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Robust Control Toolbox User's Guide

1. 0 0008 3 4 4 EML A ppliGations ostrea pets ot ely sascha Nh ee Sse le gnc vee 3 6 Sta bility enan Barats phe Bape aae a So tapas eeateh oh etwas eo takes 3 7 RMS Grain a se Seok Son ate ten eels Lee Se a eke ERE ang alia hia ha 3 8 LQG Performance 0 0 0 ee ens 3 8 Further Mathematical Background 3 10 Bibliography Teee Seances Melee eels ewe OR Fh ee 3 11 LMI Lab Tools for Specifying and Solving LMIs 4 2 Some Terminology 0 000 cece ee eens 4 2 Overview of the LMI Lab 0 0 00 00 ee eens 4 5 Specifying a System of LMIs 00045 4 7 Specify LMI System at the Command Line 4 8 Specifying LMI System 0 0 0 0 4 8 Initializing the LMI System 0 0 00 cease 4 9 Specifying the LMI Variables 00 4 10 Specifying Individual LMIs 00000 4 12 Specify LMIs with the LMI Editor GUI 4 15 Keyboard Shortcuts 0 000 ccc ee eens 4 17 EMITA ONS e arse ah eek Bar eee hn Rag ek ae ae 4 17 How Imivar and Imiterm Manage LMI Representation 4 18 Querying the LMI System Description 4 19 Wim azte ee ets Seeks ag Bh eich Ra lt 4 19 Iminbr and matnbr 0 0 0 eee eee 4 19 LMI Solvers oau er 28 ee ae ad de A ee 4 20 Minimize Linear Objectives under LMI Constraints 4 22 Conversion Between Decision and Matrix Variables
2. The first entry indicates to which LMI the term belongs The value m means left side of the m th LMI and m means right side of the m th LMI The second and third entries identify the block to which the term belongs For instance the vector 1 1 2 1 indicates that the term is attached to the 1 2 block The last entry indicates which matrix variable is involved in the term This entry is 0 for constant terms k for terms involving the k th matrix variable X and k for terms involving X es here X and S are first and second variables in the order of declaration Finally the second and third arguments of lmiterm contain the numerical data values of the constant term outer factor or matrix coefficients P and Q for variable terms PXQ or PX Q These arguments must refer to existing MATLAB variables and be real valued See Complex Valued LMIs on page 4 33 for the specification of LMIs with complex valued coefficients Some shorthand is provided to simplify term specification First blocks are zero by default Second in diagonal blocks the extra argument s allows you to specify the conjugated expression AXB B X A witha single miterm command For instance the first command specifies A X XA as the symmetrization of XA Finally scalar values are allowed as shorthand for scalar matrices i e matrices of the form al with a scalar Thus a constant term of the form al can be specifie
3. Vary the uncertainty and see how it affects the time responses Monte Carlo analysis See Simulate Uncertainty Effects on page 5 7 Analyze the effects of uncertainty on the linearized dynamics Ifthe operating point does not depend on the parameter uncertainty use ulinearize to obtain an uncertain state space model You can then use usample to sample the uncertain variables and obtain a family of LTI models Ifthe operating point depends on the parameter uncertainty use usample to sample the uncertainty and then use the Simulink Control Design linearize command to compute the linearized dynamics for each uncertainty value See How to Vary Uncertainty Values on page 5 7 and Obtain Uncertain State Space Model from Simulink Model on page 5 20 Compute an uncertain linearization i e obtain an uncertain state space model uss object that combines the uncertain variables with the linearized dynamics You can use this model to perform worst case robustness analysis See Obtain Uncertain State Space Model from Simulink Model on page 5 20 If you cannot use Uncertain State Space blocks in the Simulink model because you share the model or generate code you can still compute an uncertain linearization by specifying a block to linearize to an uncertain variable For example you can specify a gain block to linearize to an uncertain real parameter ureal See Specify Uncertain Linearization for Core or Cu
4. Functions for synthesizing robust gain scheduled H controllers hinfgs Linear Matrix Inequalities For users interested in developing their own applications the LMI Lab provides a general purpose and fully programmable environment to specify and solve virtually any LMI problem Note that the scope of this facility is by no means restricted to control oriented applications Note Robust Control Toolbox software implements state of the art interior point LMI solvers While these solvers are significantly faster than classical convex optimization algorithms you should keep in mind that the complexity of LMI computations can grow quickly with the problem order number of states For example the number of operations required to solve a Riccati equation is o n where n is the state dimension while the cost of solving an equivalent Riccati inequality LMI is o n 3 Introduction to Linear Matrix Inequalities LMIs and LMI Problems 3 4 A linear matrix inequality LMI is any constraint of the form A x oo Ao x1A1 te hs xyAyn lt 0 where x x1 Xy is a vector of unknown scalars the decision or optimization variables Ao Ay are given symmetric matrices e lt 0 stands for negative definite i e the largest eigenvalue of A x is negative Note that the constraints A x gt 0 and A x lt B x are special cases of Equation 3 1 since they can be rewritten as A x lt 0 and A x
5. Reconstruct M if need be and make a 2 by 2 selection from M a ureal a 3 b ureal b 10 pe 20 M a 1 b b at 1 b31 3 M M Uncertainty a Range 2 5 5 2 3 ans Uncertain matrix with 2 rows and 2 columns The uncertainty consists of the following blocks a Uncertain real nominal 3 variability 0 5 2 1 occurrences b Uncertain real nominal 10 variability 20 20 2 occurrences Type ans NominalValue to see the nominal value get ans to see all properties an Make a single column selection from M and use single index references to access elements of it h M 2 1 2 3 2 h Uncertain matrix with 4 rows and 1 columns The uncertainty consists of the following blocks a Uncertain real nominal 3 variability 0 5 2 1 occurrences b Uncertain real nominal 10 variability 20 20 1 occurrences Type h NominalValue to see the nominal value get h to see all properties and h h 2 ans Uncertain matrix with 1 rows and 1 columns The uncertainty consists of the following blocks b Uncertain real nominal 10 variability 20 20 1 occurrences 1 27 T Building Uncertain Models Type ans NominalValue to see the nominal value get ans to see all properties an Finally make the 3 2 entry of M uncertain M 3 2 ureal c 3 perc 40 M Uncertain matrix with 3 rows and 2 columns The uncertainty consists of the following blo
6. to simulate the model The software uses the nominal values of the uncertain variables a and input_unc during simulation After the simulation completes the MultiPlot Graph block shows the following figure 5 Analyzing Uncertainty Effects in Simulink OGes k AAU98 A a ee eee eee i i i 4 i i i lt i i J 1 1 4 1 i i i i J i 1 i 1 4 Generate random samples of uncertainty values a Findall Uncertain State Space blocks and associated uncertain variables in the model uvars ufind rct_sim_ex2 MATLAB returns the following result uvars a 1x1 ureal input_unc 1x1 ultidyn 5 16 Vary Uncertainty Values Across Multiple Uncertain State Space Blocks The uncertain variables a and input_unc are ureal and ultidyn objects respectively and the structure uvars lists them by name b Randomly sample the uncertain variables val_all usample uvars MATLAB returns the following result val_all a 1 1167 input_unc 1x1 ss The structure val_all contains sample values of the uncertain variables a and input_unc The software samples the values within the specified uncertainty ranges for a and input_unc 5 Simulate the model for the uncertainty values val_all By repeating the process inside a for loop you can assess how uncertainty affects the model responses For example perform 10 simulations using random uncertainty values for
7. B x lt 0 respectively The LMI of Equation 3 1 is a convex constraint on x since A y lt 0 and A z lt 0 imply that Al 2 2 k 0 As a result Its solution set called the feasible set is a convex subset of RN Finding a solution x to Equation 3 1 if any is a convex optimization problem Convexity has an important consequence even though Equation 3 1 has no analytical solution in general it can be solved numerically with guarantees of finding a solution when one exists Note that a system of LMI constraints can be regarded as a single LMI since A x lt 0 ae lt 0 is equivalent to A x diag Aj x AK x lt 0 where diag A x Ax x denotes the block diagonal matrix with LMIs and LMI Problems A x Ax x on its diagonal Hence multiple LMI constraints can be imposed on the vector of decision variables x without destroying convexity In most control applications LMIs do not naturally arise in the canonical form of Equation 3 1 but rather in the form LXi Xn lt Ra Xn where L and R are affine functions of some structured matrix variables Xi Xa A simple example is the Lyapunov inequality A X XA lt 0 where the unknown X is a symmetric matrix Defining x xy as the independent scalar entries of X this LMI could be rewritten in the form of Equation 3 1 Yet it is more convenient and efficient to describe it in its natural form Equation 3
8. M MoV X Xo M Mo X X lt M My Xe X M My X X X X 1 aloo X X Advanced LMI Techniques Note that X X in turn requires that X X a and X xe 0 Consequently X and X should be declared as symmetric and skew symmetric matrix variables respectively Assuming for instance that M e C the LMI system Equation 4 11 would be specified as follows Mi real M M2 imag M bigM M1 M2 M2 M1 setlmis declare bigX X1 X2 X2 X1 with X1 X1 and X2 X2 0 X1 n1 sX1 lmivar 1 5 1 X2 n2 SsX2 lmivar 3 skewdec 5 n1 bigX lmivar 3 SX1 sX2 sX2 sX1 describe the real counterpart of 1 12 lmiterm 1 1 1 0 1 lmiterm 1 1 1 bigX 1 1 Imiterm 2 1 1 bigX bigM bigM lmiterm 2 1 1 bigX 1 1 lmis getlmis Note the three step declaration of the structured matrix variable bigX ZX nex 1 X X 1 Specify X as a real symmetric matrix variable and save its structure description SX1 as well as the number n1 of decision variables used in Xj 2 Specify X as a skew symmetric matrix variable using Type 3 of lmivar and the utility skewdec The command skewdec 5 n1 creates a 5 by 5 skew symmetric structure depending on the decision variables n1 1 n1 2 3 Define the structure of bigX in terms of the structures sX1 and sX2 of X and X2 See Structured Matrix Variables on page 4 31 for more details on such structure manipulatio
9. Validating Results 0 0 00 Modify a System of LMIs 0 00 00005 Deleting an LMI neria ae ee Bae in bodes Bowe pbs Re Deleting a Matrix Variable 20 000 ce eae Instantiating a Matrix Variable 004 Advanced LMI Techniques 00055 Structured Matrix Variables 0 000000 e eee Complex Valued LMIs 0 0 00 cece eae Specifying c x Objectives for mincx 0 004 Feasibility Radius 0 00 cece ees Well Posedness Issues 0000s Semi Definite B x in gevp Problems Efficiency and Complexity Issues 0 005 Solving M P XQ QTXTP lt 0 ee eee Bibliography erei cco ese oe pes es ee gees al ge Hes Bea 4 26 4 27 4 28 4 28 4 28 4 29 4 31 4 31 4 33 4 36 4 37 4 38 4 39 4 40 4 40 4 42 Analyzing Uncertainty Effects in Simulink 5 Analyzing Uncertainty in Simulink Simulink Blocks for Analyzing Uncertainty Specify Uncertainty Using Uncertain State Space Blocks How to Specify Uncertainty in Uncertain State Space Blocks Next Steps fais Mites Pn Pha hee Soca ee eee a Oar ee Simulate Uncertainty Effects 00 How to Simulate Effects of Uncertainty How to Vary Uncertainty Values 00055 Vary Uncertainty Values Using Individual Uncertain State Space Blocks 235 6
10. basic 000 ucomplexm m 1 2 3 4 5 6 7 8 9 10 11 12 Sample the uncertain matrix and compare to the nominal value Note the element by element sizes of the difference are generally equal indicative of the default identity weighting matrices that are in place Uncertain Complex Parameters and Matrices abs usample m m NominalValue ans 0 2948 0 1001 0 2867 0 3028 0 2384 0 2508 0 3376 0 1260 0 2506 0 2200 0 3472 0 1657 Change the left and right weighting matrices making the uncertainty larger as you move down the rows and across the columns m WL diag 0 2 0 4 0 8 1 6 m WR diag 0 1 1 4 Sample the uncertain matrix and compare to the nominal value Note the element by element sizes of the difference and the general trend that the smallest differences are near the 1 1 element and the largest differences are near the 4 3 element which is completely expected by choice of the diagonal weighting matrices abs usample m m NominalValue ans 0 0091 0 0860 0 2753 0 0057 0 1717 0 6413 0 0304 0 2756 1 4012 0 0527 0 4099 1 8335 1 21 T Building Uncertain Models Systems with Unmodeled Dynamics The unstructured uncertain dynamic system Control Design Block the udyn object represents completely unknown multivariable time varying nonlinear systems For practical purposes these uncertain elements represent noncommuting symbolic variables placeholders All algebraic operations such a
11. Another issue has to do with homogeneous feasibility problems such as A P PA lt 0 P gt 0 While this problem is technically well posed the LMI optimization is likely to produce solutions close to zero the trivial solution of the nonstrict problem To compute a nontrivial Lyapunov matrix and easily differentiate between feasibility and infeasibility replace the constraint P gt 0 by P gt al with a gt 0 Note that this does not alter the problem due to its homogeneous nature Semi Definite B x in gevp Problems Consider the generalized eigenvalue minimization problem Minimize J subject to A x lt AB x B x gt 0 C x lt 0 Technically the positivity of B x for some x R is required for the well posedness of the problem and the applicability of polynomial time interior point methods Hence problems where Bf A i 0 0 with B x gt 0 strictly feasible cannot be directly solved with gevp A simple remedy consists of replacing the constraints A x lt B x B x gt 0 by aog op lt AB x Bi 2 gt 0 where Y is an additional symmetric variable of proper dimensions The resulting problem is equivalent to Equation 4 14 and can be solved directly with gevp 4 39 4 IMILab 4 40 Efficiency and Complexity Issues As explained in Tools for Specifying and Solving LMI on page 4 2 the term oriented description of LMIs used in the LMI Lab typically leads to higher efficiency than the canonical rep
12. NominalValue Uncertainty InputDelay OutputDelay Ts TimeUnit InputName InputUnit InputGroup OutputName OutputUnit OutputGroup Name Notes UserData The properties ResponseData and Frequency behave in exactly the same manner as Control System Toolbox frd objects except that ResponseData is a umat The InputGroup and OutputGroup behave in exactly the same manner as all of the Control System Toolbox objects ss zpk tf and properties InputName OutputName frd The NominalValue is a Control System Toolbox frd object and hence all methods for frd objects are available For instance plot the Bode response of the nominal system bode sysg nom 1 44 100x1 double rad TimeUnit 4 D umat 2x1 frd 1x1 struct 0 2x1 double 0 seconds 1x1 2x1 2x1 1x1 struct cell cell struct Create Uncertain Frequency Response Data Models To Out j I 8 B 6s jut 1 To lt i pa Magnitude dB Phase deg To amp i B 0 To Cunt 180 oe oo E oe oo ee oo LL dit Frequency H3 Just as with the umat and uss classes the Uncertainty property is a structure containing the uncertain elements Direct access to the elements is facilitated through Uncertainty Change the nominal value of the uncertain element named p1 within sysg to 14 and replot the Bode plot of the new nominal system sysg
13. The resulting system of two LMIs is returned in NEWSYS The LMI identifiers initial ranking of the LMI in the LMI system are not altered by deletions As a result the last LMI S gt I remains known as the third LMI even though it now ranks second in the modified system To avoid confusion it is safer to refer to LMIs via the identifiers returned by newlmi If BRL Xpos and Slmi are the identifiers attached to the three LMIs Equation 4 6 Equation 4 8 Slmi keeps pointing to S gt J even after deleting the second LMI by NEWSYS dellmi LMISYS Xpos Deleting a Matrix Variable Another way of modifying an LMI system is to delete a matrix variable that is to remove all variable terms involving this matrix variable This operation is performed by delmvar For instance consider the LMI A X XA BW W B I lt 0 with variables X X e R x and We R x This LMI is defined by setlmis Modify a System of LMIs X lmivar 1 4 1 X W lmivar 2 2 4 W lmiterm 1 1 1 X 1 A s lmiterm 1 1 1 W B 1 s lmiterm 1 1 1 0 1 LMISYS getlmis To delete the variable W type the command NEWSYS delmvar LMISYS W The resulting NEWSYS now describes the Lyapunov inequality A X XA I lt 0 Note that delmvar automatically removes all LMIs that depended only on the deleted matrix variable The matrix variable identifiers are not affected by deletions and continue to point to the same matrix variable Fo
14. a 3 Range 2 5 4 b ureal b 4 Percentage 15 c ureal c 2 Plusminus 1 3 M U a b b c MAT 2 Rows 2 Columns a real nominal 3 range 2 5 4 1 occurrence b real nominal 4 variability 15 15 2 occurrences c real nominal 2 variability 1 0 3 1 occurrence Grid the uncertain real parameter b in M with 100 points The result is a umat array with dependence on uncertain real parameters a and c Mgridi gridureal M b 100 UMAT 2 Rows 2 Columns array 100 x 1 a real nominal 3 range 2 5 4 1 occurrence c real nominal 2 variability 1 0 3 1 occurrence Operating on the uncertain matrix M grid the uncertain real parameter a with 20 points the uncertain real parameter b with 12 points and the uncertain real parameter C with 7 points The result is a 2 by 2 by20 by 12 by7 double array Mgrid3 gridureal M a 20 b 12 c 7 size Mgrid3 ans 2 2 20 12 7 1 69 T Building Uncertain Models Create Arrays with repmat 1 70 The MATLAB command repmat is used to replicate and tile arrays It works on the built in objects of MATLAB namely double char as well as the generalized container objects cell and struct The identical functionality is provided for replicating and tiling uncertain elements ureal ultidyn etc and umat objects You can create an uncertain real parameter and replicate it in a 2 by 3 uncertain matrix Compare to generat
15. a b 4 a ans 0 0 0 simplify M 3 4 5 6 ans 0 0 0 simplify M 4 a 0 0 ans 0 0 0 simplify M 1 2 3 should be 5 ans 5 simplify M 1 3 2 4 Uncertain Real Parameter Name a NominalValue 4 variability 1 1 Create Arrays with stack and cat Functions You can create a random 1 by 3 by 4 double matrix and stack this with M along the second array dimension creating a 1 by 3 by 4 by 2 umat N randn 1 3 4 M2 stack 2 M N size M2 ans 1 3 4 2 arraysize M2 ans 4 2 As expected both M and N can be recovered from M2 di simplify M2 1 M d2 simplify M2 2 N max abs d1 max abs d2 ans 0 0 It is also possible to stack M and N along the 1st array dimension creating a 1 by 3 by 8 by 1 umat M3 stack 1 M N size M3 ans 1 3 8 arraysize M3 ans 8 1 As expected both M and N can be recovered from M3 d3 simplify M3 1 4 M d4 simplify M3 5 8 N max abs d3 max abs d4 ans 0 0 1 63 T Building Uncertain Models Create Arrays by Assignment 1 64 Arrays can be created by direct assignment As with other MATLAB classes there is no need to preallocate the variable first Simply assign elements all resizing is performed automatically For instance an equivalent construction to a b M is Mequiv 1 Mequiv 1 Mequiv 1 1 1 ureal a 4 ureal b 2 stack 1 a
16. e with nominal value 10 and variability 230 230 Name e NominalValue 10 Mode Percentage Range 13 33 PlusMinus 23 23 Percentage 230 230 T Building Uncertain Models AutoSimplify full Specifying conflicting values for Range Percentage PlusMinus when creating a ureal element does not result in an error In this case the last specified property is used This last occurrence also determines the Mode unless Mode is explicitly specified in which case that is used regardless of the property value pairs ordering f ureal f 3 PlusMinus 2 1 Percentage 40 g ureal g 2 PlusMinus 2 1 Mode Range Percentage 40 g Mode f Uncertain real parameter f with nominal value 3 and variability 40 40 Uncertain real parameter g with nominal value 2 and range 1 2 2 8 ans Range Create an uncertain real parameter use usample to generate 1000 instances resulting in a 1 by 1 by 1000 array reshape the array and plot a histogram with 20 bins within the range of 2 to 4 h ureal h 3 hsample usample h 1000 hist reshape hsample 1000 1 20 1 10 Create Uncertain Real Parameters 70 Make the range unsymmetric about the nominal value and repeat the sampling and histogram plot with 40 bins over the range of 2 to 6 h Range 2 6 hsample usample h 1000 hist reshape hsample 1000 1 40 T Building Uncertain Models Note t
17. 2 which is the approach taken in the LMI Lab 3 Introduction to Linear Matrix Inequalities LMI Applications 3 6 Finding a solution x to the LMI system A x lt 0 is called the feasibility problem Minimizing a convex objective under LMI constraints is also a convex problem In particular the linear objective minimization problem Minimize c x subject to A x lt 0 plays an important role in LMI based design Finally the generalized eigenvalue minimization problem Minimize J subject to A x lt AB x B x gt 0 C x gt 0 is quasi convex and can be solved by similar techniques It owes its name to the fact that is related to the largest generalized eigenvalue of the pencil A x B x Many control problems and design specifications have LMI formulations 9 This is especially true for Lyapunov based analysis and design but also for optimal LQG control H control covariance control etc Further applications of LMIs arise in estimation identification optimal design structural design 6 7 matrix scaling problems and so on The main strength of LMI formulations is the ability to combine various design constraints or objectives in a numerically tractable manner A nonexhaustive list of problems addressed by LMI techniques includes the following Robust stability of systems with LTI uncertainty u analysis 24 21 27 Robust stability in the face of sector bounded nonlinearities Popov c
