The LRP Toolkit User`s Manual - Forge
Contents
1. 1 e lrp boundCond_xsin 1rp for x 0 of a sine wave of unit height and a period of one e lrp boundCond_xsin 1rp T for x 0 of a sine wave of unit height and a period of T For output y e lrp boundCond_y1 1rp for yo p 1 e lrp boundCond_ysin 1rp for yo p of a sine wave of unit height and a period of one e lrp boundCond_ysin 1rp T for yo p of a sine wave of unit height and a period of T 2 5 Visualization of the process dynamics The Toolkit can be used to visualise the process dynamics All the standard methods of visualisation are available 2 5 1 Along the pass plots This plot is useful for showing the dynamics of the whole pass along the p dimension It is also very important tool for comparing the shape of a few passes This plot emphasizes the p direction To draw this plot the user must first select the pass k for which the plot is to be made Upto 32 passes can be shown simultanously On the X points p are shown the value is given on the Y axis The example of such plot is shown on figure 2 2 2 THE LRP TOOLKIT 11 File Tools Slp a lt 4 Output Y_1 along the pass profile 2 3 4 5 6 7 Edit Output _1 along the pass profile 10 points 3 2 5 2 Pass to pass plot Figure 2 2 Along the pass plot example This plot is useful for showing the change of dynamics over all passes for a selected point p This plot emphasizes the k dire2. To create the CoolMatrix variable the user must provide the allowed range of indices Note here that the Scilab notation of extending the size of the matrix when called with index larger than its former size is not supported as this is very error prone 3 Example To illustrate the LRP Toolkit consider the special case of 1 1 defined by the following matrices 0 19 0 07 0 19 0 8 0 1 0 2 A 0 02 0 85 0 49 B 0 7 Bo 03 0 1 0 84 0 01 0 75 0 3 0 6 0 2 0 5 0 3 0 1 0 2 0 44 0 26 g 0 4 as one oe with pass length a 10 and boundary conditions 7 T yo p sin p sin p fr O0 lt p lt a zp 0 0 9 0 9 09 for k gt 0 Note that the argument of the sin function is given in radians By using the stability analysis options offered by the toolkit we conclude that the process is asymptotically stable but unstable along the pass The plots of the pass profiles for k 3 5 7 12 15 20 confirm this note the amplitudes of pass profile vectors increase significantly see Fig 3 1 as the process dynamics evolve Suppose now that the task is to design a control law of the form 1 6 to ensure stability along the pass This can be undertaken using the toolkit sta bilizing control law design options as detailed in Section 2 6 2 For demon stration purposes we consider the case in actual fact the theory behind this design produces a family of possible choices for K when the control law matrix K of 1 6 is g
3. if you are unsure about it try the contrib 1rp directory in the main Scilab directory You will now see the LRP Toolkit user interface shown on fig 2 1 To create a new LRP system click Create a new system button positioned in the upper left corner of the GUI This will start the new system creator You will need to give the following information e the number of states n inputs r and outputs m in the system e all the system matrices A B BO C D DO e the pass length a and the number of passes to simulate p 2 THE LRP TOOLKIT 9 Main form open loop system E Bj x File Edit Display Create a new system Change system matrices Plots Pass length alpha fi 0 Number of passes beta 20 Stability Change parameters Controlar design Initial conditions Save Tex Load Inputs Update variables Ready eal Figure 2 1 The main Toolkit GUI To simplify entering the matrix values click on the ellipsis button Al ternatively you may use the Scilab functions that return an appropriately dimensioned matrix If you use functions you cannot use the ellipsis button Moreover you are responsible for checking the dimensions of matrices Note that the resulting process will have zero input and standard boundary Standard boundary conditions conditions zk 1 0 0 k gt 0 yo p 1 You may change them using the Initial conditions button
4. y p E R denotes the pass profile vector and u p R denotes the vector of control inputs To complete the process description it is necessary to specify the boundary conditions i e the initial state vector at the beginning of each pass and the initial pass profile Here no loss of generality arises from assuming k 1 0 dep b gt 0 yo P fp where the n x 1 vector d 41 contains known constant entries and f p is an m x 1 vector whose entries are known functions of p 1 2 The stability theory RO92 for linear repetitive processes consists of two basic concepts termed asymptotic stability and stability along the pass respectively Noting again the unique control problem these properties de mand that bounded sequences of inputs produce bounded sequences of pass profiles where bounded is defined in term of the norm on the underlying function space Asymptotic stability guarantees this property over the finite and fixed pass length and as a consequence there exists the so called limit pass profile i e after the sufficient number of passes the process dynamics can be replaced by those of a 1D discrete linear system The fact that the pass length is finite however means that this limit profile could be unstable as a 1D discrete linear system Over a finite duration even an unstable 1D linear system is guaranteed to produce a bounded output Stability along the pass prevents this situation from arising by
