since both angles are identical also all higher derivatives are identical as well especially the shaft speeds of the two components Alternatively the angular velocity shall be in the connector instead of the angle Variant 2 Angular Velocity in Flange Connector model Inertia model AngleSensor parameter Modelica SIunits Inertia J Flange a flange Flange a flange a RealOutput phi Flange b flange _b equation Modelica SIunits Angle phi der phi flange w Modelica SIunits AngularVelocity w 0 flange tau equation end AngleSensor w flange a w f wW flange b w w der phi f J der w lange a tau flange b tau end Inertia The main difference to the previous formulation is that the AngleSensor contains now a differential equation in order to compute the angle Especially this means that initial conditions for this angle can be provided When the inertia and the angleSensor components are connected together this results in the following equation inertial flange b w angleSensor flange w This means that the angular velocities of the two components are identical However this does not mean that the angles of the two components are identical Especially if the initial conditions given for angleSen sor phi and for inertia phi are different the angles ar
398. tic It is straightforward to model the detailed diode curve because the current i has just to be given as ana lytic or tabulated function of the voltage drop u It is more difficult to model the ideal diode curve in the right part of Figure 10 1 because the current i at u uknee is no longer a function of u i e a mathematical description in the form i i u is no longer possible This problem can be solved by introducing a curve pa rameter s and describing the curve as i i s and u u s This description form is more general and allows defining an ideal diode uniquely in a declarative way u equation OS il i2 oT u vl v2 V1 V3 on s gt 0 r u uKnee if on then 0 else s il if on then s else 0 In order to understand the consequences of parameterized curve descriptions the ideal diode is used in the simple rectifier circuit of Figure 10 2 Draft Jan 21 2009 10 1 Parameterized Curve Descriptions 137 Figure 10 2 Simple electrical circuit to demonstrate the handling of an ideal diode model vO resistor1 ww diode v2 ca soyoedes _ 4 Z10 51S314 lt _ _ _ 5 ground Collecting the equations of all components and connections as well as sorting and simplifying the set of equations under the assumption that the input voltage vO t of the voltage source is a known time function and that the states here v2 are assumed to be known and the derivatives should be computed lead
399. tically detecting the interesting range X45 14 x44 46 x 3 56 x42 200 x 1000 0 tt Lt a 1000 l f 0 5 eles j 17 2 Linear System Descriptions At the top level of the Modelica_LinearSystems library data structures are provided as Modelica records defining different representations of linear time invariant differential and difference equation systems In the record definitions functions are provided that operate on the corresponding data structure Currently the fol lowing linear system representations are available 17 2 1 Record StateSpace This record defines a multi input multi output linear time invariant differential equation system in state space form x t A x t B u t y t C x t D u z 17 1 The data part of the record contains the constant matrices A B C D with the following definition record StateSpace Real A Real B size A 1 Real C size A 1 Real D size C 1 size B 2 function definitions end StateSpace The content of the StateSpace record definition is displayed on the right side The fromXXX functions trans form from another representation to a StateSpace record Especially fromModel linearizes a Modelcia model and provides the linear system as output argument fromFile reads a StateSpace description from file and fromTransferFunction transform a transfer function description into a StateSpace form The basic ar
400. tically placed by Dymola to graphically visualize that the three connected lines form one connection set Note this does not mean that it is possible to connect to this node Instead a connection line must always start at one con nector and end at another connector The graphically drawn connection lines are interpreted as the following Modelica equations they are shown in the Text layer of the model equation connect inductorl pin_n inductor2 pin p connect inductorl pin_n inductor3 pin_n The connect construct is interpreted as a built in operator that introduces equations and has therefore to be present in the equation section of a model When processing a model a Modelica translator re places a set of connect statements with a set of equations by the following rules 1 Corresponding connector variables in a connect statement are set equal if they do not have the flow prefix 2 Variables that have the flow prefix and belong to the same connection set are combined together in one equation stating that the sum of these flow variables is zero Applying these rules to the above two connect statements above results in the following equations which are therefore completely equivalent to these statements inductorl pin_ n v inductor2 pin p v inductorl pin_n v inductor3 pin n v 0 inductorl pin_n i inductor2 pin_p i inductor3 pin_n i The three equations s