18. 6 d 4 heen Sind eh as he Sh waeng Sh ES 5 2 5 3 5 4 5 4 5 6 5 8 ix x Contents Vary Uncertainty Values Across Multiple Uncertain State Space Blocks 0 0 ccc ce eee eens 5 14 Computing Uncertain State Space Models from Simulink Models neo neer teas ek BS SRA A RE EPEE 5 19 Obtain Uncertain State Space Model from Simulink Model 5 20 Specify Uncertain Linearization for Core or Custom Simulink BIOCKS iiss tekk ed ek ae ae oe ee ee eet 5 21 Linearize Block to Uncertain Model 5 24 Using Uncertain Linearization for Analysis or Control DESI 9 astoa Geshe ald Samet ee ela e tae tt 8 5 26 Analyzing Stability Margin of Simulink Models 5 27 How Stability Margin Analysis Using Loopmargin Differs Between Simulink and LTI Models 5 27 Stability Margin of Simulink Model 5 27 Stability Margin of a Simulink Model 5 29 Building Uncertain Models Introduction to Uncertain Elements on page 1 3 Uncertain Real Parameters on page 1 5 Create Uncertain Real Parameters on page 1 7 Uncertain LTI Dynamics Elements on page 1 14 Create Uncertain LTI Dynamics on page 1 17 Uncertain Complex Parameters and Matrices on page 1 18 Systems with Unmodeled Dynamics on page 1 22 Uncertain Matrices on page 1 23 Create Uncertain Matrices from Uncertain Elements on page 1 24 Access Properti
19. O o i 0 D KA DE cA O O cso O 72 r where each diagonal block D is square and is either zero a full symmetric matrix or a scalar matrix D dxI deR This type encompasses ordinary symmetric matrices single block and scalar variables one block of size one Type 2 Rectangular structure This corresponds to arbitrary rectangular matrices without any particular structure Type 3 General structures This third type is used to describe more sophisticated structures and or correlations between the matrix variables The principle is as follows each entry of X is specified independently as either 0 xn or xn where xn denotes the n th decision variable in the problem For details on how to use Type 3 see Structured Matrix Variables on page 4 31 as well as the lmivar entry in the reference pages In Specifying LMI System on page 4 8 the matrix variables X and S are of Type 1 Indeed both are symmetric and S inherits the block diagonal structure from Equation 4 5 of D Specifically S is of the form Specify LMI System at the Command Line a 0 0 0 0 0 0 s 2 0 0 S2 83 0 0 83 s4 After initializing the description with the command setlmis these two matrix variables are declared by Imivar 1 6 1 X Imivar 1 2 0 2 1 S In both commands the first input specifies the structure type and the second input contains additional information about the structure of the variab
20. analysis commands such as bodeplot bodemag impulse and nyquist 1 38 Feedback Around an Uncertain Plant Feedback Around an Uncertain Plant It is possible to form interconnections of uss objects A common example is to form the feedback interconnection of a given controller with an uncertain plant First create the uncertain plant Start with two uncertain real parameters gamma ureal gamma 4 tau ureal tau 5 Percentage 30 Next create an unmodeled dynamics element delta and a first order weighting function whose DC value is 0 2 high frequency gain is 10 and whose crossover frequency is 8 rad sec delta ultidyn delta 1 1 SampleStateDimension 5 W makeweight 0 2 6 6 Finally create the uncertain plant consisting of the uncertain parameters and the unmodeled dynamics P tf gamma tau 1 1 W delta You can create an integral controller based on nominal plant parameters Nominally the closed loop system will have damping ratio of 0 707 and time constant of 2 tau KI 1 2 tau Nominal gamma Nominal C tf KI 1 0 Create the uncertain closed loop system using the feedback command CLP feedback P C 1 Plot samples of the open loop and closed loop step responses As expected the integral controller reduces the variability in the low frequency response subplot 2 1 1 stepplot P 5 subplot 2 1 2 stepplot CLP 5 1 39 T Building Uncertain Models Step
21. and ipermute on page 1 72 Decomposing Uncertain Objects on page 1 73 Introduction to Uncertain Elements Introduction to Uncertain Elements Uncertain elements also called uncertain Control Design Blocks are the building blocks used to form uncertain matrix objects and uncertain system objects There are 5 classes of uncertain elements Function Description ureal Uncertain real parameter ultidyn Uncertain linear time invariant dynamics ucomplex Uncertain complex parameter ucomplexm Uncertain complex matrix udyn Uncertain dynamic system All of the elements have properties which are accessed through get and set methods This get and set interface mimics the Control System Toolbox and MATLAB Handle Graphics behavior For instance get a PropertyName is the same as a PropertyName and set b PropertyName Value is the same asb PropertyName value Functionality also includes tab completion and case insensitive partial name property matching For ureal ucomplex and ucomplexm elements the syntax is p1 ureal name NominalValue Propi vali Prop2 val2 p2 ucomplex name NominalValue Propi vali Prop2 val2 p3 ucomplexm name NominalValue Propi vali Prop2 val2 For ultidyn and udyn the NominalValue is fixed so the syntax is p4 p5 ultidyn name ioSize Propi vali Prop2 val2 udyn name ioSize Propi vali Prop2 val2 Fo
22. b 1 a b 4 a 4 5 6 a O 0 a b a 0 0 0 stack 1 a b 4 a 4 5 6 Mequiv Mequiv s il A E The easiest manner for you to verify that the results are the same is to subtract and simplify d5 simplify M Mequiv max abs d5 ans 0 Binary Operations with Arrays Binary Operations with Arrays Most operations simply cycle through the array dimensions doing pointwise operations Assume A and B are umat or uss or uf rd arrays with identical array dimensions slot 3 and beyond The operation C fcn A B is equivalent to looping on k1 k2 setting C k1 k2 fon A k1 k2 B k1 k2 The result C has the same array dimensions as A and B The user is required to manage the extra dimensions i e keep track of what they mean Methods such as permute squeeze and reshape are included to facilitate this management In general any binary operation requires that the extra dimensions are compatible The umat uss and ufrd objects allow for slightly more flexible interpretation of this For illustrative purposes consider a binary operation involving variables A and B Suppose the array dimensions of A are nj x xnj and that the array dimensions of B are my X XM By MATLAB convention the infinite number of singleton e 1 trailing dimensions are not listed The compatibility of the extra dimensions is determined by the following rule If A IB then pa
23. basic full char The properties Range Percentage and PlusMinus are all automatically synchronized If the nominal value is 0 then the Mode cannot be Percentage The Mode property controls what aspect of the uncertainty remains unchanged when NominalValue is changed Assigning to any of Range Percentage PlusMinus changes the value but does not change the mode The AutoSimplify property controls how expressions involving the real parameter are simplified Its default value is basic which means elementary methods of simplification are applied as operations are completed Other values for AutoSimplify are Off no simplification performed and full model reduction like techniques are applied See Simplifying Representation of Uncertain Objects on page 1 49 to learn more about the AutoSimplify property and the command simplify If no property value pairs are specified default values are used The default Mode is PlusMinus and the default value of PLlusMinus is 1 1 Some examples are shown 1 5 1 Building Uncertain Models below In many cases the full property name is not specified taking advantage of the case insensitive partial name property matching Create Uncertain Real Parameters Create Uncertain Real Parameters This example shows how to create uncertain real parameters modify properties such as range of uncertainty and sample uncertain parameters Create an uncertain real parameter n
24. called mincx Generalized eigenvalue minimization problem Minimize A over x R subject to C x lt D x 0 lt Bix A x lt AB x The corresponding solver is called gevp Note that A x lt B x above is a shorthand notation for general structured LMI systems with decision variables x x1 xy The three LMI solvers feasp mincx and gevp take as input the internal representation LMISYS of an LMI system and return a feasible or optimizing value x of the decision variables The corresponding values of the matrix variables X4 Xx are derived from x with the function dec2mat These solvers are C MEX implementations of the polynomial time Projective Algorithm Projective Algorithm of Nesterov and Nemirovski 3 2 LMI Solvers For generalized eigenvalue minimization problems it is necessary to distinguish between the standard LMI constraints C x lt D x and the linear fractional LMIs A x lt AB x attached to the minimization of the generalized eigenvalue When using gevp you should follow these three rules to ensure proper specification of the problem Specify the LMIs involving A as A x lt B x without the i Specify them last in the LMI system gevp systematically assumes that the last L LMIs are linear fractional if L is the number of LMIs involving A Add the constraint 0 lt B x or any other constraint that enforces it This positivity constraint is required for well posedne
25. i 1 10 val_all usample uvars sim rct_sim_ex2 10 end During each simulation the software samples values of the uncertain variables input_unc and a and plots the response for the sampled values The Mu1ltiPlot Graph block shows the following responses obtained using random sample values of uncertain variables 5 17 5 Analyzing Uncertainty Effects in Simulink a hed EJ io a a Ig IRA gt a eee eee eee ee eee ee ieee ie ce ee eee fee 5 18 Computing Uncertain State Space Models from Simulink Models Computing Uncertain State Space Models from Simulink Models When you have the Simulink Control Design software you can compute an uncertain linearization i e an uncertain state space model uss combining the uncertain variables with linearized dynamics Use the uss model to perform linear analysis and robust control design You can compute an uncertain linearization in one of the following ways Using the ulinearize command as described in Obtain Uncertain State Space Model from Simulink Model on page 5 20 Using the Simulink Control Design linearize command as described in Specify Uncertain Linearization for Core or Custom Simulink Blocks on page 5 21 5 Analyzing Uncertainty Effects in Simulink Obtain Uncertain State Space Model from Simulink Model To obtain an uncertain state space model from a model that contains Uncertain State Space blocks use
26. identical block structures each block being an affine combination of X4 Xg and their transposes Note Throughout this chapter left side refers to what is on the smaller side of the inequality and right side to what is on the larger side Accordingly X is called the right hand side and 0 the left side of the LMI 0 lt X even when this LMI is written as X gt 0 The specification of an LMI system involves two steps 1 Declare the dimensions and structure of each matrix variable X Xx 2 Describe the term content of each LMI This process creates the so called internal representation of the LMI system This computer description of the problem is used by the LMI solvers and in all subsequent manipulations of the LMI system It is stored as a single vector called LMISYS There are two ways of generating the internal description of a given LMI system 1 by a sequence of lmivar 1lmiterm commands that build it incrementally or 2 via the LMI Editor 1miedit where LMIs can be specified directly as symbolic matrix expressions Though somewhat less flexible and powerful than the command based description the LMI Editor is more straightforward to use hence particularly well suited for beginners Thanks to its coding and decoding capabilities it also constitutes a good tutorial introduction to Lmivar and lmiterm Accordingly beginners may elect to skip the subsections on lmivar and lmiterm and to conce
27. object 1 48 Simplifying Representation of Uncertain Objects Simplifying Representation of Uncertain Objects A minimal realization of the transfer function matrix H s s 1 E s 1 has only 1 state obvious from the decomposition H s s s 1 1 1 2 However a natural construction formed by sys11 sys12 sys21 sys22 sys a x1 x2 x3 x4 b x1 x2 x3 x4 y1 y2 y1 y2 ss tf ss tf ss tf ss tf 6 1 1 sys11 sys12 sy x1 1 c x2 0 1 0 0 NONON 2 1 1 4511 11 3 1 1 s2 1 x3 x4 0 0 0 0 1 0 O 1 x2 x3 x4 2 0 0 0 1 5 3 sys22 1 49 T Building Uncertain Models 1 50 Continuous time model has four states and is nonminimal In the same manner the internal representation of uncertain objects built up from uncertain elements can become nonminimal depending on the sequence of operations in their construction The command simplify employs ad hoc simplification and reduction schemes to reduce the complexity of the representation of uncertain objects There are three levels of simplification off basic and full Each uncertain element has an AutoSimplify property whose value is one of the strings off basic or full The default value is basic After nearly every operation the command simplify is automatically run on the uncertain object cycling through all of the uncertain elements and att
28. on page 3 6 These solvers can handle very general LMI systems and matrix variable structures They return a feasible or optimal vector of decision variables x The corresponding values X ee of the matrix variables are given by the function dec2mat Result Validation The solution x produced by the LMI solvers is easily validated with the functions evallmi and showlmi This allows a fast check and or analysis of the results With evallmi all variable terms in the LMI system are evaluated for the value x of the decision variables The left and right sides of each LMI then become constant matrices that can be displayed with showlmi Modification of a System of LMIs An existing system of LMIs can be modified in two ways e An LMI can be removed from the system with dellmi A matrix variable X can be deleted using delmvar It can also be instantiated that is set to some given matrix value This operation is performed by setmvar and allows for example to fix some variables and solve the LMI problem with respect to the remaining ones Specifying a System of LMls Specifying a System of LMIs The LMI Lab can handle any system of LMIs of the form N LX X N lt MT R X X M where Xi Xgare matrix variables with some prescribed structure The left and right outer factors N and M are given matrices with identical dimensions The left and right inner factors L and R are symmetric block matrices with
29. problem linear objective minimization and generalized eigenvalue minimization e Validate results This chapter gives a tutorial introduction to the LMI Lab as well as more advanced tips for making the most out of its potential Some Terminology Any linear matrix inequality can be expressed in the canonical form L x m Lo xili Pnr xyLy lt 0 where Lo Iy Ly are given symmetric matrices t x 41 xn e RN is the vector of scalar variables to be determined We refer to Xx1 Xy as the decision variables The names design variables and optimization variables are also found in the literature Tools for Specifying and Solving LMIs Even though this canonical expression is generic LMIs rarely arise in this form in control applications Consider for instance the Lyapunov inequality A X XA lt 0 where 1 2 A 0 2 and the variable xy 2 Xo X3 is a symmetric matrix Here the decision variables are the free entries x1 x2 x3 of X and the canonical form of this LMI reads 2 2 0 3 0 0 X1 X9 x3 lt 0 Clearly this expression is less intuitive and transparent than Equation 4 1 Moreover the number of matrices involved in Equation 4 2 grows roughly as n 2 if n is the size of the A matrix Hence the canonical form is very inefficient from a storage viewpoint since it requires storing 0 n 2 matrices of size n when the single n by n matrix A would be sufficient Finally workin
30. sys p1 ureal p1 10 pe 50 ureal p2 3 plusm 5 1 2 p3 ureal p3 0 A p1 p2 0 p1 xo N Il B p2 p2 p3 C 1 0 1 1 p3 D 0 0 sys ss A B C D sys Uncertain continuous time state space model with 2 outputs 1 inputs 2 states The model uncertainty consists of the following blocks pi Uncertain real nominal 10 variability 50 50 2 occurrences p2 Uncertain real nominal 3 variability 0 5 1 2 2 occurrences p3 Uncertain real nominal 0 variability 1 1 2 occurrences Type sys NominalValue to see the nominal value get sys to see all properties an Compute the uncertain frequency response of the uncertain system Use the ufrd command along with a frequency grid containing 100 points The result is an uncertain frequency response data object referred to as a ufrd model sysg ufrd sys logspace 2 2 100 Uncertain continuous time FRD model with 2 outputs 1 inputs 100 frequency points pi Uncertain real nominal 10 variability 50 50 2 occurrences p2 Uncertain real nominal 3 variability 0 5 1 2 2 occurrences p3 Uncertain real nominal 0 variability 1 1 2 occurrences Type sysg NominalValue to see the nominal value get sysg to see all properties i 1 43 T Building Uncertain Models Properties of ufrd Objects View the properties with the get command get sysg Frequency FrequencyUnit ResponseData
31. the Projective Algorithm of Nesterov and Nemirovski 20 19 In addition to its polynomial time complexity this algorithm does not require an initial feasible point for the linear objective minimization problem Equation 3 3or the generalized eigenvalue minimization problem Equation 3 4 Some LMI problems are formulated in terms of inequalities rather than strict inequalities For instance a variant of Equation 3 3 is Minimize c x subject to A x lt 0 While this distinction is immaterial in general it matters when A x can be made negative semi definite but not negative definite A simple example is Minimize c x subject to x x gt 0 Such problems cannot be handled directly by interior point methods which require strict feasibility of the LMI constraints A well posed reformulation of Equation 3 5 would be Minimize c x subject to x gt 0 Keeping this subtlety in mind we always use strict inequalities in this manual Bibliography Bibliography 1 Anderson B D O and S Vongpanitlerd Network Analysis Prentice Hall Englewood Cliffs 1973 2 Apkarian P P Gahinet and G Becker Self Scheduled H Control of Linear Parameter Varying Systems Proc Amer Contr Conf 1994 pp 856 860 3 Bambang R E Shimemura and K Uchida Mixed H H Control with Pole Placement State Feedback Case Proc Amer Contr Conf 1993 pp 2777 2779 4 Barmish B R Stabilization of Unce
32. the following steps Note If you do not have Uncertain State Space blocks in the model but still want to obtain an uncertain state space model see Specify Uncertain Linearization for Core or Custom Simulink Blocks on page 5 21 1 Prerequisite Create or open the Simulink model 2 Prerequisite In the Simulink model specify the linearization input and output points using Simulink Control Design getlinio or linio commands For more information see Specifying Portion of Model to Linearize in the Simulink Control Design documentation 3 Prerequisite If you have not already done so specify uncertainty in the Simulink model as described in Specify Uncertainty Using Uncertain State Space Blocks on page 5 4 Note The software does not evaluate the uncertain variables during linearization Thus the value of the uncertainty does not affect the linearization 4 Runulinearize to compute an uncertain linearization This command returns an uss model Note If you use the Simulink Control Designlinearize command the Uncertain State Space blocks linearize to their nominal value For more information on linearization and how to evaluate the results see Linearization Basics in the Simulink Control Design documentation For an example of how to use the Simulink Control Design linearize command see Linearization of Simulink Models with Uncertainty 5 20 Specify Uncertain Linearization for Core or Cu
33. unc p1 nom 14 UFRD 2 Outputs 1 Input Continuous System 100 Frequency points p1 real nominal 14 variability 50 50 2 occurrences p2 real nominal 3 variability 0 5 1 2 2 occurrences p3 real nominal 0 variability 1 1 2 occurrences 1 45 T Building Uncertain Models Interpreting Uncertainty in Discrete Time See Interpreting Uncertainty in Discrete Time on page 1 41 The issues are identical 1 46 Lifting an frd to a ufrd Lifting an frd to a ufrd A not uncertain frequency response object may be interpreted as an uncertain frequency response object that has no dependence on uncertain elements Use the ufrd command to lift an frd object to the uf rd class sys rss 3 2 1 sysg frd sys logspace 2 2 100 usysg ufrd sysg UFRD 2 Outputs 1 Input Continuous System 100 Frequency points Arrays of frd objects can also be lifted See Array Management for Uncertain Objects on page 1 60 for more information about how arrays of uncertain objects are handled 1 47 T Building Uncertain Models Basic Control System Toolbox and MATLAB Interconnections This list has all of the basic system interconnection functions defined in Control System Toolbox software or in MATLAB append blkdiag series parallel feedback lft stack These functions work with uncertain objects as well Uncertain objects may be combined with certain objects resulting in an uncertain
34. ureal eta 6 rho ureal rho 1 A 3 deltateta delta eta 7 rho rhotdelta eta UMAT 2 Rows 2 Columns delta real nominal 2 variability eta real nominal 6 variability rho real nominal 1 variability 2 occurrences 3 occurrences 1 occurrence 1 1 1 1 1 1 1 75 T Building Uncertain Models Note that A depends on two occurrences of delta three occurrences of eta and one occurrence of rho Decompose A into M and Delta Note that Ais a double and Delta has the same uncertainty dependence as M M Delta lftdata A class M ans double Delta UMAT 6 Rows 6 Columns delta real nominal eta real nominal rho real nominal 2 occurrences 3 occurrences 1 occurrence 2 variability 6 variability 1 variability 1 1 1 1 1 1 Sample Delta at 5 points Things to note are Itis diagonal The values range between 1 and 1 There are three independent values and duplication of the entries is consistent with the dependence of Delta and A on the three uncertain real parameters usample Delta 5 ans 1 0 7106 0 0 0 0 0 0 0 7106 0 0 0 0 0 0 0 6374 0 0 0 0 0 0 0 6374 0 0 0 0 0 0 0 6374 0 0 0 0 0 0 0 1258 ans 2 0 5850 0 0 0 0 0 0 0 5850 0 0 0 0 0 0 0 3021 0 0 0 0 0 0 0 3021 0 0 0 0 0 0 0 3021 0 0 0 0 0 0 0 0803 ans 3 0 7013 0 0 0 0 0 0 0 7013 0 0 0 0 0 0 0 6749 0 0 0 0 0 0 0 674