5. by a 1D discrete linear systems state space model is simple to check by means of Theorem 1 as it requires that all eigenvalues of the matrix Do in 1 1 have modulus strictly less than unity this task is much harder for the case when the pass initial condition is an explicit function of the previous pass profile Asymptotic stability test is implemented in the Toolkit as the stAsymptotici test Its LMI counterpart of Theorem 2 is also implemented This function is termed stAsymptoticLMI1 Stability along the pass however is a much more challenging task in this respect In which context it has been noted in the introduction that a 1D Nyquist test can be used but it has not proved a suitable basis to undertake control law design in contrast to the 1D case This has led to the use of LMI based tests see Theorem 4 which are sufficient but not necessary but can be executed using computations with constant entry matrices and crucially provide a basis for control law design To test the stability along the pass in the Toolkit by means of Theorem 4 the stAlongThePassLMI1 test can be utilised In addition to the most important stability definitions asymptotic and along the pass the Toolkit includes the tests for the so called practical stability This notion lies between the asymptotic and along the pass sta bility The tests implemented according to Theorems 5 and 6 are available as stPracticali and stPracticalLMI1 respectively Both st
6. demanding the bounded ness property uniformly i e independent of the pass length It is easy to see that asymptotic stability is a necessary condition for stability along the pass Many sets of necessary and sufficient conditions for these properties are known and some of them can be tested by direct application of 1D linear systems stability tests e g Nyquist plots A major drawback however is that these do not provide a basis on which to also address the question of control law design for stability and or performance This has led in recent years to the use of Linear Matrix Inequality LMI techniques see e g BGFB94 and there now exists a large volume of results on the design of physically implementable control laws for the detailed description refer e g to Sul06 GLR 03 and references therein The unique control problem for the Linear Repetitive Processes is that the output sequence of pass profiles yk k gt 0 can contain oscillations which can increase in amplitude in the from pass to pass direction k Hence a natural definition of stability is to request that bounded input sequences produce bounded output pass profiles sequences Stability theory Rogers and Owens 1992 for LRPs is based on an abstract model of process dynamics in a Banach space here denoted by Ea of the form Yki Layk bryk 0 1 3 In this model yk E Ea denotes the pass profile on the pass k La is a bounded linear operator which map
7. dynamics This problem has been considered previously see e g Gra99 GGGR05 but the resulting software was diffucult to extend and or based on commercial environments Some of them can only be used for educational purposes The detailed discussion about available software packages for the linear repetitive processes is given in HCG 06 The above arguments confirm that there is no high quality reliable software to support the analysis and design of control laws for use with for ex ample experimental facilities RHL 05 where ILC control laws designed in a repetitive process setting could be experimentally verified Moreover the currently available tools do not allow easy inclusion of new algorithms Also there are no software packages for analysis and synthesis based on open source software the Java based toolkit GGGR05 cannot be used for this purpose due to its limitations This causes licensing problems and reduces the areas of potential use e g by students To overcome those limitations a development of a new Toolkit has been initiated To enhance the usefulness of this tool the Scilab SG environment has been chosen The main advantage of this option over other packages is the open source license of the Scilab and a rapidly growing number of users including PSA Peugeot Citroen Renault Dassault Aviation and many others The introduction to the Scilab environment is given in GDB 98 and CCN06 The LMI component used in t
8. of a structure used by the Toolkit to represent such process gt positive definitiveness of a matrix lt negative definiteness of a matrix r M the spectral radius the largest module of an eigenvalue of matrix M Bibliography Ben00 BGFB94 CCNO6 DD03 GDBt 98 GGGRO5 GLR 03 GPS 03 Gra99 HCG 06 MC A03 S E Benton Analysis and Control of Functions of Linear Repetitive Processes PhD Dissertation University of Southamp ton UK 2000 S Boyd L E Ghaoui E Feron and V Balakrishnan Linear Matrix Inequalities In System And Control Theory volume 15 of SIAM studies in applied mathematics SIAM Philadelphia 1994 S L Campbell J P Chancelier and R Nikoukhah Modeling and Simulation in Scilab Scicos Springer 2006 R D Andrea and G E Dullerud Distributed control design for spatially interconnected systems EEE Transactions on Auto matic Control 48 1 1478 1495 2003 Claude Gomez Francois Delebecque Carey Bunks Jean Philippe Chancelier Serge Steer and Ramine Nikoukhah ed itors Engineering and Scientific Computing with Scilab with Cdrom Birkhauser Boston 1998 J Gramacki A Gramacki K Galkowski and E Rogers Java based toolbox for linear repetitive processes Proc 2nd Int Conf on Inform in Control Autom and Robotics ICINCO 1 182 187 2005 K Galkowski J Lam E Rogers S Xu B Sulikowski W Paszke and D H Owens Lmi