401. ties Therefore an integrator has to perform very small step sizes near a discontinuity in order that it is roughly approximated by polynomials As a result good integrators with variable step size are usually very slow near a discontinuity provided the discontinuity has a significant effect on the simulation result It is well known how to handle discontinuous systems in a numerical reliable and efficient way by using the following strategy Cellier 1979 1 Detect the time instant of the discontinuous change e g by determining the time instant when an indica tor signal crosses zero 2 Hold the integration at this time instant 3 Perform the discontinuous change 4 Restart the integration As a result of this approach the signals during the numerical integration are always continuous and differen tiable and the signals at the discontinuous point are precisely handled The time instants where the integra tion is halted are called events Since at an event instant variables may be discontinuous a simulation en vironment should store at least two signal values at the event time instant the value when then integration was halted and the value when the integration was restarted this is e g the case in Dymola The Modelica language is designed such that the above strategy is automatically performed i e the ba sic prerequisite of an integrator to operate on smooth equations is fulfilled between event instances Excep tio
402. tion can be carried out As will be explained in more detail in section 12 1 a Modelica model containing only continuous equa tions is mapped to the following differential algebraic equation system abbreviated as DAE 0 f x t x t y t t ER xEIR yelR fEIR 4 2 where x are nx variables that appear differentiated i e the der operator is applied on them and y are all other ny unknown variables Initialization of this DAE means to compute initial values of all vari ables so that the DAE is fulfilled at the initial time 0 f y Xo Yo 6o 4 3 with X X t t Xy x t t Yo y t t Besides of an initial value problem also called simulation problem also other operations on a model require first appropriate initialization for example the lineariza tion around a stationary point in order to compute eigenvalues or frequency responses Therefore initializa tion is usually the first operation to be carried out on a Modelica model before any other action is per formed Since f are nx ny equations and there are 2 nx ny unknowns X Xo Yo usually nx additional conditions are needed to uniquely compute all the unknowns at the predefined initial time to If the Jacobian matrix J of f of of J ty E ar 4 4 is regular at the initial time exactly nx independent additional conditions are needed If the Jacobian ma trix J is singular less th
403. tion is mapped to a set of equations in or der that the symbolic transformation algorithms see Chapter 12 can be applied This is performed by treat ing an algorithm section as a set of N equations where every variable in the algorithm section is part of the equation system and all variables appearing at the left hand side of an assignment statement are treated as variables at the left hand side of the equation All variables that appear at the left hand side of an assign ment statement are initialized to their start values for non discrete variables or to their pre values for discrete variables see Chapter 7 whenever the algorithm is invoked Due to this feature it is impossible for an algorithm section or a function to have a memory Furthermore it is guaranteed that the left hand side variables always have well defined values For example the algorithm section of block Limiter above has only one variable y appearing at the left hand side of the assignment statements Therefore the algorithm section of the Limiter block is treated as one equation of the following form for the symbolic processing 86 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 y f u uMax uMin y start Practically this means that this equation is sorted together with all other equations After the evaluation or der of the equations has been determined the original algorithm section can be re introduced at the appro
404. to the condition input as well especially blocks of the Modelica Blocks Logical library Instead of using logical blocks via the Modelica Blocks Sources SetBoolean component any type of logical expression can be defined in textual form as shown in the next figure Figure 18 4 Example demonstrating the use of the SetBoolean component initialStep transition2 SetBoolean2 step active F SetBooleant timer 1 With the block SetBoolean a time varying expression can be provided as modification to the output sig nal y see block with icon text timer y gt 1 in the figure above The output signal can be in turn con nected to the condition input of a TransitionWithSignal block The SetBoolean block can also be used to compute a Boolean signal depending on the active step In the figure above the output of the block with the icon text step active is true when step is ac tive otherwise it is false note the icon text of SetBoolean displays the modification of the output signal y This signal can then be used to compute desired actions in the physical systems model For ex ample if a valve shall be open when the StateGraph is in step1 or in step4 a SetBoolean block may be connected to the valve model using the following condition stepl active or step2 active Via the Modelica operators edge and change conditions depending on rising and falling edges of Bool