35. where x is the vector of free entries of X Since c should select the diagonal entries of X it is obtained as the decision vector corresponding to X J that is c mat2dec LMIs eye 3 Note that the function defcx provides a more systematic way of specifying such objectives see Specifying c x Objectives for mincx on page 4 36 for details Call mincx to compute the minimizer xopt and the global minimum copt c xopt of the objective options 1e 5 0 0 0 0 copt xopt mincx LMIs c options Here 1e 5 specifies the desired relative accuracy on copt The following trace of the iterative optimization performed by mincx appears on the screen Solver for linear objective minimization under LMI constraints Iterations Best objective value so far 1 2 8 511476 3 13 063640 EAR new lower bound 34 023978 4 15 768450 atte new lower bound 25 005604 5 17 123012 os new lower bound 21 306781 6 17 882558 KER new lower bound 19 819471 7 18 339853 4 23 4 IMILab ne new lower bound 19 189417 8 18 552558 sted new lower bound 18 919668 9 18 646811 Rola new lower bound 18 803708 10 18 687324 ao new lower bound 18 753903 11 18 705715 Piska new lower bound 18 732574 12 18 712175 Erk new lower bound 18 723491 13 18 714880 ie new lower bound 18 719624 14 18 716094 shies new lower
36. 1 refers to x while sY 2 2 refers to x4 this defines the variable Z 0 TX E 0 x x4 0 t 0 as confirmed by checking its entry wise dependence on the decision variables sZ 0o 1 4 0 Complex Valued LMIs The LMI solvers are written for real valued matrices and cannot directly handle LMI problems involving complex valued matrices However complex valued LMIs can be turned into real valued LMIs by observing that a complex Hermitian matrix L x satisfies L x lt 0 if and only if Re L x Im L x Im L x Re L x j This suggests the following systematic procedure for turning complex LMIs into real ones 4 33 4 IMILab 4 34 e Decompose every complex matrix variable X as X X jXe where X and X are real Decompose every complex matrix coefficient A as A A jA2 where A and A are real Carry out all complex matrix products This yields affine expressions in Xj X for the real and imaginary parts of each LMI and an equivalent real valued LMI is readily derived from the above observation For LMIs without outer factor a streamlined version of this procedure consists of replacing any occurrence of the matrix variable X X jX by Xi X X gt XY and any fixed matrix A A jAg including real ones by A Ap A2 Ay For instance the real counterpart of the LMI system M XM lt X X X gt I reads given the decompositions M M jM and X X jX_ with Mj Xj real
37. 15 class Mss ans double The above 2 step sequence can be performed in 1 step Mss values usample M b 20 a 15 class Mss ans double Create Arrays with usample In this case values is a 20 by 15 struct array with 2 fields b and a whose values are the values used in the random sampling It follows that usubs M values is the same as Mss Rather than sampling the each variable a and b independently generating a 20 by 15 grid in a 2 dimensional space the two dimensional space can be sampled Sample the 2 dimensional space with 800 points Ms values usample M a b 800 size Ms ans 3 2 800 size values ans 800 1 1 67 T Building Uncertain Models Create Arrays with usubs 1 68 Suppose Values is a Struct array with the following properties the dimensions of Value match the array dimensions of M as described in Create Arrays with usample on page 1 66 the field names of Values are some or all of the names of the uncertain elements of M and the dimensions of the contents of the fields within Values match the sizes of the uncertain elements within M Then usubs M Values will substitute the uncertain elements in M with the contents found in the respective fields of Values You can create a 3 by 2 uncertain matrix using two uncertain real parameters ureal a 4 ureal b 2 a b b b a b 1 b 1 a b a b M Create a 5 by 1 struct array with field name a M
38. 4 37 4 IMILab 4 38 Well Posedness Issues The LMI solvers used in the LMI Lab are based on interior point optimization techniques To compute feasible solutions such techniques require that the system of LMI constraints be strictly feasible that is the feasible set has a nonempty interior As a result these solvers may encounter difficulty when the LMI constraints are feasible but not strictly feasible that is when the LMI L x lt 0 has solutions while L x lt 0 has no solution For feasibility problems this difficulty is automatically circumvented by feasp which reformulates the problem Find x such that L x lt 0 as Minimize subject to Ix lt t XI In this modified problem the LMI constraint is always strictly feasible in x t and the original LMI Equation 4 12 is feasible if and only if the global minimum tmin of Equation 4 12 satisfies tmin lt 0 For feasible but not strictly feasible problems however the computational effort is typically higher as feasp strives to approach the global optimum tmin 0 to a high accuracy For the LMI problems addressed by mincx and gevp nonstrict feasibility generally causes the solvers to fail and to return an infeasibility diagnosis Although there is Advanced LMI Techniques no universal remedy for this difficulty it is sometimes possible to eliminate underlying algebraic constraints to obtain a strictly feasible problem with fewer variables
39. 785 1 4375 6 0290i 35 7428 Generating Many Samples If A is an uncertain object then usample A N generates N samples of A For example 20 samples of a ureal gives a 1 by 1 20 double array B usample A 20 size B ans 1 1 20 Similarly 30 samples of the 1 by 3 umat M yields a 1 by 3 by 30 array size usample M 30 ans 1 53 T Building Uncertain Models 1 54 1 3 30 See Create Arrays with usample on page 1 66 for more information on sampling uncertain objects Sampling ultidyn Elements When sampling an ultidyn element or an uncertain object that contains a ultidyn element the result is always a state space SS object The property SampleStateDimension of the ultidyn class determines the state dimension of the samples Create a 1 by 1 gain bounded ultidyn object with gain bound 3 Verify that the default state dimension for samples is 1 del ultidyn del 1 1 Bound 3 del SampleStateDimension ans Sample the uncertain element at 30 points Verify that this creates a 30 by 1 ss array of l input 1 output 1 state systems delS usample del 30 size delS 30x1 array of state space models Each model has 1 outputs 1 inputs and 1 states Plot the Nyquist plot of these samples and add a disk of radius 3 Note that the gain bound is satisfied and that the Nyquist plots are all circles indicative of 1st order systems nyquist dels hold on theta linspace pi pi pl
40. 9 0 0 1 76 Decomposing Uncertain Objects 0 0 0 0 0 6749 0 0 0 0 0 0 0 3967 ans 4 0 4262 0 0 0 0 0 0 0 4262 0 0 0 0 0 0 0 0795 0 0 0 0 0 0 0 0795 0 0 0 0 0 0 0 0795 0 0 0 0 0 0 0 9959 ans 5 0 8392 0 0 0 0 0 0 0 8392 0 0 0 0 0 0 0 8467 0 0 0 0 0 0 0 8467 0 0 0 0 0 0 0 8467 0 0 0 0 0 0 0 6732 In fact verify that the maximum gain of Delta is indeed 1 maxnorm wcnorm Delta maxnorm LowerBound 1 0000 UpperBound 1 0004 Finally verify that 1ft Delta M is the same as A Subtract and use the full option in simplify simplify 1ft Delta M A full ans 0 0 0 0 Advanced Syntax of Iftdata Even for the advanced user the variable Delta will actually not be that useful as it is still a complex object On the other hand its internal structure is described completely using a 3rd and 4th output argument M Delta BlkStruct NormUnc 1ftdata A The rows of BLkStruct correspond to the uncertain elements named in fieldnames A Uncertainty Note that the range bound information about each uncertain element is not included in B1LkStruct 1 77 T Building Uncertain Models 1 78 The elements of BLkStruct describe the size type and number of copies of the uncertain elements in A and implicitly delineate the exact block diagonal structure of Delta Note that the range bound information about each uncertain element is not included in BlkStruct BlkStruct 1 ans N
41. Advanced LMI Techniques Advanced LMI Techniques This last section gives a few hints for making the most out of the LMI Lab It is directed toward users who are comfortable with the basics as described in Tools for Specifying and Solving LMIs on page 4 2 Structured Matrix Variables Fairly complex matrix variable structures and interdependencies can be specified with lmivar Recall that the symmetric block diagonal or rectangular structures are covered by Types 1 and 2 of lmivar provided that the matrix variables are independent To describe more complex structures or correlations between variables you must use Type 3 and specify each entry of the matrix variables directly in terms of the free scalar variables of the problem the so called decision variables With Type 3 each entry is specified as either 0 or x where x is the n th decision variable The following examples illustrate how to specify nontrivial matrix variable structures with lmivar First consider the case of uncorrelated matrix variables Example Specifying Matrix Variable Structures Suppose that the problem variables include a 3 by 3 symmetric matrix X and a 3 by 3 symmetric Toeplitz matrix y Y2 Y3 Y Y2 Yi Yo 3 Y2 X The variable Y has three independent entries hence involves three decision variables Since Y is independent of X these decision variables should be labeled n 1 n 2 n 3 where n is the number of decision variables involved in
42. Ctrl X Copy Ctrl C Paste Ctrl V Comment Out Ctrl Shift X Delete Del Find Referenced Variables Create Subsystem from Selection Ctrl G Convert Subsystem to gt Format gt Rotate amp Flip Arrange gt Mask Library Link gt Signals amp Ports gt Requirements Linear Analysis gt Coverage gt V Specify block linearization using one of the following Enter an expression to specify the linearization of the Simulink block plant 1 wt input_unc ox cancel Help Apoy 5 22 Specify Uncertain Linearization for Core or Custom Simulink Blocks In the Block Linearization Specification dialog box select the Specify block linearization using one of the following check box Selecting this check box lets you to specify an uncertain variable for linearization This check box defaults to MATLAB Expression in the drop down menu This option lets you specify the block to linearize to an uncertain variable using a MATLAB expression containing Robust Control Toolbox functions To learn more about the options see Controlling Block Linearization in the Simulink Control Design documentation In the Enter an expression to specify the linearization of the Simulink block field enter an expression which must evaluate to an uncertain variable or uncertain model such as ureal umat ultidyn or uss Block Linearization Specification Plant rx Se V Specify block li
43. Response Amplitude 0 0 5 1 15 2 25 3 3 5 4 45 5 Time seconds Step Response Amplitude 0 0 5 1 15 2 25 3 3 5 4 45 5 Time seconds 1 40 Interpreting Uncertainty in Discrete Time Interpreting Uncertainty in Discrete Time The interpretation of a ultidyn element as a continuous time or discrete time system depends on the nature of the uncertain system USS within which it is an uncertain element For example create a scalar ultidyn object Then create two 1 input 1 output uss objects using the ultidyn object as their D matrix In one case create without specifying sample time which indicates continuous time In the second case force discrete time with a sample time of 0 42 delta ultidyn delta 1 1 sys1 uss delta USS O States 1 Output 1 Input Continuous System delta 1x1 LTI max gain 1 1 occurrence sys2 uss delta 0 42 USS O States 1 Output 1 Input Discrete System Ts 0 42 delta 1x1 LTI max gain 1 1 occurrence Next get a random sample of each system When obtaining random samples using usamp1e the values of the elements used in the sample are returned in the 2nd argument from usamp1e as a structure sysis div usample sys1 sys2s d2v usample sys2 Look at div delta Ts and d2v delta Ts In the first case since Sy 1 is continuous time the system d1v del1ta is continuous time In the second case since SyS2 is discrete time with sam
44. Robust Control Toolbbox User s Guide Gary Balas Richard Chiang Andy Packard Michael Safonov MATLAB R201 5b J MathWorks How to Contact MathWorks Latest news www mathworks com Sales and services www mathworks com sales_and_ services User community www mathworks com matlabcentral Technical support www mathworks com support contact_us Phone 508 647 7000 The MathWorks Inc 3 Apple Hill Drive Natick MA 01760 2098 Robust Control Toolbox User s Guide COPYRIGHT 2005 2015 by The MathWorks Inc The software described in this document is furnished under a license agreement The software may be used or copied only under the terms of the license agreement No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks Inc FEDERAL ACQUISITION This provision applies to all acquisitions of the Program and Documentation by for or through the federal government of the United States By accepting delivery of the Program or Documentation the government hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer software documentation as such terms are used or defined in FAR 12 212 DFARS Part 227 72 and DFARS 252 227 7014 Accordingly the terms and conditions of this Agreement and only those rights specified in this Agreement shall pertain to and govern the use modification reproduction release performanc
45. X 1 1 3rd LMI lmiterm 3 1 1 S 1 1 lmiterm 3 1 1 0 1 LMISYS getlmis Here the I1mivar commands define the two matrix variables X and S while the I1miterm commands describe the various terms in each LMI Upon completion getlmis returns the internal representation LMISYS of this LMI system The following subsections give more details on the syntax and usage of these various commands Initializing the LMI System on page 4 9 Specifying the LMI Variables on page 4 10 Specifying Individual LMIs on page 4 12 More information on how the internal representation is updated by lmivar lmiterm can also be found in How Imivar and Imiterm Manage LMI Representation on page 4 18 Initializing the LMI System The description of an LMI system should begin with setlmis and end with getlmis The function set1mis initializes the LMI system description When specifying a new system type setlmis 4 9 4 IMILab 4 10 To add on to an existing LMI system with internal representation LMISO type setlmis LMISO Specifying the LMI Variables The matrix variables are declared one at a time with lmivar and are characterized by their structure To facilitate the specification of this structure the LMI Lab offers two predefined structure types along with the means to describe more general structures Type 1 Symmetric block diagonal structure This corresponds to matrix variables of the form D
46. X To retrieve this number define the variable X Type 1 by setlmis X n lmivar 1 3 1 The second output argument n gives the total number of decision variables used so far heren 6 Given this number Y can be defined by Y lmivar 3 n 1 2 3 2 1 2 3 2 1 4 31 4 IMILab 4 32 or equivalently by Y Imivar 3 toeplitz n 1 2 3 where toeplitz is a standard MATLAB function For verification purposes we can visualize the decision variable distributions in X and Y with decinfo lmis getlmis decinfo lmis X ans n 1 2 4 ow oof decinfo lmis Y ans ooN oN OO N O The next example is a problem with interdependent matrix variables Example Specifying Interdependent Matrix Variables Consider three matrix variables X Y Z with structure x 0 z 0 0 x X Y Z 0 y 0 t 0 where x y z t are independent scalar variables To specify such a triple first define the two independent variables X and Y both of Type 1 as follows setlmis X n SX lmivar 1 1 0 1 0 Y n sY lmivar 1 1 0 1 0 The third output of lmivar gives the entry wise dependence of X and Y on the decision variables x1 x2 X3 X4 x y zZ t sX Advanced LMI Techniques 1 0 0 2 sY 3 0 0 4 Using Type 3 of lmivar you can now specify the structure of Z in terms of the decision variables x x and x t Z n sZ lmivar 3 0 sX 1 1 sY 2 2 0 Since X 1
47. ability margins computed using the linearized model 1 Open the Simulink model airframemargin Fin Deflection az Control q Control Airframe Model az Response 2 Define a margin analysis point at the output of the Airframe Model block blocki airframemargin Airframe Model 3 Specify the output az of the Airframe Model block as a margin analysis point porti 1 4 Compute stability margins cm dm mm loopmargin airframemargin block1 port1 5 View the classical gain and phase margins cm cm 5 29 5 Analyzing Uncertainty Effects in Simulink 5 30 GainMargin 4 5652 2 5055e 003 GMFrequency 7 1979 314 1593 PhaseMargin 65 1907 PMFrequency 2 1463 DelayMargin 53 0113 DMFrequency 2 1463 Stable 1 Compare the classical gain and phase margins cm with stability margins of the linearized model computed using allmargin Define linearization I O points io linio airframemargin Airframe Model 1 looptransfer Linearize the model lin_sys linearize airframemargin io Compute gain and phase margins cm_lin allmargin lin_sys cm_lin GainMargin 4 5652 2 5055e 003 GMFrequency 7 1979 314 1593 PhaseMargin 65 1907 PMFrequency 2 1463 DelayMargin 53 0113 DMFrequency 2 1463 Stable 1 The gain and phase margins cm and cm_1lin computed using the two approaches match
48. ake its values random scalars Create a 1 by 4 struct array with field name b struct a num2cell rand 5 1 struct b num2cell rand 1 4 Avalue Bvalue Substitute the uncertain real parameter a in M with the values in Avalue yielding ma Similarly substitute the uncertain real parameter b in M with the values in Avalue yielding mb ma usubs M Avalue UMAT 3 Rows 2 Columns array 5 x 1 b real nominal 2 variability 1 1 6 occurrences mb usubs M Bvalue UMAT 3 Rows 2 Columns array 1 x 4 a real nominal 4 variability 1 1 2 occurrences Continue substituting the uncertain real parameter b in ma with the values in Bvalue yielding mab Do the analogous operation for mb yielding mba Subtract and note that the difference is 0 as expected mab usubs ma Bvalue mba usubs mb Avalue thediff mab mba max abs thediff ans 4 4409e 016 Create Arrays with gridureal Create Arrays with gridureal The command gridureal enables uniform sampling of specified uncertain real parameters within an uncertain object It is a specialized case of uSubs gridureal removes a specified uncertain real parameter and adds an array dimension to the end of the existing array dimensions The new array dimension represents the uniform samples of the uncertain object in the specified uncertain real parameter range Create a 2 by 2 uncertain matrix with three uncertain real parameters a ureal
49. ame delta Size 1 1 Type ureal Occurrences 2 BlkStruct 2 ans Name eta Size 1 1 Type ureal Occurrences 3 BlkStruct 3 ans Name rho Size 1 1 Type ureal Occurrences 1 Together these mean Delta is a block diagonal augmentation of the normalized version of 3 uncertain elements The first element is named delta Itis 1 by 1 it is of class ureal and there are 2 copies diagonally augmented The second element is named eta It is 1 by 1 it is of class ureal and there are 3 copies diagonally augmented The third element is named rho It is 1 by 1 it is of class ureal and there is 1 copy The 4th output argument contains a cell array of normalized uncertain elements The cell array contains as many occurrences of each element as there are occurrences in the original uncertain object A size NormUnc ans 6 1 NormUnc 1 Uncertain Real Parameter Name deltaNormalized NominalValue 0 Decomposing Uncertain Objects variability 1 1 isequal NormUnc 2 NormUnc 1 ans 1 NormUnc 3 Uncertain Real Parameter Name etaNormalized NominalValue O variability 1 1 isequal NormUnc 4 NormUnc 3 ans 1 isequal NormUnc 5 NormUnc 3 ans 1 NormUnc 6 Uncertain Real Parameter Name rhoNormalized NominalValue O variability 1 1 Each normalized element has the string Normalized appended to its original name to avoid confusion By normaliz
50. and behavior of repsys for uss and ufrd objects are the same as the traditional repsys which operates on s and frd objects Just as in those cases the uncertain version of repsys also allows for diagonal tiling 1 71 T Building Uncertain Models Using permute and ipermute 1 72 The commands permute and ipermute are generalizations of transpose which exchanges the rows and columns of a two dimensional matrix permute A ORDER rearranges the dimensions of A so that they are in the order specified by the vector ORDER The array produced has the same values of A but the order of the subscripts needed to access any particular element are rearranged as specified by ORDER The elements of ORDER must be a rearrangement of the numbers from 1 to N All of the uncertain objects are essentially 2 dimensional output and input operators with array dependence This means that the first 2 dimensions are treated differently from dimensions 3 and beyond It is not permissible to permute across these groups For uss and ufrd the restriction is built into the syntax The elements of the ORDER vector only refer to array dimensions Therefore there is no possibility of permute across these dimensions In you need to permute the first two dimensions use the command transpose instead For umat the restriction is enforced in the software The elements of the ORDER vector refer to all dimensions However the first two elements of ORDER must be a r
51. ate Uncertain Matrices from Uncertain Elements 1 24 Access Properties of a umat 0 0 00 0 cece 1 25 Row and Column Referencing 5 1 27 Matrix Operation on umat Objects 1 29 Evaluate Uncertain Elements by Substitution 1 31 Lifting a double matrix to a umat 00005 1 32 vi Contents Uncertain State Space Models uss Properties of uss Objects Create Uncertain State Space Model Sample Uncertain Systems Feedback Around an Uncertain Plant Interpreting Uncertainty in Discrete Time Lifting asstoauss Create Uncertain Frequency Response Data Models Properties of ufrd Objects Interpreting Uncertainty in Discrete Time Lifting an frd toaufrd Basic Control System Toolbox and MATLAB Interconnections Simplifying Representation of Uncertain Objects Effect of the Autosimplify P Direct Use of simplify LOPCYUY ese ares ste es ee ee Generate Samples of Uncertain Systems Generating One Sample Generating Many Samples Sampling ultidyn Elements Substitution by usubs Specifying the Substitution with Structures Nominal and Random Values 2 0000 eee Array Management for Uncertain Objects Reference Into Arrays Create Arrays with sta