9. tem are not acceptable Hence a stronger concept of stability i e stability along the pass must be used This stronger stability demands the BIBO property to hold independently of dynamics i e in the direction along the pass p and from pass to pass k Introduce the formal definition of stability along the pass as follows Definition 2 RO92 Ben00 Sul06 In terms of the abstract model of 1 3 stability along the pass holds provided that there exist real numbers Ma gt 0 and Nx 0 1 which are independent of a such that L lt MA k gt 0 Theorem 3 GPS 03 Discrete linear repetitive process described by 1 1 and 2 1 is stable along the pass if and only if the 2D characteristic poly nomial NO I mie zA z Bo Cla 22 det 2C In E z2Do 1 4 V 21 22 E U where U diag 21 22 a lt 1 z2 lt 1 Theorem 1 4 is very difficult to use in practice Note that it requires to check an infinite number of values To overcome this problem it is recommended to use the following theorem 1 Introduction 4 Stability along the pass Practical stability 1 1 2 STABILISATION The control law Theorem 4 GPS 03 Discrete linear repetitive process described by 1 1 and 2 1 are stable along the pass if there exist matrices P PT gt 0 and Q QT gt 0 satisfying the following LMI Breed 7 Be pe AR ae EP 1 5 A gt PA Az PA Q The proof can be found in e g GLR 03 Note tha
10. toolkit does not enforce any constrains on its structure The only require ment is that there must be a field solutionExists of integer type note here the difference between lrp stability and lrp controller field where a boolean type is used instead which informs the user whether the system is stable unstable or the method used does not provide the answer The solutionExists field can have the following values e 1 test is inconclusive In this case the only alternative is to use another test e 0 the process is unstable e 1 the process is stable Obviously the SCILAB works in finite precision arithmetic and hence numer ical errors can influence the results To implement a stability test the programmer should write a function that returns an lrp structure and is given one as the only argument Note here that a better solution would be to use a by variable or by pointer passing method but this is not implemented in SCILAB This function must be placed in the stability directory of the toolkit To conserve memory the lrp stability field is dynamically created If the user does not complete any stability tests then this field does not exist 2 7 3 THE BOUNDARY CONDITIONS The boundary conditions of 2 1 can be entered either as a set of values or by providing a function that returns the appropriate value given the pass k and point number p By default if the user does not supply the boundary conditions they are a
11. 2 1 To change the pass length and the number of simulated passes click the Change parameters button If any of the values were changed outside the Toolkit GUI click Update matrices button to feed the proper values into the GUI To close the GUI click Exit 2 4 2 Creating new LRP structure by using createLRPModel This function is especially useful in scripts This function can be called with many different arguments For detailed description type help createLRPModel in the Scilab window To create a sample system use lrp createLRPMode1l The created sys tem has 2 states 3 inputs and 4 outputs The system matrices are random from 0 to 1 The pass length a 10 and number of simulated passes 3 15 The external input u equals zero To change the boundary conditions see section 2 4 4 2 THE LRP TOOLKIT 10 2 4 3 Creating new LRP structure by using createStubLRP If you need greater control over the creation process use the createStubLRP function to create the necessary fields Then fill all the stub values with data If you do not fill all the stub values the Toolkit will not function correctly To check the correctness of the LRP structure use the checkLRP function Do not forget to set the lrp controllers field 2 4 4 Setting the boundary conditions To set the boundary conditions of the LRP structure named 1rp use the appropriate generator function For state x e lrp boundCond_x1 1rp for x 0
12. ILAB script files and functions All the basic process parameters are included in the lrp structure which is implemented as a new type based on the tlist typed list native SCILAB datatype with the following fields e 1rp_Sys_Cla string name of the data type e The 1lrp mat field of type lrp_Sys_Cla_Mat used for storing the model matrices see 1 1 Currently it contains the values of A B BO C D DO e The lrp dim field of type lrp_Sys_Cla_Dim used for storing the model dimensions Currently it has the following fields alpha positive integer pass length number of points on each pass denoted a in 1 1 beta positive integer number of passes over which the simulation will run n r m positive integers dimensions of state input and output vectors respectively e The lrp ini field of type lrp_Sys_Cla_Ini used for storing the initial conditions Currently it has two fields e x0 yO real matrices boundary conditions see 2 1 Note that zk 1 0 in 2 1 is here denoted by x0 for simplicity e controller list list of known control laws for the process see below e indController positive integer index of current control law This field contains the index of currently active control law If indController 1 then no control law is applied stability list list of performed stability tests see below Note that the model of 1 1 does not impose any c