405. to the inverse plant model By solving a DAE sys tem or the symbolically transformed system the inverse plant model computes the desired measurement sig nals Yma and the desired plant inputs uy A feedback controller is used to stabilize the overall system and to improve robustness This might be a simple PID like controller It can be shown that the feedback controller has no effect as long as the plant and the inverse plant mod els are identical the plant and the inverse plant models are stable and both start at the same initial conditions In this case the reference model i e in many cases the filter determines the input output behavior i e it is the transfer function of the closed loop system If these assumptions are not fulfilled a control error occurs and the controller has to stabilize the system and cope with the imprecise inverse plant model and its initial conditions This structure has several advantages e The two controller parts inverse plant model with reference model and feedback controller can be designed independently from each other e The controller structure can be applied to unstable plants provided the inverse model is stable see section 19 8 e Since the inverse plant model is in the feedforward path the calculation of ug and of yma might be performed offline if possible so that hard real time requirements for the solution of the inverse plant model need not to be present The disadvantage of this struct
406. tro duced such as v der s in the model above The notation of expressions is a direct mapping of the usual mathematical notation with standard precedence of operators and with parenthesis The basic opera tors are For example the following two descriptions of a polynomial are identical yl x 2 x 4 y2 xX 2 2 x 8 Every model can have optionally annotation definitions These definitions do not influence the simulation result and are used e g to define the graphical positioning of the element the layout of the menu or the documentation Modelica supports the following four basic data data types Table 2 1 Basic data types of Modelica Modelica Description Example for literal element Real Floating point number 1 0 2 1234 le 14 1 4e3 usually about16 significant digits Integer Integer number ee A Boolean Logical number false true String String any number of characters File name enumeration Enumeration type Extrapolation enumeration HoldLastPoint LastTwoPoints Periodic Usage Extrapolation LastTwoPoints We have seen the usage of the Real type in the example above A variable declared as Boolean can have the values false or true A String is defined as in C by enclosing the string text in double apos trophes An enumeration is a type with an ordered set of names An en
407. true Draft Jan 21 2009 4 2 Initialization with attributes 63 menu Note that only start values are introduced that are explicitly defined within the model by using the start attribute All start values are collected together under a group with the heading Initialization In the input field the desired start value can be given At the left side of the input field a small checkbox is present that defines how the start value shall be interpreted Possible values are Table 4 1 Interpretation of check box for setting of attribute fixed ystat V true i e y y start is used as initial equation ner fixed false i e y is assigned the value y start before the l initialization computation is started y start is a guess value the fixed attribute is not set as a modifier neither fixed true E nor fixed false In grey color the current value of the fixed at tribute is shown as defined in the model class Here fixed true because in model firstOrderl fixed true isset It is also possible to include different initial conditions in the same model by setting the fixed attribute corre spondingly For example in the following model the user can either use the initial value of y as initial condi tion or he can initialize in steady state by setting the start value of der y to zero model FirstOrder2 initialize with y start or in steady state parameter Real k gain paramete
408. true If true the converted result is left justified if false it is right justified in the string Integer Defines the number of significant digits in the result string significantDigits 6 e g 12 3456 0 0123456 12345600 1 23456E 10 String format Defines the string formating according to ANSI C without and character e g 6g 14 5e 6f In particular is optional format lt flags gt lt width gt lt precision gt lt conversion gt with lt flags gt zero one or more of left adjustment of the converted number number will always be printed with a sign 0 padding to the field width with leading zeros lt width gt Minimum field width The converted number will be printed in a field at least this wide and wider if necessary If the converted number has fewer characters it will be padded on the left or the right depending on lt flags gt with blanks or 0 depending on lt flags gt lt precision gt The number of digits to be printed after the deci mal point for e E or f conversions or the number of significant digits for g or G conversions Wan lt conversion gt e Exponential notation using a lower case e E Exponential notation using an upper case E f Fixed point notation g Either Tan or up G Same as g but with upper case E Examples 232 Chapter 21 Modelica Built In Functions