52. bound 18 717986 15 18 716509 rates new lower bound 18 717297 16 18 716695 ai new lower bound 18 716873 Result feasible solution of required accuracy best objective value 18 716695 guaranteed relative accuracy 9 50e 06 f radius saturation 0 000 of R 1 00e 09 The iteration number and the best value of c x at the current iteration appear in the left and right columns respectively Note that no value is displayed at the first iteration which means that a feasible x satisfying the constraint Equation 4 10 was found only at the second iteration Lower bounds on the global minimum of c x are sometimes detected as the optimization progresses These lower bounds are reported by the message xxx new lower bound xxx 4 24 Minimize Linear Objectives under LMI Constraints Upon termination mincx reports that the global minimum for the objective Trace X is 18 716695 with relative accuracy of at least 9 5x10 This is the value copt returned by mincx mincx also returns the optimizing vector of decision variables xopt The corresponding optimal value of the matrix variable X is given by Xopt dec2mat LMIs xopt X which returns 6 3542 5 8895 2 2046 Xopt 5 8895 6 2855 2 2201 2 2046 2 2201 6 0771 This result can be compared with the stabilizing Riccati solution computed by Care Xst care a b q 1 norm Xopt Xst ans 6 5390e 05 4 25 4 IMILab Conversion Between D
53. c behavior of the performance degradation curve Robust Performance Margin Robust Performance Margin The hyperbola is used to define the performance margin Systems whose performance degradation curve intersects high on the hyperbola curve represent non robustly performing systems in that very small deviations of the uncertain elements from their nominal values can result in very large system gains Conversely an intersection low on the hyperbola represent robustly performing systems Robust Performance Margin 3 M oO ie System performance degradation curve M y 1 x curve in uncertainty size vs performance tradeoff space ww gt FZ a a a Maximum System Gain over Uncertainty oO n n 0 0 5 1 1 5 2 25 3 Bound on Normalized Uncertainty The point where the system performance degradation curve crosses the green line is used as a scalar measure of the robustness of a system to uncertainty The horizontal coordinate of the crossing point is the robust performance margin PerfMarg 2 Generalized Robustness Analysis Worst Case Gain Measure 2 6 The worst case gain measure is the maximum achievable system gain over all uncertain elements whose normalized size is bounded by 1 3 WC Gain uses bourd of 1 on normalized uncertainty x M or ho ee System performance degradation curve WC Gain 1 22 Maximum System Gain o
54. ck and Create Arrays by Assignment cat Functions 1 33 1 33 1 36 1 37 1 39 1 41 1 42 1 43 1 44 1 46 1 47 1 48 1 49 1 50 1 52 1 53 1 53 1 53 1 54 1 57 1 58 1 58 1 60 1 61 1 62 1 64 Binary Operations with Arrays 00 005 1 65 Create Arrays with usample 0005 1 66 Create Arrays with usubs 0 0 0 0 cece 1 68 Create Arrays with gridureal 00005 1 69 Create Arrays with repmat 0 000005 1 70 Create Arrays with repsys 0 0 00 eens 1 71 Using permute and ipermute 0000 0s 1 72 Decomposing Uncertain Objects 004 1 73 Normalizing Functions for Uncertain Elements 1 73 Properties of the Decomposition 0000005 1 74 Syntax of Iftdatae sewer ound ab ha bo Ae Oe ard we eds 1 75 Advanced Syntax of Iftdata 2 0 2 ce ee 1 77 Generalized Robustness Analysis 2 Generalized Robustness Analysis 5 2 2 Robust Stability Margin 0 0 0 0 ce eee 2 4 Robust Performance Margin 0 000055 2 5 Worst Case Gain Measure eee eee ees 2 6 Introduction to Linear Matrix Inequalities 3 Linear Matrix Inequalities 0005 3 2 GML Weatures 2 roede an aa 28 a dete be dean ee Sag hE hae 3 2 vii viii Contents LMIs and LMI Problems
55. cks a Uncertain real nominal 3 variability 0 5 2 2 occurrences b Uncertain real nominal 10 variability 20 20 2 occurrences c Uncertain real nominal 3 variability 40 40 1 occurrences Type M NominalValue to see the nominal value get M to see all properties and M 1 28 Matrix Operation on umat Objects Matrix Operation on umat Objects Many matrix operations are allowed such as matrix multiply transpose and inverse Combinations of certain i e not uncertain matrices and uncertain matrices are allowed Premultiply M by a 1 by 3 constant matrix resulting in a 1 by 2 umat M1 2 3 1 M M1 Uncertain matrix with 1 rows and 2 columns The uncertainty consists of the following blocks a Uncertain real nominal 3 variability 0 5 2 1 occurrences b Uncertain real nominal 10 variability 20 20 2 occurrences c Uncertain real nominal 3 variability 40 40 1 occurrences Type M1 NominalValue to see the nominal value get M1 to see all properties and Verify that the 1st entry of M1 is 2 a 3 b 1 Direct subtraction yields a umat without any dependence on uncertain elements Simplifying the class shows that the result has no uncertain elements as expected d M1 1 2 M Uncertainty a 3 M Uncertainty b 1 d Uncertain matrix with 1 rows 1 columns and no uncertain blocks Type d NominalValue to see the nominal value get d to see all propert
56. d are interpreted as array dimensions See Array Management for Uncertain Objects on page 1 60 for more information about how multidimensional arrays of uncertain objects are handled Md randn 4 5 6 M umat Md M umat Md M 6x1 array of uncertain matrices with 4 rows 5 columns and no uncertain blocks Type M NominalValue to see the nominal value get M to see all properties and M Md randn 4 5 6 7 M umat Md M 6x7 array of uncertain matrices with 4 rows 5 columns and no uncertain blocks Type M NominalValue to see the nominal value get M to see all properties and M Uncertain State Space Models uss Uncertain State Space Models uss Uncertain systems USS are linear systems with uncertain state space matrices and or uncertain linear dynamics Like their certain 1 e not uncertain counterpart the ss object they are often built from state space matrices using the ss command In the case where some of the state space matrices are uncertain elements uncertain Control Design Blocks the result will be a uncertain state space USS object Combining uncertain systems with uncertain systems with the feedback command for example usually leads to an uncertain system Not uncertain systems can be combined with uncertain systems Usually the result is an uncertain system The nominal value of an uncertain system is a SS object which is familiar to Control System Toolbox softwa
57. d as the scalar a This also applies to the coefficients P and Q of variable terms The dimensions of scalar matrices are inferred from the context and set to 1 by default For instance the third LMI S gt Jin Example Specifying Matrix Variable Structures on page 4 31 is described by Imiterm 3 1 1 2 1 1 S xS 1 lmiterm 3 1 1 0 1 1 1 I I Recall that by convention S is considered as the right side of the inequality which justifies the 3 in the first command Finally to improve readability it is often convenient to attach an identifier tag to each LMI and matrix variable The variable identifiers are returned by lmivar and the LMI identifiers are set by the function newlmi These identifiers can be used in lmiterm commands to refer to a given LMI or matrix variable For the LMI system of Specifying LMI System on page 4 8 this would look like setlmis X lmivar 1 6 1 S lmivar 1 2 0 2 1 4 13 4 IMILab BRL newlmi lmiterm BRL lmiterm BRL lmiterm BRL lmiterm BRL Xpos newlmi lmiterm Xpo Slmi newlmi Naaa S 1 X 1 A 8 1 S C C 2 X 1 B 2 S 1 1 1 1 X 1 1 lmiterm Slmi 1 1 S 1 1 lmiterm Slmi 1 1 0 1 When the LMI system is completely specified type LMISYS getlmis This returns the internal representation LMISYS of this LMI system This MATLAB description of the problem can be forwarded to other LMI Lab funct
58. d the properties of each element within a umat can be changed directly Using usubs specific values may be substituted for any of the uncertain elements within a umat The command usamp1e generates a random sample of the uncertain matrix substituting random samples within their ranges for each of the uncertain elements The command wcnorm computes tight bounds on the worst case maximum over the uncertain elements ranges norm of the uncertain matrix Standard MATLAB numerical matrices i e double naturally can be viewed as uncertain matrices without any uncertainty 1 23 1 Building Uncertain Models Create Uncertain Matrices from Uncertain Elements 1 24 You can create 2 uncertain real parameters and then a 3 by 2 uncertain matrix using these uncertain elements a ureal a 3 b ureal b 10 pe 20 M a 1 b b at 1 b 1 3 M Uncertain matrix with 3 rows and 2 columns The uncertainty consists of the following blocks a Uncertain real nominal 3 variability 1 1 2 occurrences b Uncertain real nominal 10 variability 20 20 3 occurrences Type M NominalValue to see the nominal value get M to see all properties and M Access Properties of a umat Access Properties of a umat Use get to view the accessible properties of a umat get M NominalValue 3x2 double Uncertainty 1x1 elementlist The NominalValue is a double obtained by replacing all uncertain element
59. d the shorter dimension list with trailing 1 s Now compare the extra dimensions In the k th dimension it must be that one of 3 conditions hold nk mk or nk 1 or mk 1 In other words non singleton dimensions must exactly match so that the pointwise operation can be executed and singleton dimensions match with anything implicitly through a repmat 1 65 T Building Uncertain Models Create Arrays with usample 1 66 An extremely common manner in which to generate an array is to sample in some of the uncertain elements an uncertain object Using the ureal objects a and b from above create a 2 by 3 umat M a b b b a b 1 b 1 a b UMAT 3 Rows 2 Columns a real nominal 4 variability b real nominal 2 variability size M ans 3 2 1 1 3 occurrences 1 1 6 occurrences Sample at 20 random points within its range the uncertain real parameter b in the matrix M This results in a 3 by 2 by 20 umat with only one uncertain element a The uncertain element b of M has been sampled out leaving a new array dimension in its place Ms bvalues usample M b 20 Ms UMAT 3 Rows 2 Columns array 20 x 1 a real nominal 4 variability 1 1 2 occurrences size Ms ans 3 2 20 Continue sampling at 15 random points within its range the uncertain real parameter a in the matrix Ms This results in a 3 by 2 by 20 by 15 double Mss avalues usample Ms a 15 size Mss ans 3 2 20
60. e display and disclosure of the Program and Documentation by the federal government or other entity acquiring for or through the federal government and shall supersede any conflicting contractual terms or conditions If this License fails to meet the government s needs or is inconsistent in any respect with federal procurement law the government agrees to return the Program and Documentation unused to The MathWorks Inc Trademarks MATLAB and Simulink are registered trademarks of The MathWorks Inc See www mathworks com trademarks for a list of additional trademarks Other product or brand names may be trademarks or registered trademarks of their respective holders Patents MathWorks products are protected by one or more U S patents Please see www mathworks com patents for more information Revision History September 2005 March 2006 September 2006 March 2007 September 2007 March 2008 October 2008 March 2009 September 2009 March 2010 September 2010 April 2011 September 2011 March 2012 September 2012 March 2013 September 2013 March 2014 October 2014 March 2015 September 2015 First printing Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only New for Version 3 0 2 Release 14SP3 Revised for Version 3 1 Releas
61. e 2006a Revised for Version 3 1 1 Release 2006b Revised for Version 3 2 Release 2007a Revised for Version 3 3 Release 2007b Revised for Version 3 3 1 Release 2008a Revised for Version 3 3 2 Release 2008b Revised for Version 3 3 3 Release 2009a Revised for Version 3 4 Release 2009b Revised for Version 3 4 1 Release 2010a Revised for Version 3 5 Release 2010b Revised for Version 3 6 Release 2011a Revised for Version 4 0 Release 2011b Revised for Version 4 1 Release 2012a Revised for Version 4 2 Release 2012b Revised for Version 4 3 Release 2013a Revised for Version 5 0 Release 2013b Revised for Version 5 1 Release 2014a Revised for Version 5 2 Release 2014b Revised for Version 5 3 Release 2015a Revised for Version 6 0 Release 2015b Contents Building Uncertain Models 1 Introduction to Uncertain Elements 1 3 Uncertain Real Parameters 000000 eee 1 5 Create Uncertain Real Parameters 1 7 Uncertain LTI Dynamics Elements 1 14 Time Domain of ultidyn Elements 0 1 15 Create Uncertain LTI Dynamics 1 17 Uncertain Complex Parameters and Matrices 1 18 Uncertain Complex Parameters 00055 1 18 Uncertain Complex Matrices 0 0 0 0 eee 1 19 Systems with Unmodeled Dynamics 1 22 Uncertain Matrices 0 0 eee eee 1 23 Cre
62. e Complex Structured Singular Value Automatica 29 1994 pp 71 109 22 Popov V M Absolute Stability of Nonlinear Systems of Automatic Control Automation and Remote Control 22 1962 pp 857 875 23 Scherer C Mixed H H Control to appear in Trends in Control A European Perspective volume of the special contributions to the ECC 1995 24 Stein G and J C Doyle Beyond Singular Values and Loop Shapes J Guidance 14 1991 pp 5 16 Bibliography 25 Vidyasagar M Nonlinear System Analysis Prentice Hall Englewood Cliffs 1992 26 Willems J C Least Squares Stationary Optimal Control and the Algebraic Riccati Equation IEEE Trans Aut Contr AC 16 1971 pp 621 634 27 Young P M M P Newlin and J C Doyle Let s Get Real in Robust Control Theory Springer Verlag 1994 pp 143 174 28 Zames G On the Input Output Stability of Time Varying Nonlinear Feedback Systems Part I and II IEEE Trans Aut Contr AC 11 1966 pp 228 238 and 465 476 3 13 LMI Lab Tools for Specifying and Solving LMIs on page 4 2 Specifying a System of LMIs on page 4 7 Specify LMI System at the Command Line on page 4 8 Specify LMIs with the LMI Editor GUI on page 4 15 How Imivar and lmiterm Manage LMI Representation on page 4 18 Querying the LMI System Description on page 4 19 LMI Solvers on pa
63. e form M P XQ Q X P lt 0 where M P Q are given matrices and X is an unstructured m by n matrix variable Advanced LMI Techniques It turns out that a particular solution Xc of Equation 4 16 can be computed via simple linear algebra manipulations 1 Typically Xc corresponds to the center of the ellipsoid of matrices defined by Equation 4 16 The function basiclmi returns the explicit solution Xc Xc basiclmi M P Q Since this central solution sometimes has large norm basic1lmi also offers the option of computing an approximate least norm solution of Equation 4 16 This is done by X basiclmi M P Q Xmin and involves LMI optimization to minimize X 4 41 4 IMILab Bibliography 1 Gahinet P and P Apkarian A Linear Matrix Inequality Approach to H Control Int J Robust and Nonlinear Contr 4 1994 pp 421 448 2 Nemirovski A and P Gahinet The Projective Method for Solving Linear Matrix Inequalities Proc Amer Contr Conf 1994 pp 840 844 3 Nesterov Yu and A Nemirovski Interior Point Polynomial Methods in Convex Programming Theory and Applications SIAM Books Philadelphia 1994 4 Shamma J S Robustness Analysis for Time Varying Systems Proc Conf Dec Contr 1992 pp 3163 3168 4 42 Analyzing Uncertainty Effects in Simulink Analyzing Uncertainty in Simulink on page 5 2 Simulink Blocks for Analyzing Uncertai
64. e is 1 The AutoSimplify property serves the same function as in the uncertain real parameter You can create a 2 by 3 gain bounded uncertain linear dynamics element Verify its size and check the properties f ultidyn f 2 3 size f ans 2 3 get f Name f NominalValue 2x3 double Type GainBounded Bound 1 SampleStateDim 1 AutoSimplify basic Time Domain of ultidyn Elements On its own every ultidyn element is interpreted as a continuous time system with uncertain behavior quantified by bounds gain or real part on its frequency response To see this create a ultidyn and view the sample time of several random samples of the element h ultidyn h 1 1 get usample h Ts ans 0 get usample h Ts ans 0 get usample h Ts ans 0 1 15 T Building Uncertain Models 1 16 However when a ultidyn element is an uncertain element of an uncertain state space model uss then the time domain characteristic of the element is determined from the time domain characteristic of the system The bounds gain bounded or positivity apply to the frequency response of the element This is explained and demonstrated in Interpreting Uncertainty in Discrete Time on page 1 41 Create Uncertain LTI Dynamics Create Imaginary Axis Uncertain LTI Dynamics You can create a 1 by 1 scalar positive real uncertain linear dynamics element whose frequency response always has rea
65. earrangement of the numbers 1 and 2 The remaining elements of ORDER must be a rearrangement of the numbers 3 through N If either of those conditions fail an error is generated Hence for umat arrays either permute or transpose can be used to effect the transpose operation Decomposing Uncertain Objects Decomposing Uncertain Objects Each uncertain object umat uss Ufrd is a generalized feedback connection 1ft of a not uncertain object e g double ss frd with a diagonal augmentation of uncertain elements ureal ultidyn ucomplex ucomplexm udyn In robust control jargon if the uncertain elements are normalized this decomposition is often called the M D form The purpose of the uncertain objects ureal ultidyn umat uss etc is to hide this underlying decomposition and allow the user to focus on modeling and analyzing uncertain systems rather than the details of correctly propagating the M D representation in manipulations Nevertheless advanced users may want access to the familiar M D form The command 1ftdata accomplishes this decomposition Since ureal ucomplex and ucomplexm do not have their NominalValue necessarily at zero and in the case of ureal objects are not symmetric about the NominalValue some details are required in describing the decomposition Normalizing Functions for Uncertain Elements Associated with each uncertain element is a normalizing function The normalizing function maps the unc
66. ecision and Matrix Variables While LMIs are specified in terms of their matrix variables X4 Xx the LMI solvers optimize the vector x of free scalar entries of these matrices called the decision variables The two functions mat2dec and dec2mat perform the conversion between these two descriptions of the problem variables Consider an LMI system with three matrix variables X X Xz Given particular values X1 X2 X3 of these variables the corresponding value xdec of the vector of decision variables is returned by mat2dec xdec mat2dec LMISYS X1 X2 X3 An error is issued if the number of arguments following LMISYS differs from the number of matrix variables in the problem see matnbr Conversely given a value xdec of the vector of decision variables the corresponding value of the k th matrix is given by dec2mat For instance the value X2 of the second matrix variable is extracted from xdec by X2 dec2mat LMISYS xdec 2 The last argument indicates that the second matrix variable is requested It could be set to the matrix variable identifier returned by lmivar The total numbers of matrix variables and decision variables are returned by matnbr and decnbr respectively In addition the function decinfo provides precise information about the mapping between decision variables and matrix variable entries 4 26 Validating Results Validating Results The LMI Lab offers two functions to analyze and validate
67. ed ureal objects have nominal value of 0 and range from 1 to 1 e ultidyn objects are norm bounded with norm bound of 1 ucomplex objects have nominal value of 0 and radius 1 ucomplexm objects have nominal value of 0 and identity matrices for each of the WL andWR weights The possible behaviors of Delta and blkdiag NormUnc are the same Consequently the possible behaviors of A and 1ft blkdiag NormUnc M are the same Hence by manipulating M BlkStruct and NormUnc a power user has direct access to all of the linear fractional transformation details and can easily work at the level of the theorems and algorithms that underlie the methods 1 79 Generalized Robustness Analysis Generalized Robustness Analysis on page 2 2 Robust Stability Margin on page 2 4 Robust Performance Margin on page 2 5 Worst Case Gain Measure on page 2 6 2 Generalized Robustness Analysis Generalized Robustness Analysis The performance of a nominally stable uncertain system model will generally degrade for specific values of its uncertain elements Moreover the maximum possible degradation increases as the uncertain elements are allowed to further and further deviate from their nominal values The graph below shows the typical tradeoff curve between allowable deviation of uncertain elements from their nominal values and the worst case degradation in system performance Here system performa
68. ed in terms of LMIs a problem can be solved exactly by efficient convex optimization algorithms see LMI Solvers on page 4 20 e While most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations they often remain tractable in the LMI framework This makes LMI based design a valuable alternative to classical analytical methods See 9 for a good introduction to LMI concepts Robust Control Toolbox software is designed as an easy and progressive gateway to the new and fast growing field of LMIs e For users who occasionally need to solve LMI problems the LMI Editor and the tutorial introduction to LMI concepts and LMI solvers provide for quick and easy problem solving For more experienced LMI users LMI Lab offers a rich flexible and fully programmable environment to develop customized LMI based tools LMI Features Robust Control Toolbox LMI functionality serves two purposes e Provide state of the art tools for the LMI based analysis and design of robust control systems Offer a flexible and user friendly environment to specify and solve general LMI problems the LMI Lab Examples of LMI based analysis and design tools include Functions to analyze the robust stability and performance of uncertain systems with varying parameters popov quadstab quadperf Functions to design robust control with a mix of H Ha and pole placement objectives h2hinfsyn
69. empting to simplify without error the representation of the effect of that uncertain object The AutoSimplify property of each element dictates the types of computations that are performed In the off case no simplification is even attempted In basic fairly simple schemes to detect and eliminate nonminimal representations are used Finally in full numerical based methods similar to truncated balanced realizations are used with a very tight tolerance to minimize error Effect of the Autosimplify Property Create an uncertain real parameter view the AutoSimplify property of a and then create a 1 by 2 umat both of whose entries involve the uncertain parameter a ureal a 4 a AutoSimplify ans basic m1 a 4 6 a UMAT 1 Rows 2 Columns a real nominal 4 variability 1 1 1 occurrence Note that although the uncertain real parameter a appears in both two entries of the matrix the resulting uncertain matrix m1 only depends on 1 occurrence of a Set the AutoSimplify property of ato off from basic Recreate the 1 by 2 umat Now note that the resulting uncertain matrix m2 depends on 2 occurrences of a a AutoSimplify off m2 at 4 6 a Simplifying Representation of Uncertain Objects UMAT 1 Rows 2 Columns a real nominal 4 variability 1 1 2 occurrences The basic level of autosimplification often detects and simplifies duplication created by linear terms in the var