13. The LRP Toolkit User s Manual L Hladowski B Cichy K Galkowski E Rogers May 7 2009 1 Introduction The LRP Toolkit is a powerful set of utilities to simplify the development of new algorithms based on the Linear Repetitive Process LRP settings The Toolkit is ment to be used within the Scilab environment see www scilab org for details Before exploring the Toolkit functions it is necessary to introduce some theoretical background 1 1 The Linear Repetitive Process theory The state space model Repetitive processes are a distinct class of 2D systems i e information propagation in two independent directions of both system theoretic and applications interest The essential unique characteristic of such a process is a series of sweeps termed passes through a set of dynamics defined over a fixed finite duration known as the pass length denoted by a lt 00 On each pass an output termed the pass profile is produced which acts as a forcing function on and hence contributes to the dynamics of the next pass profile This in turn leads to the unique control problem in that the output sequence of pass profiles generated can contain oscillations that increase in amplitude in the pass to pass direction Hence these processes propagate information in two separate directions i e from pass to pass and along a pass respectively Physical examples include long wall coal cutting and metal rolling opera tions see for exam
14. ability properties can be investigated using the LMI technique One of the reasons of selecting the Scilab as the host platform for the Toolkit was the excellent LMI solver available for this platform as LMITOOL a Package for LMI Optimization in Scilab NDG Note also that provision is available to easily include existing or newly developed tests by simply implementing a single SCILAB function with no need to change the GUI which is shown on fig 2 4 2 6 2 CONTROL LAW DESIGN As outlined in the Background section the stabilisation of the linear repetive process is not always easy In developing new techniques it is beneficial to compare the newly developed method with the ones already known One approach would be to use the LMI tools already available in SCILAB and 2 THE LRP TOOLKIT 14 2 6 3 TeX EXPORT ies f Provide system malices Enter the system matrices k 0 1 2 3 maz 1 Pes vomber P 0 1 2 3 p max 1 Points meachpass ie At p Bug P Eye p Meal P C 2 Dy a p 2 P Matrix A In 2 34 rand hd triv R fina 25 6 41 23 221 rand f matrix D ioj x 331 rand 148 79 Enter values for matrix D pf f pa Spa 0 0011 ze fz54 4 l are 4 2 x trand 369 41 rand 137 132 rand Randomize al Nen Figure 2 5 Matrix wizard windows main and the m
15. ation the main directory of the Toolkit is empty it will be deleted as well The installer uses the highly reliable Nullsoft Installation System NSIS For your convinience the installer will add an Uninstall information into the Windows registry 2 3 4 Automatic deinstallation 2 3 5 Manual installation The config file does not have any extension To uninstall the Toolkit run the uninstall exe program After using the provided uninstaller all Windows registry information is no longer needed and is removed This program removes the Toolkit without leaving any leftover files Note that if you change anything in any Toolkit file except bin and html files which can be easily recreated during uninstallation phase you will be asked if you want to remove such file This protects you from accidently deleting your custom functions To install the Toolkit manually unpack the ZIP file into any directory The Scilab team recommends the contrib subdirectory of the Scilab After copying all the files modify the config file with any text editor In this file set the LRPHomeDir to point to the directory where the Toolkit is created Modify other entries in this file as well If you do not modify the config file the Toolkit will not work properly 2 THE LRP TOOLKIT 2 3 6 Manual deinstallation To uninstall the Toolkit simply delete its directory Do not forget to save the files you modified if you need them 2 4 Creating t
16. atrix entry code the methods by hand However this approach requires substantial knowledge about the internal data structures used by the LMI solver and this is not ideal Consequently a set of ready to use control law design templates have been incorporated into the toolkit Moreover expanding it by adding additional control law design methods is a simple task Presentation of simulation results can be a time consuming task especially when considering large matrices which are often encountered in this area To simplify this tedious task the toolkit has been equipped with BTEX export capabilities An essential novelty is the fact that the user needs to write the ATEX file adding the tags that will be replaced by the simulation results instead of using a complicated syntax of the previous Toolkit version It is possible to include any number of plots both 2D and 3D On each plot any number of points passes can be displayed which is an essential difference with the interactive plots discussed earlier Note however that 32 points passes on each plot can be displayed in unique colors a great improvement over the previous version The process of plot selection is simplified by the use of an interactive wizard It is also possible to export simulation data from the script 2 6 4 USABILITY ENCHANCEMENTS One of the design goals was to make this new toolkit as user friendly as possible To achieve this a new system wizard for enteri