409. tsseseeeeeeeeeeeeeees 207 19 3 Inverse Model in Feedforward Path of Controller cccccccceeesscccccceessssceeeecccesssseeeeceeceesseeeees 210 19 4 Inverse Model in Feedback Path of Controller cccccccccccesccccccceesesscceececeeesssseeeeceeesssseeeeeeees 211 19 5 Inverse Model in Feedback Linearization Controller 000 ccccccccccccceesssscecceeeesssseeecceeeseeseeeeees 212 19 6 Inverse Model in Robust Controller Disturbance Observer cccccccccsscceessseccsesseeeeesseeeeeeeees 213 19 7 Example AppliGati Ofi sic cc5 coiccssasdeaiavebes ss soaks osescuavada old sbsducdussdeodsecsadsbasiecdedesaovea colahsuesthbadecesdeaseie 214 19 8 Difficulties with Inverse Models cccccccccccccsssssscceccecessssceecccccssssseececeessssseeceeeessstseeeeeceeees 219 ACTIONS sessssacassseresadsensisscnensansarntennienwsesmnsinismemnnaunielensneetiontn Do Chapter 20 Contributors to Modelica sccsssssccsscsssesscscecsssccescsssceccsscscessscccessssccesssscccssssceeeeseeees 226 Chapter 21 Modelica Built In Functions and Operators scccccssscsscssscsssssecsscssescsccsceeeeseeesees 227 211 Elementary Operators iieccic cccecvsvianssveccdes Pecesdecdssest idea sec cessasita sesh eee Wik pee dave sbate daveev EEEE ETRE 227 21 2 Mathematical Functroms scissss c2e5 cetheshesated scbebaheins ia teahios dele desdsleeasd avian ae nid eee ilaleis 229 21 3 Conversion Fun Cti On 3s cairccedecesheeacentdesiasenesd seve ate v
410. uates to the base type of v The following example shows a combination of Real and Boolean equations to define a basic Hysteresis block Draft Jan 21 2009 6 2 Discrete equations and the pre operator 111 lt model Hysteresis import IF Modelica Blocks Interfaces IF RealInput u IF RealOutput y Boolean high start true fixed true equation high not pre high and u gt 1 or pre high and u gt 1 y if high then 1 else 1 end Hysteresis gt The Boolean variable high defines whether the output y is on the upper high true or on the lower high false branch of the characteristic Since high is a Boolean variable its value can only change at an event instant because relations only change at event instants The Boolean equation for high contains pre high i e the value of high just before the actual event instant This is the value of high from the previous event instant because the value of high can change only at event in stants Events occur when one of the relations i e u gt 1 or u gt 1 changes their values i e whenever there is a potential change of the corresponding branch It is a quality of implementation of the used Modelica simulation environment whether unnecessary events are generated For example a simple implementation will trigger an event whenever u gt 1 or u
411. uations to compute one variable The user is forced to reformulate the state machine by explicitly defining priorities For example if filll and fill2 are steps that are executed in parallel there might be a Set Boolean block that defines openValve if filll active then 1 else if fil112 active then 0 else 2 Therefore step i111 has a higher priority as step 1112 In a Statechart or Graphchart it is difficult to modularize a sub chart if the used actions reference vari ables in an outer scope Assume for example that a state machine control has the following hierarchy control superstatel stepl Within step1 a Statechart would e g access variable control openValve say as entry action 194 Chapter 18 Hierarchical State Machines StateGraph Library Draft Jan 21 2009 openValve true This typical usage has the drawback that it is difficult to use the hierarchical state superstatel as component in another context because step1 references a particular name outside of this component In a StateGraph there would be typically a SetBoolean component in the control compo nent stating openValve superstatel stepl active As a result the superstate1 component can be used in another context because it does not depend on the environment where it is used The disadvantage of the StateGraph approach is that the user might not be able to formulate the net work directly as
412. umeration literal value consists of the name of the enumeration type appended with a dot and the element name For engineering applications it is important to support units of the physical quantities In Modelica at tributes can be defined in a declaration including the actually used units The example above is then refined to model MovingMass2 Mass pulled with a constant force parameter Real m unit kg min 0 2 Mass of block parameter Real f unit N 6 Pulling force Real s unit m Position of block Real v unit m s Speed of block annotation Diagram Rectangle extent lt other definitions gt equation 18 Chapter 2 Basic Modelica Language Constructs Draft Jan 21 2009 v der s m der v f end MovingMass2 The unit attribute defines that the equations are only correct if the variable values are with respect to the given unit Additionally with attribute min it is defined that the variable value cannot be negative A unit is not sufficient to define a physical quantity This can be easily seen by expressing the units of torque and of energy in the SI base units Table 2 2 Torque and energy in SI base units quantity unit in SI base units Torque Nm kgm s Energy J kgm s Although Torque and Energy are different physical quantities they have identical units In Modelica several unit related attributes can be d