70. ertain element into a normalized uncertain element If pis an uncertain real parameter with range L R and nominal value N then the normalizing function F is A B rae Dp with the property that for all p satisfying L lt p lt R it follows that 1 lt F p lt 1 moreover F L 1 F N 0 and F R 1 If the nominal value is centered in the range then it is easy to conclude that a EtL R L B 2 R L C 1 D 0 1 73 T Building Uncertain Models 1 74 It is left as an algebra exercise for the user to work out the various values forA B C and D when the nominal value is not centered If E is an uncertain gain bounded linear time invariant dynamic uncertainty with gain bound B then the normalizing function F is F E E If E is an uncertain positive real linear time invariant dynamic uncertainty with positivity bound B then the normalizing function F is r B 1 a 2 81 0 2 Di where a 2 6 1 The normalizing function for an uncertain complex parameter with nominal value C and radius y is F g 6 C 1 f The normalizing function for uncertain complex matrices H with nominal value N and weights W and Wp is F H W H N Wp In each case as the uncertain element varies over its range the absolute value of the normalizing function or norm in the matrix case varies from 0 and 1 Properties of the Decomposition Take an uncertain object A dependent on uncertain
71. es Type A NominalValue to see the nominal value get A to see all properties and A Use usubs to substitute the uncertain element named delta in A with the value 2 8 leaving all other uncertain elements intact Note that the result B is an uncertain matrix with dependence only on eta and rho B usubs A delta 2 3 B Uncertain matrix with 2 rows and 2 columns The uncertainty consists of the following blocks eta Uncertain real nominal 6 variability 1 1 3 occurrences rho Uncertain real nominal 1 variability 1 1 1 occurrences Type B NominalValue to see the nominal value get B to see all properties and B To set multiple elements list individually or group the values in a data structure For instance the following are the same 1 57 T Building Uncertain Models 1 58 B1 usubs A delta 2 3 eta A Uncertainty rho S delta 2 3 S eta A Uncertainty rho B2 usubs A S In each case delta is replaced by 2 3 and eta is replaced by A Uncertainty rho Any superfluous substitution requests are ignored Hence the following returns an uncertain model that is the same as A B4 usubs A fred 5 Specifying the Substitution with Structures An alternative syntax for uSubs is to specify the substituted values in a structure whose fieldnames are the names of the elements being substituted with values Create a structure NV with 2 fields delta and eta Set the va
72. es and simulate the model for various values in the uncertainty range Monte Carlo simulation For more information see How to Vary Uncertainty Values on page 5 7 You can view and compare the simulation results for various sample values of uncertainty using the Mu1ltiPlot Graph block How to Vary Uncertainty Values There are two ways to control the uncertainty values using the Uncertainty value struct or to use nominal value field of the Uncertain State Space block parameters dialog box For simple models with few uncertain variables or one Uncertain State Space block type the value in the Uncertain State Space block itself For more information see Vary Uncertainty Values Using Individual Uncertain State Space Blocks on page 5 8 For complex models with large number of uncertain variables or Uncertain State Space blocks use a single data structure for all uncertain variables referenced by the model Using this approach you can collectively control the values of all or a subset of uncertain variables and toggle between nominal and user defined values from the MATLAB prompt For more information see Vary Uncertainty Values Across Multiple Uncertain State Space Blocks on page 5 14 5 7 5 Analyzing Uncertainty Effects in Simulink Vary Uncertainty Values Using Individual Uncertain State Space Blocks This example shows the workflow for varying uncertainty values using individual Uncertain State Space bloc
73. es of a umat on page 1 25 Row and Column Referencing on page 1 27 Matrix Operation on umat Objects on page 1 29 Evaluate Uncertain Elements by Substitution on page 1 31 Uncertain State Space Models uss on page 1 33 Create Uncertain State Space Model on page 1 36 Sample Uncertain Systems on page 1 37 Feedback Around an Uncertain Plant on page 1 39 Interpreting Uncertainty in Discrete Time on page 1 41 Lifting a ss to a uss on page 1 42 Create Uncertain Frequency Response Data Models on page 1 43 Interpreting Uncertainty in Discrete Time on page 1 46 Lifting an frd to a ufrd on page 1 47 Basic Control System Toolbox and MATLAB Interconnections on page 1 48 Simplifying Representation of Uncertain Objects on page 1 49 Generate Samples of Uncertain Systems on page 1 53 1 Building Uncertain Models Substitution by usubs on page 1 57 Array Management for Uncertain Objects on page 1 60 Reference Into Arrays on page 1 61 Create Arrays with stack and cat Functions on page 1 62 Create Arrays by Assignment on page 1 64 Binary Operations with Arrays on page 1 65 Create Arrays with usample on page 1 66 Create Arrays with usubs on page 1 68 Create Arrays with gridureal on page 1 69 Create Arrays with repmat on page 1 70 Create Arrays with repsys on page 1 71 Using permute
74. f defcx are the identifiers X and P of the variables involved in the objective and the outputs Xj and Pj are the corresponding values 2 Evaluate the objective expression for X Xj and P Pj This yields the j th entry of c by definition In our example the result is C N O Other objectives are handled similarly by editing the following generic skeleton n decnbr LMI system c zeros n 1 for j tin matrix values defcx LMI system j matrix identifiers C j objective matrix values end Feasibility Radius When solving LMI problems with feasp mincx or gevp it is possible to constrain the solution x to lie in the ball xix lt R where R gt 0 is called the feasibility radius This specifies a maximum Euclidean norm magnitude for x and avoids getting solutions of very large norm This may also speed up computations and improve numerical stability Finally the feasibility radius bound regularizes problems with redundant variable sets In rough terms the set of scalar variables is redundant when an equivalent problem could be formulated with a smaller number of variables The feasibility radius R is set by the third entry of the options vector of the LMI solvers Its default value is R 109 Setting R to a negative value means no rigid bound in which case the feasibility radius is increased during the optimization if necessary This flexible bound mode may yield solutions of large norms
75. g with the canonical form is also detrimental to the efficiency of the LMI solvers For these various reasons the LMI Lab uses a structured representation of LMIs For instance the expression A X XA in the Lyapunov inequality Equation 4 1 is explicitly described as a function of the matrix variable X and only the A matrix is stored In general LMIs assume a block matrix form where each block is an affine combination of the matrix variables As a fairly typical illustration consider the following LMI drawn from H theory A X XA XC B No Ox yI D N lt 0 BT D yI 4 IMILab where A B C D and N are given matrices and the problem variables are X X e RP and y R We use the following terminology to describe such LMIs Niscalled the outer factor and the block matrix ATX XA XC B L X y CX yI D BT D yI is called the inner factor The outer factor needs not be square and is often absent Xand y are the matrix variables of the problem Note that scalars are considered as 1 by 1 matrices The inner factor L X y is a symmetric block matrix its block structure being characterized by the sizes of its diagonal blocks By symmetry L X y is entirely specified by the blocks on or above the diagonal Each block of L X y is an affine expression in the matrix variables X and y This expression can be broken down into a sum of elementary terms For instance the block 1 1 contains two elementary ter
76. ge 4 20 e Minimize Linear Objectives under LMI Constraints on page 4 22 Conversion Between Decision and Matrix Variables on page 4 26 e Validating Results on page 4 27 Modify a System of LMIs on page 4 28 Advanced LMI Techniques on page 4 31 Bibliography on page 4 42 4 IMILab Tools for Specifying and Solving LMIs The LMI Lab is a high performance package for solving general LMI problems It blends simple tools for the specification and manipulation of LMIs with powerful LMI solvers for three generic LMI problems Thanks to a structure oriented representation of LMIs the various LMI constraints can be described in their natural block matrix form Similarly the optimization variables are specified directly as matrix variables with some given structure Once an LMI problem is specified it can be solved numerically by calling the appropriate LMI solver The three solvers feasp mincx and gevp constitute the computational engine of the LMI portion of Robust Control Toolbox software Their high performance is achieved through C MEX implementation and by taking advantage of the particular structure of each LMI The LMI Lab offers tools to Specify LMI systems either symbolically with the LMI Editor or incrementally with the lmivar and lmiterm commands Retrieve information about existing systems of LMIs Modify existing systems of LMIs Solve the three generic LMI problems feasibility
77. hat the distribution is skewed However the number of samples less than the nominal value and the number of samples greater than the nominal value are equal on average Verify this length find hsample lt h NominalValue ans 482 length find hsample gt h NominalValue Create Uncertain Real Parameters ans 518 The distribution used in usample is uniform in the normalized description of the uncertain real parameter See Decomposing Uncertain Objects to learn more about the normalized description There is no notion of an empty ureal or any other uncertain element for that matter ureal by itself creates an element named UNNAMED with default property values 1 13 1 Building Uncertain Models Uncertain LTI Dynamics Elements 1 14 Uncertain linear time invariant objects ultidyn are used to represent unknown linear time invariant dynamic objects whose only known attributes are bounds on their frequency response Uncertain linear time invariant objects have an internal name the Name property and are created by specifying their size number of outputs and number of inputs The property Type specifies whether the known attributes about the frequency response are related to gain or phase The property Type may be GainBounded or PositiveReal The default value is GainBounded The property Bound is a single number which along with Type completely specifies what is kno
78. he command usample randomly samples the uncertain system at a specified number of points Randomly sample an uncertain system at 20 points in its modeled uncertainty range This gives a 20 by 1 SS array Consequently all analysis tools from Control System Toolbox are available ureal p1 10 Percentage 50 ureal p2 3 PlusMinus 5 1 2 p3 ureal p3 0 A p1 p2 0 p1 B p2 p2 p3 C 1 0 1 1 p3 D 0 0 TT po to sys ss A B C D Create uncertain state space model sys Uncertain continuous time state space model with 2 outputs 1 inputs 2 states The model uncertainty consists of the following blocks pi Uncertain real nominal 10 variability 50 50 2 occurrences p2 Uncertain real nominal 3 variability 0 5 1 2 2 occurrences p3 Uncertain real nominal 0 variability 1 1 2 occurrences Type sys NominalValue to see the nominal value get sys to see all properties an manysys usample sys 20 size manysys 20x1 array of state space models Each model has 2 outputs 1 inputs and 2 states stepplot manysys 1 37 T Building Uncertain Models Step Response To Out 1 Amplitude Time seconds The command stepplot can be called directly on a uss object The default behavior samples the uss object at 20 instances and plots the step responses of these 20 models as well as the nominal value The same features are available for other
79. he structure to an element s nominal value and then following or preceeding the call to usubs with a call to usample to generate the random samples is acceptable and achieves the same effect 1 59 T Building Uncertain Models Array Management for Uncertain Objects 1 60 All of the uncertain system classes uss ufrd may be multidimensional arrays This is intended to provide the same functionality as the LTI arrays of the Control System Toolbox software The command size returns a row vector with the sizes of all dimensions The first two dimensions correspond to the outputs and inputs of the system Any dimensions beyond are referred to as the array dimensions Hence if szM size M then ZM 3 end are sizes of the array dimensions of M For these types of objects it is clear that the first two dimensions system output and input are interpreted differently from the 3rd 4th 5th and higher dimensions which often model parametrized variability in the system input output behavior umat objects are treated in the same manner The first two dimensions are the rows and columns of the uncertain matrix Any dimensions beyond are the array dimensions Reference Into Arrays Reference Into Arrays Suppose M is a umat uss or ufrd and that Yidx and Uidx are vectors of integers Then M Yidx Uidx selects the outputs rows referred to by Yidx and the inputs columns referred to by Uidx preserving all of the array di
80. he uncertainty from all Uncertain State Space blocks Parameters Uncertain system variable uss Uncertainty value struct or to use nominal value Q Initial states nominal dynamics a Initial states uncertain dynamics 0 Lok cancel Help Apply The Uncertain System block uses the uncertain system variable unc_ sys This variable is defined in the Model workspace as unc_sys 5 Analyzing Uncertainty Effects in Simulink ss ureal a 1 Range 2 5 1 5 0 1 0 1 input_unc The uncertain model depends on a single uncertain variable named a The Uncertainty value field specifies to use nominal value of the uncertain variable a Click OK to close the dialog box Click to simulate the model The software uses the nominal value of a during simulation After the simulation completes the MultiPlot Graph shows the following plot File Edit View Insert Tools Desktop Window Help DSGES k AAVRRKL 2 08 aQ BB fee PAS A ee ee Input 1 gt gt aL a A co Time 4 To simulate the model using a manually defined value of a a Double click the Uncertain State Space block and enter struct a 0 3 inthe Uncertainty value field 5 10 Vary Uncertainty Values Using Individual Uncertain State Space Blocks Uncertain system variable uss unc_syg Uncertainty value struct or to use nominal value struct a 0 3 b Click iC to sim
81. icking the load button Read a sequence of lmivar lmiterm commands from a file read button You can then click on describe the matrix variables or describe the LMIs to visualize the symbolic expression of the LMI system specified by these commands The file should describe a single LMI system but may otherwise contain any sequence of MATLAB commands This feature is useful for code validation and debugging Write in a file the sequence of Lmivar lmiterm commands needed to describe a particular LMI system write button Specify LMls with the LMI Editor GUI This is helpful to develop code and prototype MATLAB functions based on the LMI Lab Generate the internal representation of the LMI system by clicking create The result is written in a MATLAB variable named after the LMI system if the name of the LMI system is set to my1mi the internal representation is written in the MATLAB variable my1mi Note that all LMI related data should be defined in the MATLAB workspace at this stage The internal representation can be passed directly to the LMI solvers or any other LMI Lab function Keyboard Shortcuts As with lmiterm you can use various shortcuts when entering LMI expressions at the keyboard For instance zero blocks can be entered simply as 0 and need not be dimensioned Similarly the identity matrix can be entered as 1 without dimensioning Finally upper diagonal LMI blocks need not be fully specified Rather you ca
82. ies and d simplify d class ans Uncertain matrix with 1 rows 1 columns and no uncertain blocks Transpose M form a product an inverse and sample the uncertain result As expected the result is the 2 by 2 identity matrix H M M K inv H usample K H 3 ans 1 1 29 T Building Uncertain Models 1 0000 0 0000 0 0000 1 0000 ans 2 1 0000 0 0000 0 0000 1 0000 ans 3 1 0000 0 0000 0 0000 1 0000 1 30 Evaluate Uncertain Elements by Substitution Evaluate Uncertain Elements by Substitution Uncertain elements can be substituted for using uSubs For more information see Substitution by usubs on page 1 57 Here we illustrate a few special cases Substitute all instances of the uncertain real parameter named a with the number 4 This results in a umat with dependence on the uncertain real parameters b and c M2 usubs M a 4 M2 Uncertain matrix with 3 rows and 2 columns The uncertainty consists of the following blocks b Uncertain real nominal 10 variability 20 20 2 occurrences c Uncertain real nominal 3 variability 40 40 1 occurrences Type M2 NominalValue to see the nominal value get M2 to see all properties and Similarly we can substitute all instances of the uncertain real parameter named b with M Uncertainty a resulting in a umat with dependence on the uncertain real parameters a and C M3 usubs M b M Uncertai
83. ime classical gain and phase margins Loop at a time disk margins e Multi loop disk margin To learn more about these stability margins see the Algorithms section of the loopmargin reference page The loopmargin command computes the stability margin based on linearization of Simulink models To compute stability margins of a Simulink model 1 Specify the block where you want to define a margin analysis point 2 Specify the output port of the block where you want the margin analysis point The software performs the analysis by opening the loop at all specified margin analysis point 3 Use the loopmargin command to compute the stability margins at the margin analysis point Optionally you can compare the classical gain and phase margins obtained using loopmargin with the stability margins computed for the linearized model The results using the two approaches should match for simple SISO models For MIMO models the loopmargin command provides richer robustness information For an example see Stability Margin of a Simulink Model on page 5 29 Additionally you can compute stability margins by specifying multiple margin analysis points and multiple operating points For an example see Loop Margins for an Airframe Autopilot Stability Margin of a Simulink Model Stability Margin of a Simulink Model This example illustrates how to compute the stability margins of the airframemargin model and compare the results with st
84. imulink model and affects both simulation and linearization Use this approach to vary the uncertainty and analyze the effects on simulation or linearization To specify uncertainty in the Uncertain State Space block 1 Drag and drop an Uncertain State Space block from the Robust Control Toolbox block library into a Simulink model For more information on how to open the block library see Simulink Blocks for Analyzing Uncertainty on page 5 3 Pyret usim ee Sea File Edit View Display Diagram Simulation Analysis Code Pa gt oa BE 4 D gt 7 7 rct_usim_ex ajrct_usim_ex w Q EJ Uncertain State Space Ready 100 ode45 5 4 Specify Uncertainty Using Uncertain State Space Blocks 2 In the Simulink model double click the Uncertain State Space block to open the Function Block Parameters Uncertain State Space dialog box as shown in the following figure Uncertain State Space Block mask link This block models linear systems with uncertain parameters and uncertain dynamics see USS You can simulate how uncertainty affects system performance by generating randomized values for the uncertain variables Use UFIND to find all uncertain variables in the model and USAMPLE to generate sample values for the Uncertainty value mask parameter Each uncertainty value represents one possible behavior of the uncertain system You can also investigate uncertainty effects in
85. ing the same uncertain matrix through multiplication a ureal a 5 Amat repmat a 2 3 UMAT 2 Rows 3 Columns a real nominal 5 variability 1 1 1 occurrence Amat2 a ones 2 3 simplify Amat Amat2 ans 0 0 0 0 0 0 Create a 4 by 1 umat array by stacking four 1 by 3 umat objects with the stack command Use repmat to tile this 1 by 3 by 4 by 1 umat into a 2 by 3 by 8 by 5 umat a ureal a 4 b ureal b 2 M stack 1 a b 1 a b 4 a 4 5 6 a 0 0 size M ans 1 3 4 Mtiled repmat M 2 1 2 5 UMAT 2 Rows 3 Columns array 8 x 5 a real nominal 4 variability 1 1 1 occurrence b real nominal 2 variability 1 1 1 occurrence Verify the equality of M and a few certain tiles of Mtiled di simplify M Mtiled 2 5 8 3 d2 simplify M Mtiled 1 1 4 2 d3 simplify M Mtiled 2 1 4 5 max abs d1 max abs d2 max abs d3 ans 0 0 0 Note that repmat never increases the complexity of the representation of an uncertain object The number of occurrences of each uncertain element remains the same regardless of the extent of the replication and tiling Create Arrays with repsys Create Arrays with repsys Replicating and tiling uncertain state space systems USS and uncertain frequency response data ufrd is done with repsys The syntax and behavior are the same as the manner in which repmat is used to replicate and tile matrices The syntax