17. based stability analysis and robust controller design for discrete linear repetitive processes Int J Robust Nonlinear Control 13 1195 1211 2003 K Galkowski W Paszke B Sulikowski E Rogers and D H Owens LMI based stability analysis and robust controller design for discrete linear repetitive processes International Journal of Robust and Nonlinear Control 13 13 1195 1211 2003 J Gramacki Metody badania stabilno ci i stabilizacja liniowych dyskretnych procesow powtarzalnych PhD thesis Politechnika Zielonogorska 1999 L Hladowski B Cichy K Galkowski B Sulikowski and E Rogers Scilab compatible software for analysis and control of repetitive processes Proc IEEE International Symposium on Computer Aided Control Systems Design CACSD 2006 2006 H Melkote B Cloke and V Agarwal Modeling and compen sator designs for self servowriting in disk drives Proc American Control Conference 1 737 742 jun 4 6 2003 Bibliography 24 NDG Nul OARF00 pL RHL 05 RO92 SG SGROO5 Sul06 R Nikoukhah F Delebecque and L E Ghaoui LMITOOL a package for LMI optimization in scilab Nullsoft Nsis users manual D H Owens N Amann E Rogers and M French Analysis of linear iterative learning control schemes a 2D systems repeti tive processes approach Multidimensional Systems and Signal Processing 11 1 2 125 177 2000 Matthew IGx89 Lieder Md5 plugin d
18. ction To draw this plot the user must first select the point p for which the plot is to be made Upto 32 points can be shown simultanously On the X passes k are shown the value is given on the Y axis The example of such plot is shown on figure 2 3 2 6 IMPROVEMENTS Since the release of the first version of the LRP Toolkit see HCG 06 a vast number of bugs have been fixed Currently the new release of the Toolkit is available The most important change is that the functions that do not depend on the considered model structure have been rewritten to accept a much more general parameters This makes extension of the Toolkit much easier in practice After the initial release it became obvious that much stronger type checks are required This is motivated by a fact that the linear repetitive process model contains many variables that are error prone This has lead to a development of new functions for dealing with this task 2 THE LRP TOOLKIT 12 File Tools Slp a lt 4 Output Y_1 along the pass profile 2 3 4 5 6 7 Edit Output _1 along the pass profile 10 points 3 Figure 2 3 Pass to pass plot example Currently the great effort is made towards new vastly improved help sys tem This is based on standard Scilab templates but contains many more illustrative examples Moreover a number of potential pitfalls is explained All such cases have a suggested valid solution Very m
19. e of LMIs First define _ A Bo o fo o 2 1 Do D and D 1 Introduction In the Toolkit the following methods of obtaining K and K 2 are imple mented Theorem 7 GLR 03 The linear repetitive process of 1 1 with the ap plied control law of 1 6 is stable along the pass if there exist matrices Y Y T gt 0 Z Z 0 and N such that the following LMI is feasible Z Y 0 0 7 A Y B N s Y B N Y x T NT x BT YxAZ 4 NTx BF 0 Y where a i A 0 hela If the LMI is feasible the controller K is given by K NY 1 7 Proof of this theorem can be found in GLR 03 In the Toolkit this ap proach is implemented as LMIAlongThePass1 Another variant of this method is to use Theorem 8 GLR 03 The linear repetitive process of 1 1 with the ap plied control law of 1 6 is stable along the pass if there exist matrices P PP gt 0 P PP gt 0 P PT diag Pi P2 gt 0 Ni and Na such that the following LMI is feasible _p bP RN PoT NTRE P 0 where A Ao Ny No oF a If the above LMI is feasible the controller K is given by K NP 1 8 Again the proof can be found in GLR 03 This method is implemented as LMIAlongThePass2 2 THE LRP TOOLKIT 2 1 Alternatives 2 2 The LRP Toolkit Why Scilab The main functions of the Toolkit A main problem encountered during the control related analysis of repetitive processes is how to visualize the process
20. he LRP structure The LRP structure used by the Toolkit is quite complex To simplify cre ation of new structures you can use three different methods given from the simplest to the most complex e use the graphics user interface e use the createLRPModel function e use the createStubLRP function and fill all the fields manually Each of those options will be covered in the corresponding subsections 2 4 1 Creating new LRP structure by using the graphics user interface This method is the preferred method of creating the LRP structure It is the simplest and least error prone approach To create the structure run the graphics user interface fig 2 1 by selecting the Start GUI from the LRP menu shown as the item in the Scilab main menu If there are no structure named 1rp it will be automatically created The created structure is named 1rp Any other structure of this name is deleted If you need to preserve the old system assingn it a different name e g myLrp lrp If you cannot find the LRP menu close the Scilab and run the Toolkit again If it does not work or you do not want to close the Scilab session run the install sci script file by typing global LRP_OPTIONS cd LRP_OPTIONS path exec install sci The first two lines set the Toolkit directory If you have problems with them use the cd PATH_TO_LRP_TOOLKIT where PATH_TO_LRP_TOOLKIT is the directory where you installed the LRP Toolkit