413. ure 19 7 Step responses of normalized low pass filters of order 4 step Criticalbamping Bessel Butterworth Chebyshev 1 4 1 24 1 04 0 8 0 64 0 4 4 0 2 4 0 0 0 2 Since the Bessel filter is faster as the CriticalDamping filter the first choice for R is the Bessel filter The design of the reference transfer function is therefore rather simple 1 Use a normalized low pass Bessel filter with unit gain Modelica_LinearSystems Sampled Filter with parameters analogFilter Bessel filterType LowPass gain 1 normalized true 2 Use the relative degree of the plant model P as filter order order for example if P is a PT2 block use order 2 3 Adjust the cut off frequency f cut of the filter manually or by parameter variation optimization un til the controller performance is satisfactory Start the tuning with a value of f cut which is in the or der of the frequency up to which the plant transfer function P is valid Use either a more detailed plant model e g a non linear model for the tuning or perform the tuning directly at the real plant 19 2 Inversion of Nonlinear Models In the previous section 19 1 it was shown how inverse plant models can be utilized for an important con troller structure There are also other useful linear controller structures that are based on inverse models In general all controllers that are
414. ure is that for some applications the feedback controller may still have to be scheduled as a function of the operating conditions Inverse model based feedforward control will be demonstrated on the basis of an example in section 19 7 19 4 Inverse Model in Feedback Path of Controller The disadvantage of the inverse feedforward controller can be avoided by moving the inverse model into the feedback path This is shown in Figure 19 10 The feedback controller now only sees the combined inverse and plant model The structure is a generalization of the linear compensation controller described in 212 Chapter 19 Nonlinear Inverse Models for Advanced Controllers Draft Jan 21 2009 Follinger 1994 page 266 Figure 19 10 Compensation controller using an inverse plant model in the feedback path Plant must be stable and minimum phase Plant can be unstable and non minimum phase when the inverse model is approximated as the inverse of the steady state plant model u 0 y 0 e t i i y y O a feedback inverse u Z piani controller plant model controller Di a he ee et eee ee For linear plant models the feedback controller see Figure 19 10 must have a relative degree that is equal or larger than the relative degree of the plant in order that the system is proper For single input single out put systems a useful feedback controller is ue e 19 17 where 1 r s is
415. urns result in the range n 7 cos x cosine x is in rad cosh x hyperbolic cosine exp x exponential of x base e log x natural base e logarithm x gt 0 1og10 x base 10 logarithm x gt 0 sign x Is expanded into noEvent if x gt 0 then 1 else if x lt 0 then 1 else 0 230 Chapter 21 Modelica Built In Functions and Operators Draft Jan 21 2009 Operator Description sin x sine x is in rad sinh x hyperbolic sine sqrt x Returns the square root of x if x gt 0 otherwise an error occurs tan x tangent x is in rad and shall not be 2 2 2 2 37 2 tanh x hyperbolic tangent 21 3 Conversion Functions Table 21 7 Conversion functions All functions are vectorizable and trigger events Operator Description changes discontinuously Examples ceil 2 3 3 0 ceil 2 3 2 0 ceil x Returns the smallest integer not less than x Result and argument shall have type Real Note outside of a when clause state events are triggered when the return value triggered when the return value changes discontinuously Examples div 4 3 1 div 4 3 1 div 7 5 2 3 0 div x y Returns the algebraic quotient x y with any fractional part discarded also known as truncation toward zero Note this is defined for in C99 in C89 the result for nega tive numbers is implementation defined so the st
416. utput Complex y algorithm y re ul re u2 re y im ul im u2 im end add model TestComplex Complex cl re 2 im 1 annotation Complex c2 annotation Complex c3 annotation equation c2 add cl Complex sin time cos time c3 add cl Complex re sin time im cos time end TestComplex Above a record for complex numbers is defined The record extends from Modelica Icons Record in order that a default icon is available for the record Function add is used to add two complex numbers The input and argument arguments of this function are records of type Complex Finally the test model TestComplex shows a simple usage of the complex number record In the declaration section three com plex numbers are declared They are constructed here by dragging them from the package browser in to the diagram layer of the model The advantage is that a user can click on a record instance in the diagram layer and the parameter menu of the record pops up to e g define the actual values of complex number c1 see Table 5 2 Table 5 2 Graphical view of a record instance in the diagram layer Diagram layer of TestComplex Parameter menu of TestComplex cl Parameters c1 c2 es Ea E re 2 Real part im Imaginary part The complex numbers c2 and c3 are constructed by adding c1 with another complex number that is on the fly constructed with the recor