86. ink Models Analyzing Stability Margin of Simulink Models Robust Control Toolbox provides the loopmargin command to analyze the stability margins of LTI models created in MATLAB and Simulink models To use loopmargin with Simulink models you must have the Simulink Control Design software This section describes the difference between the MATLAB and Simulink approaches of using loopmargin and the workflow for computing the stability margin of Simulinkmodels For more information on how to analyze the stability margins of LTI models see MIMO Robustness Analysis How Stability Margin Analysis Using Loopmargin Differs Between Simulink and LTI Models When analyzing stability margins of LTI models using the syntax cm dm mm loopmargin P C the software assumes the input and output of the linear plant P as the margin analysis points as shown in the following figure Margin analysis points Linear Controller Analyzing stability margin of Simulink models differs from analyzing stability margin of LTI models because you can enter specific margin analysis points in the Simulink model For more information on how to assign margin analysis points in Simulink models see the Usage with Simulink section of the loopmargin reference page Stability Margin of Simulink Model The loopmargin command computes the following types of stability margins 5 27 5 Analyzing Uncertainty Effects in Simulink 5 28 e Loop at a t
87. ion of an LMI system is stored as a single vector called the internal representation The user should not attempt to read or retrieve information directly from this vector Robust Control Toolbox software provides three functions called 1lmiinfo lminbr and matnbr to extract and display all relevant information in a user readable format Imiinfo lminbr is an interactive facility to retrieve qualitative information about LMI systems This includes the number of LMIs the number of matrix variables and their structure the term content of each LMI block etc To invoke 1miinfo enter 1miinfo LMISYS where LMISYS is the internal representation of the LMI system produced by getlmis Iminbr and matnbr These two functions return the number of LMIs and the number of matrix variables in the system To get the number of matrix variables for instance enter matnbr LMISYS 4 19 4 IMILab LMI Solvers 4 20 LMI solvers are provided for the following three generic optimization problems here x denotes the vector of decision variables i e of the free entries of the matrix variables Xi Xx e Feasibility problem Find x RY or equivalently matrices Xj Xx with prescribed structure that satisfies the LMI system A x lt B x The corresponding solver is called feasp Minimization of a linear objective under LMI constraints Minimize c x over x R subject to A x lt B x The corresponding solver is
88. ions for subsequent processing The command getlmis must be used only once and after declaring all matrix variables and LMI terms Here the identifiers X and S point to the variables X and S while the tags BRL Xpos and Slmi point to the first second and third LMI respectively Note that Xpos refers to the right hand side of the second LMI Similarly X would indicate transposition of the variable X Specify LMls with the LMI Editor GUI Specify LMIs with the LMI Editor GUI The LMI Editor lmiedit is a graphical user interface GUI to specify LMI systems in a straightforward symbolic manner Typing lmiedit calls up a window with several editable text areas and various buttons o ici x cant Wirin E sareta watt ar able ewe er ha arn wt ae as ee Use the view Declore each motrix vorioble 7 commands buttons nome ond structure here 3 to visualize the sequence of Lmivar lmiterm j zj commands needed to Se ase describe this LMI EAE aoe 5 system Specify the LMIs os MATLAB expressions here ly Reod o sequence of Imivar imiterm com mands from o file by clicking Read Gene rote the internal representation of the LMI system by clicking Create Sove Load the symbolic description of the LMI system os a MATLAB string jum reece amans wadhe win Al bras ii AnA rew ws ues In more detail to specify your LMI system 1 Declare each matrix variab
89. ious entries Higher order quadratic bilinear etc duplication is often not detected by the basic autosimplify level For example reset the AutoSimplify property of a to basic from off Create an uncertain real parameter and a 1 by 2 umat both of whose entries involve the square of the uncertain parameter a AutoSimplify basic m3 a at4 6 a a UMAT 1 Rows 2 Columns a real nominal 4 variability 1 1 4 occurrences Note that the resulting uncertain matrix m3 depends on 4 occurrences of a Set the AutoSimplify property of ato full from basic Recreate the 1 by 2 umat Now note that the resulting uncertain matrix m4 depends on 2 occurrences of a a AutoSimplify full m4 a at4 6 a a UMAT 1 Rows 2 Columns a real nominal 4 variability 1 1 2 occurrences Although m4 has a less complex representation 2 occurrences of a rather than 4 as in m3 some numerical variations are seen when both uncertain objects are evaluated at say 0 usubs m3 a 0 ans 0 0 usubs m4 a 0 ans 1 0e 015 0 4441 0 Small numerical differences are also noted at other evaluation points The example below shows the differences encountered evaluating at a equal to 1 usubs m3 a 1 ans 5 6 1 51 T Building Uncertain Models 1 52 usubs m4 a 1 ans 5 0000 6 0000 Direct Use of simplify The simplify command can be used to override all uncertain eleme
90. iple Uncertain State Space blocks in a Simulink model Use this approach for complex models with large number of uncertain variables or Uncertain State Space blocks This section uses a Simulink model to provide step by step instructions for toggling between nominal and user defined uncertainty values at the MATLAB prompt 1 Open the Simulink model rct_sim_ex2 rcet_sim_ex2 The model contains two Uncertain State Space blocks as shown in the following figure Unmodeled dynamics Step The Unmodeled dynamics and First order with uncertain pole blocks depend on the uncertain variables input_unc and a 2 Double click the Unmodeled dynamics block to open the block parameters dialog box The Uncertainty value field contains the variable val_al11 Similarly the Uncertainty value field in the First order with uncertain pole block parameters dialog contains the variable val_all You use this variable to vary the uncertain variable values across both the Uncertain State Space blocks Vary Uncertainty Values Across Multiple Uncertain State Space Blocks Uncertain system variable uss Uncertainty value struct or to use nominal value val_all Note When defining val_al1l1 you can enter only a subset of uncertain variables referenced by the model in the structure When you do not specify some uncertain variables the software uses their nominal value during simulation At the MATLAB prompt specify val_all and click O
91. ks in a Simulink model Use this approach for simple models with few uncertain variables or one Uncertain State Space block This section uses a simple Simulink model to provide step by step instructions for Toggling between nominal manually defined and randomly generated uncertainty values associated with the Uncertain State Space block Simulating the model s responses for these uncertainty values 1 Open the Simulink model rct_sim_ex1 rcet_sim_ex1 The model contains an Uncertain State Space block called Uncertain System as shown in the following figure E a Uncertain System ep yst MultiPlot Graph 2 Double click the Uncertain System block to open the Function Block Parameters dialog box 5 8 Vary Uncertainty Values Using Individual Uncertain State Space Blocks Uncertain State Space Block mask link This block models linear systems with uncertain parameters and uncertain dynamics see USS You can simulate how uncertainty affects system performance by generating randomized values for the uncertain variables Use UFIND to find all uncertain variables in the model and USAMPLE to generate sample values for the Uncertainty value mask parameter Each uncertainty value represents one possible behavior of the uncertain system You can also investigate uncertainty effects in the frequency domain by linearizing the Simulink model with ULINEARIZE This computes an uncertain state space model USS that aggregates t
92. l part greater than 0 5 Set the SampleStateDimension property to 5 Plot a Nyquist plot of 30 instances of the element g ultidyn g 1 1 Type Positivereal Bound 0 5 g SampleStateDimension 5 nyquist usample g 30 xlim 2 10 ylim 6 6 Nyquist Diagram 1 17 T Building Uncertain Models Uncertain Complex Parameters and Matrices 1 18 Uncertain Complex Parameters The ucomp1lex element is the Control Design Block that represents an uncertain complex number The value of an uncertain complex number lies in a disc centered at NominalValue with radius specified by the Radius property of the ucomplex element The size of the disc can also be specified by Percentage which means the radius is derived from the absolute value of the NominalValue The properties of ucomplex objects are Properties Meaning Class Name Internal Name char NominalValue Nominal value of element double Mode Range Percentage char Radius Radius of disk double Percentage Additive variation percent of Radius double AutoSimplify off basic full char The simplest construction requires only a name and nominal value Displaying the properties shows that the default Mode is Radius and the default radius is 1 a ucomplex a 2 j a Uncertain complex parameter a get a Name a NominalValue 2 0000 1 0000i Mode Radius Radius 1 Percentage 44 7214 Au
93. le For a matrix variable X of Type 1 this second input is a matrix with two columns and as many rows as diagonal blocks in X The first column lists the sizes of the diagonal blocks and the second column specifies their nature with the following convention 1 full symmetric block 0 scalar block 1 zero block In the second command for instance 2 0 2 1 means that S has two diagonal blocks the first one being a 2 by 2 scalar block and the second one a 2 by 2 full block For matrix variables of Type 2 the second input of lmivar is a two entry vector listing the row and column dimensions of the variable For instance a 3 by 5 rectangular matrix variable would be defined by lmivar 2 3 5 For convenience lmivar also returns a tag that identifies the matrix variable for subsequent reference For instance X and S in Specifying LMI System on page 4 8 could be defined by X S lmivar 1 6 1 lmivar 1 2 0 2 1 The identifiers X and S are integers corresponding to the ranking of X and S in the list of matrix variables in the order of declaration Here their values would be X 1 and S 2 4 11 4 IMILab Note that these identifiers still point to X and S after deletion or instantiation of some of the matrix variables Finally lmivar can also return the total number of decision variables allocated so far as well as the entry wise dependence of the matrix variable on these decision variables see the 1miva
94. le name and structure in the upper half of the worksheet The structure is characterized by its type S for symmetric block diagonal R for unstructured and G for other structures and by an additional 4 15 4 IMILab 4 16 structure matrix This matrix contains specific information about the structure and corresponds to the second argument of 1mivar see Specifying the LMI Variables on page 4 10 for details Please use one line per matrix variable in the text editing areas Specify the LMIs as MATLAB expressions in the lower half of the worksheet For instance the LMI ATX XA XB lt 0 BTX I is entered by typing a x x a x b b x 1 lt 0 if x is the name given to the matrix variable X in the upper half of the worksheet The left and right hand sides of the LMIs should be valid MATLAB expressions Once the LMI system is fully specified the following tasks can be performed by clicking the corresponding button Visualize the sequence of lmivar lmiterm commands needed to describe this LMI system view commands button Conversely the LMI system defined by a particular sequence of lmivar lmiterm commands can be displayed as a MATLAB expression by clicking on the describe buttons Beginners can use this facility as a tutorial introduction to the lmivar and lmiterm commands Save the symbolic description of the LMI system as a MATLAB string Save button This description can be reloaded later on by cl
95. lues of these fields to be the desired substituted values Then perform the substitution with usubs NV delta 2 3 NV eta A Uncertainty rho B6 usubs A NV Here B6 is the same as B1 and B2 above Again any superfluous fields are ignored Therefore adding an additional field gamma to NV and substituting does not alter the result NV gamma 0 B7 usubs A NV Here B7 is the same as B6 The commands wegain robuststab and usample all return substitutable values in this structure format More discussion can be found in Create Arrays with usubs on page 1 68 Nominal and Random Values If the replacement value is the partial and case independent string Nominal then the listed element are replaced with their nominal values Therefore B8 usubs A fieldnames A Uncertainty nom Substitution by usubs B8 11 0000 0 3333 6 0000 11 0000 B9 ll A NominalValue B9 11 0000 0 3333 6 0000 11 0000 are the same It is possible to only set some of the elements to NominalValues and would be the typical use of usubs with the nominal argument Within A set eta to its nominal value delta to a random value within its range and rho to a specific value say 6 5 B10 usubs A eta nom delta rand rho 6 5 B10 10 5183 0 2531 13 5000 15 6100 Unfortunately the nominal and Random specifiers may not be used in the structure format However explicitly setting a field of t
96. mensions For example if size M equals 4 5 3 6 7 then for example the size of M 4 2 1 2 4 is 2 3 3 6 7 If size M 1 1 or ize M 2 1 then single indexing on the inputs or outputs rows or columns is allowed If Sidx is a vector of integers then M Sidx selects the corresponding elements All array dimensions are preserved If there are K array dimensions and idx1 idx2 idxK are vectors of integers then G M Yidx Uidx idx1 idx2 idxK selects the outputs and inputs referred to by Yidx and Uidx respectively and selects from each array dimension the slices referred to by the idx1 idx2 idxK index vectors Consequently size G 1 equals length Yidx size G 2 equals length Uidx size G 3 equals length idx1 size G 4 equals length idx2 and size G K 2 equals length idxk If M has K array dimensions and less than K index vectors are used in doing the array referencing then the MATLAB convention for single indexing is followed For instance suppose ize M equals 3 4 6 5 7 4 The expression G M 1 3 1 4 2 3 4 5 3 1 8 10 12 2 4 20 18 is valid The result has ize G equals 2 2 3 3 7 The last index vector 8 10 12 2 4 20 18 is used to reference into the 7 by 4 array preserving the order dictated by MATLAB single indexing e g the 10th element of a 7 by 4 array is the element in the 8 2 position in the array Note that if M has either one output row or one input colum
97. ms A X and XA Terms are either constant or variable Constant terms are fixed matrices like B and D above Variable terms involve one of the matrix variables like XA XC and yl above The LMI Equation 4 3 is specified by the list of terms in each block as is any LMI regardless of its complexity As for the matrix variables X and y they are characterized by their dimensions and structure Common structures include rectangular unstructured symmetric skew symmetric and scalar More sophisticated structures are sometimes encountered in control problems For instance the matrix variable X could be constrained to the block diagonal structure Another possibility is the symmetric Toeplitz structure Tools for Specifying and Solving LMIs ol ae X _ xX Xo X3 Xo xy Summing up structured LMI problems are specified by declaring the matrix variables and describing the term content of each LMI This term oriented description is systematic and accurately reflects the specific structure of the LMI constraints There is no built in limitation on the number of LMIs that you can specify or on the number of blocks and terms in any given LMI LMI systems of arbitrary complexity can therefore be defined in the LMI Lab Overview of the LMI Lab The LMI Lab offers tools to specify manipulate and numerically solve LMIs Its main purpose is to Allow for straightforward description of LMIs in their natural block matrix f
98. n and M has array dimensions then it is not allowable to combine single indexing in the output input dimensions along with indexing in the array dimensions This will result in an ambiguity in how to interpret the second index vector in the expression i e does it correspond to the input output reference or does it correspond to the first array dimension 1 61 T Building Uncertain Models Create Arrays with stack and cat Functions 1 62 An easy manner to create an array is with stack Create a 4 by 1 umat array by stacking four 1 by 3 umat objects with the stack command The first argument of stack specifies in which array dimension the stacking occurs In the example below the stacking is done is the 1st array dimension hence the result is a 1 by 3 by 4 by 1 umat referred to as a 4 by 1 umat array ureal a 4 ureal b 2 stack 1 a b 1 a b 4 a 4 5 6 a O 0 MAT 1 Rows 3 Columns array 4 x 1 a real nominal 4 variability 1 1 1 occurrence b real nominal 2 variability 1 1 1 occurrence a b M U size M ans 1 3 4 arraysize M ans 4 1 Check that result is valid Use referencing to access parts of the 4 by 1 umat array and compare to the expected values The first 4 examples should all be arrays full of 0 zeros The last two should be the value 5 and the uncertain real parameter a respectively simplify M 1 a b 1 ans 0 0 0 simplify M 2
99. n just type in place of each such block Limitations Though fairly general 1miedit is not as flexible as Lmiterm and the following limitations should be kept in mind e Parentheses cannot be used around matrix variables For instance the expression a xtb c c a x b is invalid when x is a variable name By contrast atb x x atb is perfectly valid Loops and if statements are ignored When turning lmiterm commands into a symbolic description of the LMI system an error is issued if the first argument of lmiterm cannot be evaluated Use the LMI and variable identifiers supplied by newlmi and lmivar to avoid such difficulties 4 17 4 IMILab How Imivar and Imiterm Manage LMI Representation 4 18 Users familiar with MATLAB may wonder how 1lmivar and lmiterm physically update the internal representation LMISYS since LMISYS is not an argument to these functions In fact all updating is performed through global variables for maximum speed These global variables are initialized by setlmis cleared by get1mis and are not visible in the workspace Even though this artifact is transparent from the user s viewpoint be sure to Invoke getlmis only once and after completely specifying the LMI system Refrain from using the command clear global before the LMI system description is ended with getlmis Querying the LMI System Description Querying the LMI System Description Recall that the full descript
100. nal values of the uncertain parameters g and tau Tip You can also use this field to vary the uncertainty values for performing Monte Carlo simulation For more information see Simulate Uncertainty Effects on page 5 7 Optional Specify the initial states of the nominal and uncertain dynamics in the Initial states nominal dynamics and Initial states uncertain dynamics fields respectively For more information on the block parameters see the Uncertain State Space block reference page Next Steps After you specify uncertainty in Uncertain State Space blocks you can perform one of the following Simulate the model using nominal manually defined or random values as described in Simulate Uncertainty Effects on page 5 7 Perform an uncertain linearization as described in Obtain Uncertain State Space Model from Simulink Model on page 5 20 Simulate Uncertainty Effects Simulate Uncertainty Effects How to Simulate Effects of Uncertainty on page 5 7 e How to Vary Uncertainty Values on page 5 7 How to Simulate Effects of Uncertainty As described in Specify Uncertainty Using Uncertain State Space Blocks on page 5 4 the uncertain state space model you specify in the Uncertain State Space block depends on a set of uncertain variables ureal or ultidyn objects You can simulate the model using nominal value of these uncertain variables Additionally you can sample these uncertain variabl
101. nce is characterized by system gain e g peak magnitude on Bode plot Interpreting the system as the relationship mapping disturbances commands to errors small system gains are desirable and large gains are undesirable wo Maximum System Gain due to varying amounts of uncertainty Nm on NO k System performance degradation curve 1 72 Nominal System Gain 0 8 R System Gain as large as 1 72 if Maximum System Gain over Uncertainty on 0 5 uncertain elements can deviate from their nominal values by 1 5 units 0 0 0 5 1 15 2 25 3 Bound on Normalized Uncertainty When all uncertain elements are set to their nominal values i e zero deviation from their nominal values the input output gain of the system is its nominal value In the figure the nominal system gain is about 0 8 As the uncertainties are allowed to deviate from nominal the maximum over the uncertain elements system gain increases The Generalized Robustness Analysis heavy blue line represents the maximum system gain due to uncertainty of various sizes the horizontal axis and is called the system performance degradation curve It is monotonically increasing Determining specific attributes of the system performance degradation curve are referred to as robustness computations Generally robustness computations refer to determining specific attributes of the system performance degradation curve The commands robuststab robustpe
102. nearization using one of the following MATLAB Expression v Enter an expression to specify the linearization of the Simulink block ss ureal a 5 5 1 1 ok J _cancel_ Help Apply Click OK to save the changes Note You can also specify a block to linearize to an uncertain variable at the command line For an example see Linearize Block to Uncertain Model on page 5 24 Run the linearize command to compute an uncertain linearization This command returns an USS model For more information on linearization and how to validate linearization results see Linearization Basics in the Simulink Control Design documentation For an example of how to use the linearize command to compute an uncertain linearization see Linearization of Simulink Models with Uncertainty 5 23 5 Analyzing Uncertainty Effects in Simulink Linearize Block to Uncertain Model This example demonstrates how to specify a core Simulink block to linearize to an uncertain variable and compute an uncertain linearization at the command line To learn how to perform this task using the user interface see Specify Uncertain Linearization for Core or Custom Simulink Blocks on page 5 21 1 Open the Simulink model slexAircraftExample 2 Double click the Aircraft Dynamics Model block to view the subsystem wdot ft sec 2 e D Alpha deg Vertical Channel Vertical Gust w Gust ft sec Pitch Rate q rad