21. he Toolkit is described in NDG The main functions of the implemented toolkit include e Visualization of the process dynamics 3D plots and 2D plots both along the pass and pass to pass e Stability analysis asymptotic and along the pass using both classic i e Nyquist diagram based and LMI settings e Control law design based on the use of LMIs Sul06 GLR 03 e Adding new analysis and control law design methods easily The new methods are automatically supported by the GUI 2 THE LRP TOOLKIT 2 3 Installation 2 3 1 Requirements 2 3 2 Windows The LRP Toolkit requires e Scilab www scilab org Recommended version is Scilab 4 1 or later e To use graphics user interface the Scilab version with Tcl Tk support almost all versions have this feature e About 4MB megabyte apprx 1 million bytes free disc space e For I4TRX support any TeX distribution must be installed For Windows the MikTeX www miktex org is recommended e The gui directory of the Toolkit must be granted the read and write rights The Toolkit is distributed in two forms as a Windows only installer or as a multi platform ZIP file After installation both files yield identical results It is possible to move the Toolkit sci files between platforms The LRP Toolkit can be installed either manually or automatically 2 3 3 Automatic installation It is recommended to use this method to install the Toolkit If after deinstal l
22. iven by K 0 4887 0 5174 0 6102 0 4543 0 2739 f Note here that the user does not have to execute any commands to compute the resulting controlled process state space model this is automatically done The resulting controlled stable along the pass process state space model is of the form 1 1 with 0 201 0 3439 0 2982 0 8 A 0 3221 0 4878 0 06287 B 0 7 0 6934 0 1452 0 5669 0 3 0 2634 0 01911 Ba 0 01802 0 09172 Do 0 4637 0 1178 0 4023 0 1965 E i D D 0 3491 0 2052 0 1017 0 03956 C 0 004519 0 193 0 1441 0 4 3 Example 20 4 Output Y_1 along the pass profile 3 5 7 12 15 20 File Tools Edit Output Y_1 along the pass profile 10 points 4 Output Y_1 along the pass profile 2 3 4 5 6 7 _ 5 x Output Y_4 along the pass profile 40 points Figure 3 2 Stability along the pass controlled process output 1 along the pass All the matrices obtained during the control law computations are available to the user by using the options supplied and they can be also exported to the IATfX based source file A representative plot of the evolution of the pass profile sequence for the controlled process is given in Fig 3 1 and the stability along the pass property is evident by inspection 3 Example 21 3 1 CONCLUSIONS AND FUTURE WORK The SCILAB toolkit whose development has been de
23. ll J D Ratcliffe J J Hatonen P L Lewin E Rogers T J Harte and D H Owens P type iterative learning control for systems that contain resonance International Journal of Adap tive Control and Signal Processing 19 10 769 796 2005 E Rogers and D H Owens Stability Analysis for Linear Repet itive Processes volume 175 of Lecture Notes in Control and In formation Sciences Springer Verlag 1992 INRIA Meta2 Project ENPC Cergrene Scilab Group Introduc tion to scilab B Sulikowski K Galkowski E Rogers and D H Owens Control and disturbance rejection for discrete linear repetitive processes Multidimensional Systems and Signal Processing 16 2 199 216 2005 B Sulikowski Computational aspects in analysis and synthesis of repetitive processes PhD thesis University of Zielona Gora 2006 Index 25 Index ILC see Iterative Learning Control Iterative Learning Control 1 Linear Repetitive Process 1 Boundary conditions 2 Model 1 Stability 2 Along the pass 2 Asymptotic 2 Linear Repetitive process Applications 1 Repetitive processes 1 Standard boundary conditions 9
24. mportant drawback it is not possible without changes to the core Toolkit files to simulate controlled pro cess to asses the effects of varying the control law matrix K LMI designs produce a family of such K The addition of this feature is a subject for future work e setcontrollerExample set controller name change the system matrices to depict the controlled process e g replace the A matrix into A B K matrix where K is the calculated constant control law matrix e writecontrollerExample write controller name a SCILAB func tion for exporting the results e g controller matrices parameters etc to AT X can be blank if no export is required e describecontrollerExample tex describe controller name in troductory text in TRX to be inserted when using the ATEX export capabilities can be left blank It must be stressed that the Toolkit files written by the user must be placed in the appropriate directories this is explained in the help file Note also that the SCILAB enforces a maximum function name length of 25 characters 2 7 2 THE Irp stability FIELD The stability field of the lrp_Sys_Cla datatype is a dynamically increas ing list that contains a number of tlist structures Each element repre sents a preformed stability test 2 THE LRP TOOLKIT 17 When the user checks a new stability condition a new tlist is added to the lrp stability field This field is defined by the programmer and the