417. utput yc The model DiscreteFirstOrderBlockVersion1 is not yet satisfactory One issue is that initialization is not explicitly defined For example assume that the input u is 1 at the beginning and after 1 s it jumps to 2 i e u if time lt 1 then 1 else 2 118 Chapter 7 Instantaneous Equations Draft Jan 21 2009 Figure 7 1 Simulation results of model DiscreteFirstOrderBlock1 for a step input DiscreteFirstOrderBlock1 T 0 1 a 0 8 b 0 2 As can be seen there is an undesired initial vibration of the output y of the controller due to the default ini tialization Another issue is that the controller is modeled in a quite idealized form because no computation time of the control algorithm is taken into account and therefore the output y of the controller acts instanta neously on a given measurement input u of the controller which is not possible in reality Since a controller might be constructed from a set of connected discrete blocks it is not useful to include the computational de lay in a discrete block but implement a separate delay block that is connected between the control system and the actuator input of the plant The above considerations lead to the following implementation For simplici ty a computational delay of one sample interval is taken into account and is implemented in the first order block and not in a separate block block Dis
418. utputl The disadvantage is that blocks with such interfaces can be connected in an arbitrary way e g two RealOut puts could be connected although this is not allowed in a block diagram A better solution is to use the pre fixes input and output Do not use not a good design for block diagram interfaces connector ReallInput2 connector RealOutput2 input Real signal output Real signal end ReallInput2 end RealOutput2 Connectors that have input output prefixes can only be connected according to block diagram semantics This means that the following restrictions hold 1 Two or more inputs cannot be connected together 2 Two or more outputs cannot be connected together 3 An input must be connected to exactly one non input which is usually an output variable however a pre fix need not necessarily be present or an equation must be provided for the input This allows e g that errors can be earlier detected with better diagnostics if the modeler wants to model block diagrams In Dymola an error is triggered immediately when a connection is drawn and one of the con nection restrictions above is violated For example in the next Figure 3 7 four input output blocks are con nected together using the above connector definitions Since the outputs of firstOrderl and firstOrder2 are connected the model is wrong Figure 3 7 Wrong connection of input output blocks ramp1 firstOrder1 duration 2
419. vatives SIAM Journal of Scientific and Statistical Computing Vol 14 pp 677 692 Mattsson S E H Olsson and H Elmqvist 2000 Dynamic Selection of States in Dymola Proceedings of the Modelica Workshop 2000 pp 61 67 http www modelica org Workshop2000 papers Mattsson pdf Modelica Association 2007 Modelica A Unified Object Oriented Language for Physical Systems Modeling Language Specification Version 3 0 http www modelica org documents ModelicaSpec30 pdf Mosterman P and G Biswas 1996 A Formal Hybrid Modeling Scheme for Handling Discontinuities in Physical System Models Proceedings of AAAI 96 Portland OR USA pp 985 990 Mosterman P M Otter and H Elmqvist 1998 Modeling Petri Nets as Local Constraint Equations for Hybrid Systems using Modelica Proceedings of SCSC 98 Reno Nevada USA Society for Computer Simulation International pp 314 319 Mugica F and F E Cellier 1994 Automated synthesis of a fuzzy controller for cargo ship steering by means of qualitative simulation Proceedings of the European Simulation MultiConference ESM 94 Barcelona Spain pp 523 528 Olsson H M Otter S E Mattsson and H Elmqvist 2008 Balanced Models in Modelica 3 0 for Increased Model Quality Proceedings of the 6 International Modelica Conference Bielefeld Germany ed B Bachmann pp 21 33 http www modelica org events modelica2008 Proceedings sessions session1a3 pdf OpenModelica 20