103. ns 4 35 4 IMILab 4 36 Specifying c x Objectives for mincx The LMI solver mincx minimizes linear objectives of the form c x where x is the vector of decision variables In most control problems however such objectives are expressed in terms of the matrix variables rather than of x Examples include Trace X where X is a symmetric matrix variable or u Xu where u is a given vector The function def cx facilitates the derivation of the c vector when the objective is an affine function of the matrix variables For the sake of illustration consider the linear objective Trace X xd Pxo where X and P are two symmetric variables and x is a given vector If lmsisys is the internal representation of the LMI system and if xo X P have been declared by xO 131 setlmis X lmivar 1 3 0 P Imivar 1 2 1 lmisys getlmis the c vector such that cT x Trace X xd Px can be computed as follows n decnbr lmisys Cc zeros n 1 for j tin Xj Pj defcx lmisys j X P c j trace Xj x0 Pj x0 end The first command returns the number of decision variables in the problem and the second command dimensions c accordingly Then the for loop performs the following operations 1 Evaluate the matrix variables X and P when all entries of the decision vector x are set to zero except xj 1 This operation is performed by the function def cx Apart Advanced LMI Techniques from lmisys and j the inputs o
104. nt s AutoSimplify property The first input to the Simplify command is an uncertain object The second input is the desired reduction technique which can either basic or full Again create an uncertain real parameter and a 1 by 2 umat both of whose entries involve the square of the uncertain parameter Set the AutoSimplify property of a to basic a AutoSimplify basic m3 a at4 6 a a UMAT 1 Rows 2 Columns a real nominal 4 variability 1 1 4 occurrences Note that the resulting uncertain matrix m3 depends on four occurrences of a The simplify command can be used to perform a full reduction on the resulting umat m4 simplify m3 full UMAT 1 Rows 2 Columns a real nominal 4 variability 1 1 2 occurrences The resulting uncertain matrix m4 depends on only two occurrences of a after the reduction Generate Samples of Uncertain Systems Generate Samples of Uncertain Systems The command usamp1e is used to randomly sample an uncertain object giving a not uncertain instance of the uncertain object Generating One Sample If Ais an uncertain object then usample A generates a single sample of A For example a sample of a ureal is a scalar double A ureal A 6 B usample A B 5 7298 Create a 1 by 3 umat with A and an uncertain complex parameter C A single sample of this umat is a 1 by 3 double C ucomplex C 2 6j M A C A A usample M ans 5 9
105. ntrate on the GUI based specification of LMIs with lmiedit 4 7 4 IMILab Specify LMI System at the Command Line The following tutorial example is used to illustrate the specification of LMI systems with the LMI Lab tools Specifying LMI System Consider a stable transfer function G s C sI A B with four inputs four outputs and six states and consider the set of input output scaling matrices D with block diagonal structure d 0 0 0 0 d 0 0 D 0 0 d ds 0 0 d ds The following problem arises in the robust stability analysis of systems with time varying uncertainty 4 Find if any a scaling D of structure Equation 4 5 such that the largest gain across frequency of D G s D is less than one This problem has a simple LMI formulation there exists an adequate scaling D if the following feasibility problem has solutions Find two symmetric matrices X R and S D D e R such that A X XA C SC XB lt 0 BTX S X gt 0 S gt l 4 8 Specify LMI System at the Command Line The LMI system Equation 4 6 Equation 4 7 and Equation 4 8 can be described with the LMI Editor as outlined below Alternatively its internal description can be generated with lmivar and lmiterm commands as follows setlmis X lmivar 1 6 1 S lmivar 1 2 0 2 1 1st LMI lmiterm 1 1 1 X 1 A s lmiterm 1 1 1 S C C lmiterm 1 1 2 X 1 B lmiterm 1 2 2 S 1 1 2nd LMI lmiterm 2 1 1
106. nty on page 5 3 Specify Uncertainty Using Uncertain State Space Blocks on page 5 4 Simulate Uncertainty Effects on page 5 7 e Vary Uncertainty Values Using Individual Uncertain State Space Blocks on page 5 8 e Vary Uncertainty Values Across Multiple Uncertain State Space Blocks on page 5 14 Computing Uncertain State Space Models from Simulink Models on page 5 19 e Obtain Uncertain State Space Model from Simulink Model on page 5 20 Specify Uncertain Linearization for Core or Custom Simulink Blocks on page 5 21 Tinearize Block to Uncertain Model on page 5 24 Using Uncertain Linearization for Analysis or Control Design on page 5 26 Analyzing Stability Margin of Simulink Models on page 5 27 Stability Margin of a Simulink Model on page 5 29 5 Analyzing Uncertainty Effects in Simulink Analyzing Uncertainty in Simulink Robust Control Toolbox software provides tools to model uncertainty in Simulink Using these tools you can analyze how the uncertainty impacts the time and frequency domain behavior of the Simulink model The Uncertain State Space block included in the Robust Control Toolbox block library is a convenient way to incorporate uncertainty information in a Simulink model For more information see Specify Uncertainty Using Uncertain State Space Blocks on page 5 4 Using this block you can perform the following types of analysis
107. nty a M3 Uncertain matrix with 3 rows and 2 columns The uncertainty consists of the following blocks a Uncertain real nominal 3 variability 0 5 2 4 occurrences c Uncertain real nominal 3 variability 40 40 1 occurrences Type M3 NominalValue to see the nominal value get M3 to see all properties and Nominal and or random instances can easily be specified M4 usubs M a nominal b random M4 Uncertain matrix with 3 rows and 2 columns The uncertainty consists of the following blocks c Uncertain real nominal 3 variability 40 40 1 occurrences 1 31 T Building Uncertain Models 1 32 Type M4 NominalValue to see the nominal value get M4 to see all properties and The command usamp1e also generates multiple random instances of a umat and uss and uf rd See Generate Samples of Uncertain Systems on page 1 53 for details Lifting a double matrix to a umat A not uncertain matrix may be interpreted as an uncertain matrix that has no dependence on uncertain elements Use the umat command to lift a double to the umat class d 1 2 3 4 5 6 M M umat Md Uncertain matrix with 2 rows 3 columns and no uncertain blocks Type M NominalValue to see the nominal value get M to see all properties and M High dimensional double matrices can also be lifted Note from the display that once the matrix is interpreted as a umat the third dimension and beyon
108. ominal value 3 with default values for all unspecified properties including plus minus variability of 1 a ureal a 3 Uncertain real parameter a with nominal value 3 and variability 1 1 View the properties and their values and note that the Range and Percentage descriptions of variability are automatically maintained get a Name a NominalValue 3 Mode PlusMinus Range 2 4 PlusMinus 1 1 Percentage 33 3333 33 3333 AutoSimplify basic Create an uncertain real parameter nominal value 2 with 20 variability Again view the properties and note that the Range and PlusMinus descriptions of variability are automatically maintained b ureal b 2 Percentage 20 get b Uncertain real parameter b with nominal value 2 and variability 20 20 Name b T Building Uncertain Models NominalValue 2 Mode Percentage Range 1 6000 2 4000 PlusMinus 0 4000 0 4000 Percentage 20 20 AutoSimplify basic Change the range of the parameter All descriptions of variability are automatically updated while the nominal value remains fixed Although the change in variability was accomplished by specifying the Range the Mode is unaffected and remains Percentage b Range 1 9 2 3 get b Name b NominalValue 2 Mode Percentage Range 1 9000 2 3000 PlusMinus 0 1000 0 3000 Percentage 5 0000 15 0000 AutoSimplify basic As mentioned the Mode
109. orm e Provide easy access to the LMI solvers optimization codes Facilitate result validation and problem modification The structure oriented description of a given LMI system is stored as a single vector called the internal representation and generically denoted by LMISYS in the sequel This vector encodes the structure and dimensions of the LMIs and matrix variables a description of all LMI terms and the related numerical data It must be stressed that you need not attempt to read or understand the content of LMISYS since all manipulations involving this internal representation can be performed in a transparent manner with LMI Lab tools The LMI Lab supports the following functionalities Specification of a System of LMIs LMI systems can be either specified as symbolic matrix expressions with the interactive graphical user interface lmiedit or assembled incrementally with the two commands lmivar and lmiterm The first option is more intuitive and transparent while the second option is more powerful and flexible 4 IMILab Information Retrieval The interactive function 1miinfo answers qualitative queries about LMI systems created with lmiedit or lmivar and lmiterm You can also use lmiedit to visualize the LMI system produced by a particular sequence of lmivar lmiterm commands Solvers for LMI Optimization Problems General purpose LMI solvers are provided for the three generic LMI problems defined in LMI Applications
110. ot del Bound exp sqrt 1 theta r hold off Generate Samples of Uncertain Systems Imaginary Axis Nyquist Diagram Real Axis Change SampleStateDimension to 4 and repeat entire procedure The Nyquist plots satisfy the gain bound and as expected are more complex than the circles found in the 1st order sampling del SampleStateDimension 4 delS usample del 30 nyquist delS hold on theta linspace pi pi plot del Bound exp sqrt 1 theta r hold off 1 55 Substitution by usubs Substitution by usubs If an uncertain object umat uss ufrd has many uncertain parameters it is often necessary to freeze some but not all of the uncertain parameters to specific values The usubs command accomplishes this and also allows more complicated substitutions for an element usubs accepts a list of element names and respective values to substitute for them You can create three uncertain real parameters and use them to create a 2 by 2 uncertain matrix A delta ureal delta 2 eta ureal eta 6 rho ureal rho 1 A 3 deltateta delta eta 7 rho rhotdelta eta A Uncertain matrix with 2 rows and 2 columns The uncertainty consists of the following blocks delta Uncertain real nominal 2 variability eta Uncertain real nominal 6 variability 1 rho Uncertain real nominal 1 variability 1 1 2 occurrences 3 occurrences 1 1 occurrenc
111. ple time 0 42 the system d2v delta is discrete time with sample time 0 42 div delta Ts ans 0 d2v delta Ts ans 0 4200 Finally in the case of a discrete time USS object it is not the case that ultidyn objects are interpreted as continuous time uncertainty in feedback with sampled data systems This very interesting hybrid theory is beyond the scope of the toolbox 1 41 T Building Uncertain Models Lifting a ss to a uss A not uncertain state space object may be interpreted as an uncertain state space object that has no dependence on uncertain elements Use the uss command to lift a ss to the uss class sys rss 3 2 1 usys uss sys USS 3 States 2 Outputs 1 Input Continuous System Arrays of SS objects can also be lifted See Array Management for Uncertain Objects on page 1 60 for more information about how arrays of uncertain objects are handled 1 42 Create Uncertain Frequency Response Data Models Create Uncertain Frequency Response Data Models Uncertain frequency responses uf rd arise naturally when computing the frequency response of an uncertain state space uss They also arise when frequency response data in an frd model object is combined added multiplied concatenated etc to an uncertain matrix umat The most common manner in which a uf rd arises is taking the frequency response of a uss To do this use the uf rd command Construct an uncertain state space model
112. property signifies what aspect of the uncertainty remains unchanged when NominalValue is modified Hence if a real parameter is in Percentage mode then the Range and PlusMinus properties are determined from the Percentage property and NominalValue Changing NominalValue preserves the Percentage property and automatically updates the Range and PlusMinus properties b NominalValue 2 2 get b Name b NominalValue 2 2000 Mode Percentage Range 2 0900 2 5300 PlusMinus 0 1100 0 3300 Percentage 5 0000 15 0000 AutoSimplify basic Create an uncertain parameter with an asymmetric variation about its nominal value Examine the properties to confirm the asymmetric range Create Uncertain Real Parameters c ureal c 5 Percentage 20 30 get c Name c NominalValue 5 Mode Percentage Range 6 3 5000 PlusMinus 1 1 5000 Percentage 20 30 AutoSimplify basic Create an uncertain parameter specifying variability with Percentage but force the Mode to be Range d ureal d 1 Mode Range Percentage 40 60 get d Name d NominalValue 1 Mode Range Range 1 4000 0 4000 PlusMinus 0 4000 0 6000 Percentage 40 60 AutoSimplify basic Finally create an uncertain real parameter and set the AutoSimplify property to full e ureal e 10 PlusMinus 23 Mode Percentage AutoSimplify Full get e Uncertain real parameter
113. r entry in the reference pages for more details Specifying Individual LMIs After declaring the matrix variables with lmivar we are left with specifying the term content of each LMI Recall that LMI terms fall into three categories The constant terms i e fixed matrices like Jin the left side of the LMI S gt I The variable terms i e terms involving a matrix variable For instance ATX and C SC in Equation 4 6 Variable terms are of the form PXQ where X is a variable and P Q are given matrices called the left and right coefficients respectively The outer factors The following rule should be kept in mind when describing the term content of an LMI Note Specify only the terms in the blocks on or above the diagonal The inner factors being symmetric this is sufficient to specify the entire LMI Specifying all blocks results in the duplication of off diagonal terms hence in the creation of a different LMI Alternatively you can describe the blocks on or below the diagonal LMI terms are specified one at a time with lmiterm For instance the LMI A X XA C SC XB lt BTX S is described by lmiterm 1 1 1 1 1 A s lmiterm 1 1 1 2 C C lmiterm 1 1 2 1 1 B lmiterm 1 2 2 2 1 1 These commands successively declare the terms ATX XA CTSC XB and S In each command the first argument is a four entry vector listing the term characteristics as follows Specify LMI System at the Command Line
114. r subsequent manipulations it is therefore advisable to refer to the remaining variables through their identifier Finally note that deleting a matrix variable is equivalent to setting it to the zero matrix of the same dimensions with setmvar Instantiating a Matrix Variable The function setmvar is used to set a matrix variable to some given value As a result this variable is removed from the problem and all terms involving it become constant terms This is useful for instance to fixsetmvar some variables and optimize with respect to the remaining ones Consider again Specifying LMI System on page 4 8 and suppose we want to know if the peak gain of G itself is less than one that is if IGI lt 1 This amounts to setting the scaling matrix D or equivalently S D D to a multiple of the identity matrix Keeping in mind the constraint S gt J a legitimate choice is S 2 Bw I To set S to this value enter 4 29 4 IMILab NEWSYS setmvar LMISYS S 2 The second argument is the variable identifier S and the third argument is the value to which S should be set Here the value 2 is shorthand for 2 by I The resulting system NEWSYS reads A X 4XA 2CT XB R BTX 21 X gt 0 2I gt l Note that the last LMI is now free of variable and trivially satisfied It could therefore be deleted by NEWSYS dellmi NEWSYS 3 or NEWSYS dellmi NEWSYS Slmi if Slmi is the identifier returned by newlmi 4 30
115. r ureal ultidyn ucomplex and ucomplexm elements the command usample will generate a random instance i e not uncertain of the element within its modeled range For example usample p1 creates a random instance of the uncertain real parameter p1 With an integer argument whole arrays of instances can be created For instance 1 Building Uncertain Models 1 4 usample p4 100 generates an array of 100 instances of the ultidyn object p4 See Generate Samples of Uncertain Systems on page 1 53 to learn more about usample Uncertain Real Parameters Uncertain Real Parameters An uncertain real parameter is the Control Design Block that represents a real number whose value is uncertain Uncertain real parameters have a name the Name property and a nominal value the NominalValue property Several other properties PlusMinus Range Percentage describe the uncertainty in parameter values All properties of a ureal can be accessed through get and set The properties are Properties Meaning Class Name Internal name char NominalValue Nominal value of element double Mode Signifies which description char from PlusMinus Range Percentage of uncertainty is invariant when NominalValue is changed PlusMinus Additive variation scalar or 1x2 double Range Numerical range 1x2 double Percentage Additive variation of absolute value of scalar or 1x2 double nominal AutoSimplify off
116. re users Properties of uss Objects Create a USS object and view the properties p1 ureal p1 10 pe 50 p2 ureal p2 3 plusm 5 1 2 p3 ureal p3 0 A p1 p2 0 p1 B p2 p2 p3 C 1 031 1 p3 D 0 0 sys ss A B C D Create uss model get sys a 2x2 umat b 2x1 umat c 2x2 umat d 2x1 double e StateName 2x1 cell StateUnit 2x1 cell NominalValue 2x1 ss Uncertainty 1x1 struct InternalDelay 0x1 double InputDelay 0 1 33 T Building Uncertain Models OutputDelay 2x1 double Ts 0 TimeUnit seconds InputName InputUnit InputGroup 1x1 struct OutputName 2x1 cell OutputUnit 2x1 cell OutputGroup 1x1 struct Name Notes UserData SamplingGrid 1x1 struct The properties a b c d and StateName behave in exactly the same manner as in Control System Toolbox ss objects The properties InputName OutputName InputGroup and OutputGroup behave in exactly the same manner as all of the Control System Toolbox objects ss zpk tf and frd The NominalValue is a Control System Toolbox ss object and hence all methods for ss objects are available For instance compute the poles and step response of the nominal system pole sys NominalValue ans 10 10 step sys NominalValue 1 34 Uncertain State Space Models uss Amplitude 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 Time seconds Just as wi
117. real parameters 0n uncertain complex parameters x uncertain complex matrices H4 Hpg uncertain gain bounded linear time invariant dynamics Ep and uncertain positive real linear time invariant dynamics P P9 Decomposing Uncertain Objects Write A o H E P to indicate this dependence Using 1ftdata A can be decomposed into two separate pieces M and A p H E P with the following properties e Mis certain i e if A is uss then Mis ss if A is umat then M is double if A is ufrd then M is frd e Ais always a umat depending on the same uncertain elements as A with ranges bounds weights etc unaltered The form of A is block diagonal with elements made up of the normalizing functions acting on the individual uncertain elements F p 0 0 0 0 0 F E 0 0 0 A p H E P 0 0 F H 0 0 0 0 0 F E 0 0 0 0 0 F P A p H E P is given by a linear fractional transformation of M and A o H E P 1 A p Mog Ma A p H E P 1 MyA p H E P M32 The order of the normalized elements making up A is not the simple order shown above It is actually the same order as given by the command fieldnames M Uncertainty See Advanced Syntax of lftdata on page 1 77 for more information Syntax of Iftdata The decomposition is carried out by the command 1ftdata You can create a 2 by 2 umat named A using three uncertain real parameters delta ureal delta 2 eta
118. resentation Ap x14 xyAy lt 0 This is no longer true however when the number of variable terms is nearly equal to or greater than the number N of decision variables in the problem If your LMI problem has few free scalar variables but many terms in each LMI it is therefore preferable to rewrite it as Equation 4 15 and to specify it in this form Each scalar variable x is then declared independently and the LMI terms are of the form x Aj If M denotes the total row size of the LMI system and N the total number of scalar decision variables the flop count per iteration for the feasp and mincx solvers is proportional to N when the least squares problem is solved via Cholesly factorization of the Hessian matrix default 2 M by N when numerical instabilities warrant the use of QR factorization instead While the theory guarantees a worst case iteration count proportional to M the number of iterations actually performed grows slowly with M in most problems Finally while feasp and mincx are comparable in complexity gevp typically demands more computational effort Make sure that your LMI problem cannot be solved with mincx before using gevp Solving M P XQ Q X P lt 0 In many output feedback synthesis problems the design can be performed in two steps 1 Compute a closed loop Lyapunov function via LMI optimization 2 Given this Lyapunov function derive the controller state space matrices by solving an LMI of th
119. rf and wcgain all compute single scalar attributes of the system performance degradation curve Redraw the system performance degradation curve with 3 additional curves a hyperbola defined by xy 1 a vertical line drawn at the uncertainty bound 1 and a vertical line tangent to the asymptotic behavior of the performance degradation curve at large uncertainty bounds These are used to define three robustness measures explained next 3 WCGain uses bound of 1 on normalized uncertainty Nw M oO Mh Uncertainty level at which system can become unstable h ol y 1 x Curve in uncertainty size vs performance tradeoff space Maximum System Gain over Uncertainty f ol r t t i j t SN StabMarg 1 9 0 0 5 1 1 5 2 2 5 3 Bound on Normalized Uncertainty 2 Generalized Robustness Analysis Robust Stability Margin 2 4 The robust stability margin StabMarg is the size of the smallest deviation from nominal of the uncertain elements that leads to system instability 3 Robust Stability Margin nh oO ht System performance Uncertainty level degradation curve at which system can EN become unstable Maximum System Gain over Uncertainty un 1 0 5 StabMarg 1 9 0 0 0 5 1 1 5 2 2 5 3 Bound on Normalized Uncertainty System instability is equivalent to the system gain becoming arbitrarily large and hence characterized by the vertical line tangent to the asymptoti