25. ng various process parameters has been implemented Since most of the model parameters are matrices the basic method to define these is the SCILAB convention for entering matrices exactly the same as in the MATLAB To simplify this process it is also possible to enter the matrix element by element see Fig 2 5 2 THE LRP TOOLKIT 15 To make the end product available for a broader audience the Windows op erating system version employs an easy to use multilingual currently Polish and English installer based on Nullsoft Install System NSIS Nul This system is widely regarded to be a very reliable free solution that produces a small overhead code Additionally the NSIS can package and verify all the files included into the prepared compilation Moreover as an additional safety measure for each installed file the Message Digest Algorithm 5 bet ter known as MD5 checksum is calculated using the MD5 library pL This value is used when upgrading and uninstalling the toolkit if any change is detected the user can choose to leave the file intact Essentially this feature provides protection against accidental deletion of manual changes During the installation phase an existing ATEX installation is automatically detected Currently the most popular MiKTeX distribution is supported by the installer but any standard ATEX can be used 2 7 IMPLEMENTATION DETAILS The toolkit consists of a TCL TK GUI frontend and a number of SC
26. onstrains on the number of passes and hence to simulate the process response it is necessary to bound 2 THE LRP TOOLKIT 16 it by some finite value selected by the user hence the parameter beta in the 1rp structure 2 7 1 THE Irp controller FIELD The controller field of the lrp_Sys_Cla datatype is a dynamically in creasing list that contains a number of tlist structures Each element holds the results of control law calculations By design the first element of the controller list i e lrp controller 1 is a copy of all the system matrices This controller is necessary for retaining the matrices for the open loop system When new control law matrices are computed a new tlist is added to the lrp controller field This field is defined by the user and the toolkit does not enforce any constraints on its structure The only requirement is that there must be a field solutionExists of the boolean type which informs whether or not it is possible to obtain control law matrices for the example under consideration by the design method being considered In order to introduce a new control law the user must 1 give it a name e g controllerExample and 2 implement a set of 3 functions and one tex file e controllerExample the same name as a control law name the main function used for calculating the constant control law matrices given a process state space model This approach allows faster calcu lations but also imposes an i
27. ple RO92 Also in recent years applications have arisen where adopting a repetitive process setting for analysis has distinct advantages over alternatives For example they can be used to analyze an important class of iterative learning control ILC schemes OARF00 More recently another application has arisen in the context of self servo writing in disk drives MCA03 and there are as yet unexploited links with one ap proach to the analysis of spatially interconnected systems DD03 Attempts to control these using standard classical 1D systems theory algorithms fail except in a few very restrictive special cases precisely because such approaches ignore the inherent 2D structure of repetitive processes i e in formation propagation occurs in two distinct directions Here we consider so called discrete linear repetitive processes which can arise either from di rect modelling of a physical process or as a result of sampling the dynamics of a differential process in the along the pass direction The state space model RO92 of such a process has the following form over 0 lt p lt a k gt 0 where k denotes the pass number or index Lryi p 1 Aatesi p Bursi p Boye p 1 1 yrti p Car i p Duk p Doye p f 1 This manual covers the LRP toolkit version 0 32 built on 14 54 52 2009 05 07 1 Introduction Boundary conditions 1 1 1 Stability Here on pass k zp p R denotes the state vector
28. s Ea into itself and bg41 Wa where 1 Introduction 3 Wa is a linear subspace of Ea Also the term Layk describes the contri bution of the pass k to the pass k 1 and b41 represents the inputs and other effects which enter on the current pass Definition 1 RO92 Ben00 Sul06 Suppose that denotes the norm on Ea Then the so called asymptotic stability holds provided there exist real numbers Ma gt 0 and Aa 0 1 such that LE lt MAX k gt 0 where is also used to denote the induced operator norm Theorem 1 RO92 The linear repetitive process described by 1 1 is asymp Asymptotic stability totically stable if and only if r Do lt 1 where r denotes the spectral radius The equivalent theorem using the Linear Matrix Inequalities LMI tech niques can be formulated as Theorem 2 SGRO05 The linear repetitve process of 1 1 is asymptoti cally stable if and only if there exists a matrix Q gt 0 of appropriate dimen sions such that the following LMI holds Asymptotic stability LMI D5 QD Q x0 The asymptotic stability guarantees the existence of a limit profile but it does not guarantee that this limit profile treated as a 1D system under the assumption that oo is stable The reason for that is the fact that asymptotic stability does not concern dynamics along the pass along the p dimension Mostly the cases where the limit pass profile is unstable as a 1D linear sys