420. vel In or der to be useful a Modelica modeling and simulation environment is needed see Figure below to graphi cally edit and browse a textually defined Modelica schematic to transform the model in a form that is better suited for reliable integration to simulate the model to visualize the simulation results and to import Modeli ca models in other simulation environments such as Simulink A growing number of such Modelica environ ments are available commercially and also in the public domain For an actual overview see http www Modelica org tools In this document the commercial Modelica environment Dymola is used for demonstra tion purposes Reuse is a key issue for handling complexity There have been several attempts to define object oriented languages for modeling of technical systems However the ability to reuse and exchange models relies on a standardized format It was thus important to bring this expertise together to unify concepts and notations Hilding Elmqvist from Dynasim initiated the Modelica design effort in September 1996 headed it until the Modelica Association was founded in 2000 and is the key architect of the language until today Still many other people have contributed see Chapter 20 on page 226 for details The language and the free Modelica Standard Library have been designed by the developers of the object oriented modeling languages Allan Dy mola NMF ObjectMath Omola SIDOPS Smile by computer s
421. ving torque that changes discontinuously at time 1s 7 is the gear ratio and t m are the constraint torques in the left and right flange of the gear It is possible to solve this system with an implicit integration algorithm see Chapter 13 without symbolic pre processing because equations need to be differentiated only once in order to transform algebraically to state space form if more differentiations would be needed numerical integra tion algorithms are usually no longer reliable Under the assumption that the differentiated variables are known i e 1 2 these are 4 equations in the 4 unknowns 2 T T2 At time 1s an event occurs and Tarive changes discontinuously The values of all variables are known just before the event occurs At the event instant the driving torque changes The important question is how to initialize this DAE to restart the integration The usual assump tion is that the differentiated variables remain continuous over an event We have therefore 4 equations in 4 Draft Jan 21 2009 12 5 Hybrid DAEs need Symbolic Preprocessing 167 unknowns 2 T T at the event instant For every singular DAE this system of equations is singular this is the definition of a singular DAE A well posed singular DAE has an infinite number of solutions For example in equation 12 26 the angular acceleration of inertia2 can be selected arbitrarily and the other variables can then be computed uniquely For e
422. vior in the feedback path Additionally the same filter is present at two places The stan dard disturbance observer uses a linear model for the inverse plant model A nonlinear desired plant model provides more freedom since it might be impossible that a physical system can be forced to have the same linear behavior in its whole operating range Note there is the requirement that the number of measurement signals is identical to the number of plant inputs dim y dim u For a single input single output system where all parts are linear the transfer function from ue to Ym is given by Ym T F s Fls 19 21 where F s is the filter P s is the plant and Pz s is the desired plant transfer function in the feedback loop For low frequencies F s 1 and therefore y P s u For high frequencies F s 0 and then y P s u The effect of the disturbance observer is therefore that it enforces the desired plant behavior for low frequencies In other words if there are modeling errors or disturbances then the disturbance observ er enforces a desired plant behavior below the cut off frequency of the filter i e the controller designed for the desired plant will usually work considerably better The disturbance observer is usually combined with other controller structures For example with the structure from section 19 3 e An inverse plant model from y to u in the feedforward path is used for command following and
423. w actualStream port Xi outflow not present if ns 1 The volume contains especially the mass balance for the substance mass flow rates If ns 1 both the vector of substance masses ms and the vector of independent mass fractions port Xi_outflow have zero dimensions and therefore the balance equation der ms is not present 78 Chapter 5 Advanced Modelica Language Constructs Draft Jan 21 2009 5 1 4 Example General transfer function model The introduced operations on arrays shall be demonstrated on the basis of a more complicated example A single input single output transfer function is defined as bs b s b s b 1 2 m m 1 u 5 1 n a a S a S a 8 a 4 where u is the input y is the output vector b contains the coefficients of the numerator polynomial and vector a contains the coefficients of the denominator polynomial In order to simulate this system it has to be transformed to a differential equation system There is no unique transformation We will use the con troller canonical form For n m 5 it has the following structure a a a a 1 ay ay ay ay ay x 1 0 0 0 X 10 u 0 1 0 0 0 5 2 0 0 1 0 0 bi bi bi b b y b a b a by a b a X 4 u a a a ay a For n 0 we have the special case of a pure gain y a u Since several single input single output blocks are present it is useful to define super class SISO that defines the