120. riterion 22 28 13 16 Quadratic stability of differential inclusions 15 8 Lyapunov stability of parameter dependent systems 12 LMI Applications Input state output properties of LTI systems invariant ellipsoids decay rate etc 9 Multi model multi objective state feedback design 4 17 8 9 10 Robust pole placement Optimal LQG control 9 Robust H control 11 14 Multi objective H synthesis 18 23 10 18 Design of robust gain scheduled controllers 5 2 Control of stochastic systems 9 e Weighted interpolation problems 9 To hint at the principles underlying LMI design let s review the LMI formulations of a few typical design objectives Stability The stability of the dynamic system x Ax is equivalent to the feasibility of the following problem Find P P such that A P PA lt 0 P gt I This can be generalized to linear differential inclusions LDI x A t x where A t varies in the convex envelope of a set of LTI models n N A t e Co Ayj An X ajA Qj 2 0 a 1 i 1 i 1 A sufficient condition for the asymptotic stability of this LDI is the feasibility of Find P P such that A P PA lt 0 P gt TI 3 7 3 Introduction to Linear Matrix Inequalities RMS Gain The random mean squares RMS gain of a stable LTI system x Ax Bu y Cx Du is the largest input output gain over all bo
121. rtain Systems via Linear Control IEEE Trans Aut Contr AC 28 1988 pp 848 850 5 Becker G and Packard P Robust Performance of Linear Parametrically Varying Systems Using Parametrically Dependent Linear Feedback Systems and Control Letters 23 1994 pp 205 215 6 Bendsoe M P A Ben Tal and J Zowe Optimization Methods for Truss Geometry and Topology Design to appear in Structural Optimization 7 Ben Tal A and A Nemirovski Potential Reduction Polynomial Time Method for Truss Topology Design to appear in SIAM J Contr Opt 8 Boyd S and Q Yang Structured and Simultaneous Lyapunov Functions for System Stability Problems Int J Contr 49 1989 pp 2215 2240 9 Boyd S L El Ghaoui E Feron and V Balakrishnan Linear Matrix Inequalities in Systems and Control Theory SIAM books Philadelphia 1994 10 Chilali M and P Gahinet H Design with Pole Placement Constraints an LMI Approach to appear in IEEE Trans Aut Contr Also in Proc Conf Dec Contr 1994 pp 5538 558 11 Gahinet P and P Apkarian A Linear Matrix Inequality Approach to H Control Int J Robust and Nonlinear Contr 4 1994 pp 421 448 12 Gahinet P P Apkarian and M Chilali Affine Parameter Dependent Lyapunov Functions for Real Parametric Uncertainty Proc Conf Dec Contr 1994 pp 2026 2081 3 11 3 Introduction to Linear Matrix Inequali
122. s addition subtraction multiplication i e cascade operate properly and substitution with usubs is allowed However all of the analysis tools e g robuststab do not handle these types of uncertain elements As such these elements do not provide a significant amount of usability and their role in the user s guide is small You can create a 2 by 3 udyn element Check its size and properties m m udyn m 2 3 Uncertain dynamics m with 2 outputs and 3 inputs get m NominalValue AutoSimplify TS TimeUnit InputName InputUnit InputGroup OutputName OutputUnit OutputGroup Name Notes UserData 1 22 2x3 ss basic 0 seconds 3x1 cell 3x1 cell 1x1 struct 2x1 cell 2x1 cell 1x1 struct im Uncertain Matrices Uncertain Matrices Uncertain matrices class umat are built from doubles and uncertain elements using traditional MATLAB matrix building syntax Uncertain matrices can be added subtracted multiplied inverted transposed etc resulting in uncertain matrices The rows and columns of an uncertain matrix are referenced in the same manner that MATLAB references rows and columns of an array using parenthesis and integer indices The NominalValue of a uncertain matrix is the result obtained when all uncertain elements are replaced with their own NominalValue The uncertain elements making up a umat are accessible through the Uncertainty gateway an
123. s with their nominal values M NominalValue ans 3 0000 0 1000 10 0000 3 1000 1 0000 3 0000 The Uncertainty property is a structure whose fields are the uncertain elements Control Design Blocks of M M Uncertainty ans a 1x1 ureal b 1x1 ureal M Uncertainty a ans Uncertain real parameter a with nominal value 3 and variability 1 1 Direct access to the elements is facilitated through Uncertainty Check the Range of the uncertain element named a within M then change it M Uncertainty a Range ans 2 4 M Uncertainty a Range 2 5 5 M UMAT 3 Rows 2 Columns a real nominal 3 variability 0 5 2 2 occurrences b real nominal 10 variability 20 20 3 occurrences 1 25 1 Building Uncertain Models 1 26 The change to the uncertain real parameter a only took place within M Verify that the variable a in the workspace is no longer the same as the variable a within M isequal M Uncertainty a a ans 0 Note that combining elements which have a common internal name but different properties leads to an error For instance subtracting the two elements gives an error not 0 M Uncertainty a a Error using gt ndlft 1ftmask Elements named a have different properties Row and Column Referencing Row and Column Referencing Standard Row Column referencing is allowed Note however that single indexing is only allowed if the umat is a column or a row
124. sec Rotary Gust qGust rad sec Pitch Accel Pitch Channel qdot radsec2 3 Specify uncertain variables for the gain blocks Zd and Mw in the Aircraft Dynamics Model subsystem spec Type Expression spec Specification ureal Zd 63 9979 Percentage 20 BlockSubs 1 struct Name slexAircraftExample Aircraft Dynamics Model Gain5 Value spec 5 24 Linearize Block to Uncertain Model spec Specification ureal Mw 63 9979 Percentage 20 BlockSubs 2 struct Name slexAircraftExample Aircraft Dynamics Model Gain4 Value spec 4 Compute the uncertain linearization mdl slexAircraftExample sys linearize md1 BlockSubs sys Uncertain continuous time state space model with 1 outputs 1 inputs 7 states The model uncertainty consists of the following blocks Mw Uncertain real nominal 64 variability 20 20 1 occurrences Zd Uncertain real nominal 64 variability 20 20 1 occurrences Type sys NominalValue to see the nominal value get sys to see all properties and sys Uncertainty to interac sys is an uncertain state space USS model 5 25 5 Analyzing Uncertainty Effects in Simulink Using Uncertain Linearization for Analysis or Control Design After computing an uncertain linearization you can perform one of the following Perform robustness analysis Perform robust control design 5 26 Analyzing Stability Margin of Simul
125. ss of the problem and is not automatically added by gevp An initial guess xinit for x can be supplied to mincx or gevp Use mat2dec to derive Xinit from given values of the matrix variables Xj Xx The following example illustrates the use of the mincx solver 4 21 4 IMILab Minimize Linear Objectives under LMI Constraints Consider the optimization problem Minimize Trace X subject to ATX XA XBB X Q lt 0 with data 1 2 1 1 1 1 0 A 3 2 1i B 0 Q 1 3 12l 1 2 1 1 0 12 36 It can be shown that the minimizer X is simply the stabilizing solution of the algebraic Riccati equation ATX XA XBB X Q 0 This solution can be computed directly with the Riccati solver care and compared to the minimizer returned by mincx From an LMI optimization standpoint the problem specified in Equation 4 9 is equivalent to the following linear objective minimization problem Minimize Trace X subject to A X XA Q XB lt Bx I Since Trace X is a linear function of the entries of X this problem falls within the scope of the mincx solver and can be numerically solved as follows 1 Define the LMI constraint of Equation 4 9 by the sequence of commands setlmis X Ilmivar 1 3 1 variable X full symmetric lmiterm 111 lmiterm 1 1 1 4 22 Minimize Linear Objectives under LMI Constraints lmiterm 1 2 2 0 Imiterm 1 2 1 X LMIs getlmis Write the objective Trace X as c x
126. stom Simulink Blocks Specify Uncertain Linearization for Core or Custom Simulink Blocks In some cases you cannot use Uncertain State Space blocks in the Simulink model because you share the model or generate code You can still account for uncertainty in your linear analysis without specifying uncertainty using Uncertain State Space blocks Robust Control Toolbox lets you specify a core or custom Simulink block to linearize to an uncertain variable The linearization produces an uncertain state space uss model The specified uncertainty associates only with the block and does not affect the model simulation For more information see Controlling Block Linearization in the Simulink Control Design documentation Note If you have Uncertain State Space blocks in the model and want to obtain an uncertain state space model see Obtain Uncertain State Space Model from Simulink Model on page 5 20 To specify blocks to linearize to uncertain variables and obtain an uncertain state space model 1 Prerequisites Create or open the Simulink model Specify linearization input and output points using the Simulink Control Design getlinio or linio commands For this example you can open the model rct_ulinearize_ builtin 2 Specify a block to linearize to an uncertain variable a Right click the block and select Linear Analysis gt Specify Selected Block Linearization 5 21 5 Analyzing Uncertainty Effects in Simulink
127. stom Simulink Blocks on page 5 21 You can then use the uncertain state space model to analyze robustness in the linear operating range Simulink Blocks for Analyzing Uncertainty Simulink Blocks for Analyzing Uncertainty Robust Control Toolbox software provides an Uncertain State Space block to model parametric and dynamic uncertainty in Simulink The block library also contains a MultiPlot Graph block that you use with the Uncertain State Space block to plot and visualize Monte Carlo simulation responses To open the Robust Control Toolbox block library type the following command at the MATLAB prompt RCTblocks The block library opens as shown in the following figure ian 7 ie aes ee e Py Library RCTblocks S File Edit View Display Diagram A A amp o gt BB o RCTblocks PalRCTblocks ii Q ss urealfa 5 5 1 1 p a Uncertain State Space MultiPlot Graph Ready 100 Alternatively on the Home tab click Simulink Library to launch to Library Browser In the Library Browser select Robust Control Toolbox 5 3 5 Analyzing Uncertainty Effects in Simulink Specify Uncertainty Using Uncertain State Space Blocks How to Specify Uncertainty in Uncertain State Space Blocks on page 5 4 e Next Steps on page 5 6 How to Specify Uncertainty in Uncertain State Space Blocks Specifying uncertainty in the Uncertain State Space block makes the uncertainty a part of the S
128. th the umat class the Uncertainty property is a structure containing the uncertain elements Direct access to the elements is facilitated through Uncertainty Check the Range of the uncertain element named p2 within sys then change its left endpoint sys Uncertainty p2 range ans 2 5000 4 2000 sys Uncertainty p2 range 1 2 1 35 T Building Uncertain Models Create Uncertain State Space Model 1 36 Uncertain systems class uss are built from of certain and or uncertain state space matrices usually using the SS command In the example below the A B and C matrices are made up of uncertain real parameters Packing them together with the SS command results in a continuous time uncertain system You can create three uncertain real parameters Then create 3 uncertain matrices A B and C and one double matrix D p1 ureal p1 10 pe 50 p2 ureal p2 3 plusm 5 1 2 p3 ureal p3 0 A p1 p2 0 p1 B p2 p2tp3 C 1 031 1 p3 D 0 0 Pack the 4 matrices together using the ss command This results in a continuous time 2 output 1 input 2 state uncertain system sys ss A B C D USS 2 States 2 Outputs 1 Input Continuous System p1 real nominal 10 variability 50 50 2 occurrences p2 real nominal 3 variability 0 5 1 2 2 occurrences p3 real nominal 0 variability 1 1 2 occurrences Sample Uncertain Systems Sample Uncertain Systems T
129. the frequency domain by linearizing the Simulink model with ULINEARIZE This computes an uncertain state space model USS that aggregates the uncertainty from all Uncertain State Space blocks Parameters Uncertain system variable uss a 5 5 1 1 Uncertainty value struct or to use nominal value 0 Initial states nominal dynamics 0 Initial states uncertain dynamics o ox cancel Help Apply 5 Analyzing Uncertainty Effects in Simulink Specify an uncertain state space model in the Uncertain system variable uss field The model must be an uss object or any other model that can be converted to uss such as umat ureal and ultidyn The model depends on a set of uncertain variables ureal or ultidyn and you can specify the model as one of the following e Function or expression that evaluates to an USS model For example ss ureal a 5 5 1 1 Variable defined in the MATLAB workspace For example unc_sys where unc_sys is defined as ss ureal a 5 5 1 1 in the workspace Specify values for the uncertain variables that the uncertain state space model you specify in step 3 uses For example if you specify the uncertain system as ureal g 2 tf 1 ureal tau 1 then you must specify values for the uncertain variables g and tau To do so enter a structure with fields g and tau in the Uncertainty value struct or to use nominal value field You can also enter to use the nomi
130. the results of an LMI optimization The function evallmi evaluates all variable terms in an LMI system for a given value of the vector of decision variables for instance the feasible or optimal vector returned by the LMI solvers Once this evaluation is performed the left and right sides of a particular LMI are returned by showlmi In the LMI problem considered in Minimize Linear Objectives under LMI Constraints on page 4 22 you can verify that the minimizer xopt returned by mincx satisfies the LMI constraint Equation 4 10 as follows evlmi evallmi LMIs xopt lhs rhs showlmi evlmi 1 The first command evaluates the system for the value xopt of the decision variables and the second command returns the left and right sides of the first and only LMI The negative definiteness of this LMI is checked by eig lhs rhs ans 2 0387e 04 3 9333e 05 1 8917e 07 4 6680e 01 4 27 4 IMILab Modify a System of LMIs 4 28 Once specified a system of LMIs can be modified in several ways with the functions dellmi delmvar and setmvar Deleting an LMI The first possibility is to remove an entire LMI from the system with dellmi For instance suppose that the LMI system of Specifying LMI System on page 4 8 is described in LMISYS and that we want to remove the positivity constraint on X This is done by NEWSYS dellmi LMISYS 2 where the second argument specifies deletion of the second LMI
131. ties 13 Haddad W M and D S Berstein Parameter Dependent Lyapunov Functions Constant Real Parameter Uncertainty and the Popov Criterion in Robust Analysis and Synthesis Part 1 and 2 Proc Conf Dec Contr 1991 pp 2274 2279 and 2632 2633 14 Iwasaki T and R E Skelton All Controllers for the General H Control Problem LMI Existence Conditions and State Space Formulas Automatica 30 1994 pp 1307 1317 15 Horisberger H P and P R Belanger Regulators for Linear Time Varying Plants with Uncertain Parameters IEEE Trans Aut Contr AC 21 1976 pp 705 708 16 How J P and S R Hall Connection between the Popov Stability Criterion and Bounds for Real Parameter Uncertainty Proc Amer Contr Conf 1993 pp 1084 1089 17 Khargonekar P P and M A Rotea Mixed H H Control a Convex Optimization Approach IEEE Trans Aut Contr 39 1991 pp 824 837 18 Masubuchi I A Ohara and N Suda LMI Based Controller Synthesis A Unified Formulation and Solution submitted to Int J Robust and Nonlinear Contr 1994 19 Nemirovski A and P Gahinet The Projective Method for Solving Linear Matrix Inequalities Proc Amer Contr Conf 1994 pp 840 844 20 Nesterov Yu and A Nemirovski Interior Point Polynomial Methods in Convex Programming Theory and Applications SIAM Books Philadelphia 1994 21 Packard A and J C Doyle Th
132. toSimplify basic with nominal value 2 1i and radius 1 Uncertain Complex Parameters and Matrices Sample the uncertain complex parameter at 400 values and plot in the complex plane Clearly the samples appear to be from a disc of radius 1 centered in the complex plane at the value 2 j asample usample a 400 plot asample 0 xlim 0 5 4 5 ylim 3 1 0 5 0 0 5 1 15 2 2 5 3 3 5 4 45 Uncertain Complex Matrices The uncertain complex matrix class ucomplexm represents the set of matrices given by the formula 1 Building Uncertain Models 1 20 N W AWp where N Wz and Wr are known matrices and A is any complex matrix with A lt 1 All properties of a ucomplexm are can be accessed with get and set The properties are Properties Meaning Class Name Internal Name char NominalValue Nominal value of element double WL Left weight double WR Right weight double AutoSimplify off basic full char The simplest construction requires only a name and nominal value The default left and right weight matrices are identity You can create a 4 by 3 ucomplexm element and view its properties m Uncertain Complex get m Name NominalValue WL WR AutoSimplify m NominalValue ans 1 2 4 5 7 8 10 11 1 m WL ans 1 0 0 1 0 0 0 0 NOD Ww o 00 Matrix Name m 4x3 m 4x3 double 4x4 double 3x3 double
133. ulate the model The MultiPlot Graph shows the following responses corresponding to the nominal and manually defined values of a 5 11 5 Analyzing Uncertainty Effects in Simulink OG See Soe a ee Ee 5 Pick a random value of a in its uncertainty range To do so double click the Uncertain State Space block and type usample ufind unc_sys in the Uncertainty value field Uncertain system variable uss Uncertainty value struct or to use nominal value usample ufind unc_sys 6 Analyze the uncertainty effects by simulating the model ten times using the following commands 5 12 Vary Uncertainty Values Using Individual Uncertain State Space Blocks for i 1 10 sim rct_sim_ex1 10 end During simulation the software samples the uncertain variable a in its uncertainty range 2 0 5 and shows the simulated response for each sample value The plots cycle through seven different colors and the last response appears in red File Edit View Insert Tools Desktop Window Help QUGMSs h ASAV90RC4 Aa 08 ao 10 possesses ngs eee a a fa en E a lt i i aaa aan EEA Sees I e 2 4 6 8 10 a Tip You can clear the plots in the MultiPlot Graph block before you run the simulation 5 Analyzing Uncertainty Effects in Simulink Vary Uncertainty Values Across Multiple Uncertain State Space Blocks This example shows the workflow for varying uncertainty values across mult
134. unded inputs u t This gain is the global minimum of the following linear objective minimization problem 1 25 26 Minimize y over X X and y such that ATX XA XB CT BX yI D lt 0 C D yI and X gt O LQG Performance For a stable LTI system x Ax Bw y Cx where w is a white noise disturbance with unit covariance the LQG or H performance IIGll is defined by T G Jim al fosoa 0 gt f G jw G jo do 3 8 LMI Applications It can be shown that G int Trace CPC7 AP PAT BB lt 0 Hence lek is the global minimum of the LMI problem Minimize Trace Q over the symmetric matrices P Q such that AP PA BB lt 0 and Q CP T gt 0 PC P Again this is a linear objective minimization problem since the objective Trace Q is linear in the decision variables free entries of P Q 3 9 3 Introduction to Linear Matrix Inequalities Further Mathematical Background 3 10 Efficient interior point algorithms are now available to solve the three generic LMI problems Equation 3 2 Equation 3 4 defined in LMI Applications on page 3 6 These algorithms have a polynomial time complexity That is the number Me of flops needed to compute an accurate solution is bounded by M N log Vie where M is the total row size of the LMI system N is the total number of scalar decision variables and Vis a data dependent scaling factor Robust Control Toolbox software implements
135. ver Uncertainty a b ol S Worst Case Gain Measure 0 0 5 1 1 5 2 2 5 3 Bound on Normalized Uncertainty On the graph this is the vertical coordinate of the performance degradation curve as it crosses the vertical line drawn at the uncertainty bound 1 Each measure captures a single scalar attribute of the system performance degradation curve Mathematically they are independent quantities answering subtlely different questions Consequently for two uncertain systems SySA and sysB it is possible that the StabMarg of sySA is larger than the StabMarg of sysB though the PerfMarg of sySA is smaller than the PerfMarg of sysB Nevertheless they are useful metrics for concise description of the robustness of a system USS or Ufrd due to various uncertain elements Introduction to Linear Matrix Inequalities Linear Matrix Inequalities on page 3 2 e LMIs and LMI Problems on page 3 4 LMI Applications on page 3 6 e Further Mathematical Background on page 3 10 Bibliography on page 3 11 3 Introduction to Linear Matrix Inequalities Linear Matrix Inequalities 3 2 Linear Matrix Inequalities LMIs and LMI techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design Three factors make LMI techniques appealing e A variety of design specifications and constraints can be expressed as LMIs Once formulat
136. wn about the uncertain frequency response Specifically if Ais an ultidyn element and if y denotes the value of the Bound property then the element represents the set of all stable linear time invariant systems whose frequency response satisfies certain conditions If Type is GainBounded G A lt for all frequencies When Type is GainBounded the default value for Bound e y is 1 The NominalValue of A is always the 0 matrix If Type is PositiveReal A A gt 2y for all frequencies When Type is PositiveReal the default value for Bound i e y is 0 The NominalValue is always y 1 2 y I All properties of a ultidyn are can be accessed with get and set although the NominalValue is determined from Type and Bound and not accessible with set The properties are Properties Meaning Class Name Internal Name char NominalValue Nominal value of element See above Type GainBounded PositiveReal char Bound Norm bound or minimum real scalar double SampleStateDimen State space dimension of random samples scalar double of this uncertain element Uncertain LTI Dynamics Elements Properties Meaning Class SampleMaxFrequen Maximum natural frequency for random scalar double sampling AutoSimplify off basic full char The SampleStateDim property specifies the state dimension of random samples of the element when using usamp1e The default valu

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