29. scribed in this paper has already been useful in the analysis and control law design for discrete linear repetitive processes of the form considered here Its basic functions include the simulation engine and the stability analysis control law design abilities A key point to note is that this toolkit removes the limitations present in others Moreover the user is able to export the results to the valid BTEX compatible format text file Software development for this toolkit is an ongoing process there are many options which remain to be implemented A representative sample of on going development work is the following topics support for new classes of processes such as wave or semi linear and for a differential process model where the dynamics along the pass are governed by a linear matrix differential equation couple along with discretization methods addition new control law design algorithms control law design for stability and performance e g ensuring that a reference signal is tracked minimization of the influence of the external disturbances handling of processes with uncertainty in the state space model an installer for the Linux Unix version Results from this work will be reported in due course 4 Appendix 4 1 Abbreviations and notation Throughout the manual the following abbreviations and notations are used Notation Meaning GUI Graphics user interface LRP Linear Repetitive Process or type
30. ssumed as zk 1 0 0 yo p 1 2 8 THE COOLMATRIX TYPE The conducted usability tests have shown that due to differences in indexing between the theoretical results and the Scilab requirements the implemen tation of any function can very easily lead to the famous fencepost error for example if the user wants to see pass number 3 he has to enter the in dex 4 in the Scilab matrix While this problem may seem simple it can lead to many difficult to detect errors especially for non standard LRP systems like the wave processes where negative indices are often used To overcome this difficulty the new version of the Toolkit includes the CoolMatriz matrix type that allows the user to index the arrays as required from 0 or any other value including negative number The functions used for this type are designed for fast prototyping and hence provide a strict error checking any attempt to use a wrong index causes an error Great effort has been made to make this type compatible with the standard Scilab 2 THE LRP TOOLKIT 18 matrix type The disadvantage of this addition to the Toolkit is the over head caused by this type This fact does not very significant as the type is aimed at the prototyping phase where a small to medium problems are tested Moreover the experience shows that the efficiency of the Toolkit is greatly dependent on the LMI solver which has the greatest impact in the practical applications
31. t theorem 4 is only sufficient condition In addition to the most important stability definitions asymptotic and along the pass the Toolkit includes the tests for the so called practical stability This notion lies between the asymptotic and along the pass stability The conditions for this type of stability to hold can be summarized as the fol lowing theorem Theorem 5 Gra99 The linear repetitive process described by 1 1 is prac tically stable if and only if r Do lt 1 r A lt 1 where r denotes the spectral radius The LMI version of this theorem can be formulated as Theorem 6 The linear repetitive process described by 1 1 is practically stable if and only if there exist matrices P gt 0 and Q gt 0 of appropriate dimensions such that the following LMI is feasible ATPA P 0 20 0 DEQD This notion of stability is seldom used and hence will not be discussed in full detail see Gra99 If the system described by 1 1 is unstable along the pass it is necessery to stabilise it by means of an appropriate control action One of the con trol laws considered to date for discrete linear repetitive processes has the following structure for the background see for example GLR 03 and the relevant cited references uk 1 p K zk p Koye p K n 1 6 where K and K are matrices to be computed Currently the only effective approach for the computation of the controller matrices is through the us
32. uch attention has been paid to presentation of the results In the new version of the Toolkit the drawing engine has been rewritten to allow much easier use in scripts all the functions have a much clearer syntax Moreover the plotting routines have been extended to handle the degener ate cases like a plot of single point on a single pass Due to readibility a maximum of 32 passes can be drawn simultaneously using different colors for clarity Additionally due to the extended TFX support creating mul tiple plots is much faster from O n to O 1 calculations of plot surfaces moreover the plot surface is calculated only for data required for plotting when user requests an along the pass plot for points 7 18 for a 100 only the first 18 points are calculated If necessary it can be requested to calculate the entire surface Compared to the previous version the 3D plots are now made in full color to better visualize the range of values in the plot 2 THE LRP TOOLKIT 13 ich Select stability test r st symptotic1 st symptoticLMI1 stPracticall stPracticalLMl1 C C Calculate C Calculate C caosa Calculate Cose st4longT hePassLMI1 Figure 2 4 Stability analysis window 2 6 1 STABILITY ANALYSIS In terms of stability analysis for discrete linear repetitive processes of the form considered here asymptotic stability and hence the construction of the resulting limit profile described
Download Pdf Manuals
Related Search