424. when the algorithm section is entered stepSize is just the difference between the actual time and the initial time i e it is not possible to compute the step size In a second conceptual mapping phase the algorithm section after the first mapping phase is trans formed in a function call e g the algorithm section above is transformed in the function call vl v2 v3 algFunction v_expr pre vl v2 start pre v3 where the output arguments are constructed from all variables occurring on the left hand side of the equal signs and all other used variables are the input arguments to the function v_expr symbolizes all variable references from expr1 expr2 expr3 and condition A modeler cannot perform such a map ping manually because Modelica functions are pure functions e g it is not allowed to access any of the discrete operators such as pre initial etc in a Modelica function Finally this function call is treated as a set of equations that is solved together with all other equations of a model We have now discussed all language elements in Modelica to model discrete systems In the following chapters it is shown how these language elements can be used to mod el different important situations but no new language elements are introduced with the exception of reinit in Chapter 11 7 4 Time synchronization multi rate delays range limited sampling Modelica has no language
425. without any problems until step 5 In this step the first two unknowns zz and z3 cannot be assigned because these variables are already assigned in previous steps So the only choice is to assign un known Zs It is not possible to perform such a simple assignment in the next equation f This equation has only one unknown z and this unknown has already been assigned The idea is now to perform a systematic reassign ment of all previous unknowns until it is possible to make an assignment to z2 in fs The process is sketched below Draft Jan 21 2009 12 2 Sorting BLT 159 Unknown z is already assigned in equation f4 This equation has the two unknowns z and z2 The assignment is changed to z However z is already assigned in f This equation has the three unknowns 2 Ze and z7 The assignment is changed to z since Z is already assigned in fs The result is shown below together with the fi nal step 7 step 6 step 7 0 f i z3 24 O z z 0 fa Zi Z6 Z7 0 f2 2Z1 26 Z7 0 f3 Z3 Zs 0 f3 2Z3 Zs gt 0 f 2 gt 0 fizz 0 F Z3 Z3 Z6 0 f Zo Zase 0 felz 0 f z 0 f7 23 27 0 f7 23 2 Z4 In step 7 again the two unknown variables z and z are already assigned It is possible to perform a reas signment of f for the other unknown z and we have finally reached an assignment where every variable is assigned exactly once It might be that this is not possible
426. xample the following values fulfill the DAE w 1 T drive IJ 1 i 0 ee 12 27 T 0 tT 0 Although these values fulfill the DAE 12 26 they are not correct because the mathematical solution re quires that i 12 28 and this equation is not fulfilled from the values above The trouble is that this equation is not present in the DAE but is only implicitly defined by differentiating the second equation of 12 26 To summarize since a singular DAE has an infinite number of solutions at an event instant it is impos sible for any numerical method to derive the mathematically correct solution without taking into account the hidden constraint equations that can be derived by the Pantelides algorithm or another method As a result a simulation system that does not take into account the hidden constraint equations of a singular DAE when a variable changes discontinuously either fails at an event instant or produces a wrong solution around the event point As a consequence a reliable modeler should not solve singular DAEs with such a simulation system even if it seems to be that the solver can handle it in some cases When building models with components from the Modelica Standard Library the results are often sin gular DAEs because the components are designed for maximal flexibility This is especially the case for the Modelica Mechanics MultiBody Modelica Mechanics Rotational Modelica Mechanics Translational
427. zed as variables x Zero voltage drop and or zero current flow are mean ingful initial conditions because no currents are flowing and therefore the circuit is at rest For thermal or fluid systems the Dymola heuristics would be completely useless without taking appro priate actions The reason is that absolute pressure temperature in Kelvin or density are used as variables x and zero values of these variables are completely outside of the operating range of usual models For this reason in every medium model Modelica types with medium specific start values are defined for the rele vant variable categories and these types are used in the declaration section of a model For example the fol lowing default types for media are defined in Modelica Media Interfaces PartialMedium type AbsolutePressure Modelica SIunits AbsolutePressure min 0 max 1 0e8 nominal 1 0e5 start 1 0e5 Draft Jan 21 2009 4 4 Too many or not enough initial conditions 69 type Temperature Modelica SIunits Temperature min 1 max 1 0e4 nominal 300 start 300 As a result all relevant variables have start values that are meaningful for the corresponding medium and therefore there is a chance that the heuristics of Dymola or of another Modelica tool works The heuristics of Dymola can be changed by providing the following setting in the command window of Dymola Advanced DefaultSteadyStateInitialization true

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