Multivariable H-infinity Control Design Toolbox: User manual
Contents
1. Fig 2 1 Multivariable Control Design Configuration Floating Platform After deriving the various relations between the inputs and outputs it follows that in terms of the standard H problem the generalized plant G is defined by W V W RV WyP e 0 0 Wi 2 na m T y V RV P T a u 2 2 d 0 V 0 Gui Gy w i t i u Gy The control acting on the outputs y is represented by Ym u u ca 2 3 This yields the closed loop system F G K mapping w z which has H norm 2 1 Structure Definition 7 Fy 12 Fy Ea 2 4 W 1PC Vy W I PCa R PCg Va W Cg I PCg V W I PCg CAR C Here the various transfer functions are named as follows F Sensitivity 91 Disturbance Attenuation Fa Control Sensitivity Fa Saturation Until now we have only described the standard approach to define the H control problem for the floating platform to achieve disturbance attenuation and avoiding saturation of the control input This results in the closed loop system G K of Eq 2 4 This configuration however is specific to the floating platform and can be significantly different for other control problems To avoid this procedure of building and implementing different configurations every time when new control problems occur we will generalize this setup 2 1 Structure Definition Basicly the de2. US Hesearch Report an University of lechnoldoy oden TEUEDE Netherlands Faculty of Electrical Engineering Multivariable H infinity Control Design Toolbox by H M Falkus A A H Damen EUT Report 94 E 282 ISBN 90 6144 282 6 April 1994 Eindhoven University of Technology Research Reports EINDHOVEN UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering Eindhoven The Netherlands ISSN 0167 9708 Coden TEUEDE Multivariable H infinity Control Design Toolbox User Manual by H M Falkus A A H Damen EUT Report 94 E 282 ISBN 90 6 144 282 6 Eindhoven Apr l 1994 li Multivariable H Control Design Toolbox CIP DATA KONINKLIJKE BIBLIOTHEEK DEN HAAG Falkus H M Multivariable H infinity control design toolbox user manual by H M Falkus A A H Damen Eindhoven Eindhoven University of Technology Faculty of Electrical Engineering Fig EUT report ISSN 0167 9708 94 E 282 With ref ISBN 90 6144 282 6 NUGI 832 Subject headings robust control multivariable control systems control simulation softwarc Abstract Multivariable H infinity Control Design Toolbox User manual H M Falkus and A A H Damen A MATLAB toolbox is presented for solving the multivariable H control design problem Algorithms are available Robust control toolbox of MATLAB which solve the problem once the control design configuration including process model and weighting functions has been
3. h3l m mhc hcb m mhc_map m mhc mss m mhc rbal m mhc csc m mhe_slrc m mhc are m 1 mhc_d2ss m mhc kgjd m mhc are m L mhc d255 m mhc djnl m mhc are m L mhc_d2ss m mhc_hin m mhc crm m mhc mss m mho ccl m mhc sem m mhc h4 m mhc sbp m meta m mhc pcl m mhc meta m mhc sim m m mhc_meta m mhc opt m mhc h5 m mhc disk m mhc h6 m Fig B 1 Global program structure C Function Description A brief description of all functions in alphabetical order presented in the overview of Appendix A will be given Function name Description mhc m Initialization script file showing the main menu mhc are m Computes the algebraic Riccati equation for the H control problem mhc_c2o m State space transformation from controller canonical form to observer canonical form mhe ccl m Calculates the closed loop system consisting of the process blocks P amp P and the H controller mhc_cm m Function to change rows in a matrix 35 36 Function name mhe_crm m mhc_csc m mhc_disk m mhc djnl m mhc dtf m mhc hl m mhc h2 m h21 m mhc h3 m mhc_h31 m mhc h4 m mhc h5 m mhc h6 m mhc hcb m mhc hcm m mhc_hco m mhc_hin m Function Description Description Computing controller reduction according to several methods Checks the conditions to solve the H control problem Disk options menu H all solution formulae derived by and Kasenally Define transfer functi
4. 4 Relative Schur reduction Type of controller reduction for method 2 3 amp 4 Variable order amp Fixed error bound 2 Fixed order amp Variable error bound Variables freq gamtol num den num P2 den P2 num V den V num W den W time tol 41 Description Array defining frequency information 1 Lower bound 2 Upper bound 3 Number of frequency points Tolerance margin for y minimization Numerator denominator transfer matrix of process block P1 Numerator denominator transfer matrix of process block P2 Numerator denominator transfer matrix of design block V Numerator denominator transfer matrix of design block W Time information End of time interval continuous mode or Sample time discrete mode Tolerance margin for minimal state space realization 42 List of Variables References Bouwels J P H M Ontwikkeling en beproeving van verschillende regelaars LQG voor de horizontale afregeling van een drijvend platform met een draaiende kraan als verstoring Measurement and Control Section Faculty of Electrical Engineering Eindhoven University of Technology The Netherlands 1991 M Sc graduation report Bruinsma N A and M Steinbuch A fast algorithm to compute the H norm of a transfer function matrix Systems amp Control Letters Vol 14 1990 p 287 293 Chiang R C and M G Safonov Robust control toolbox User s guide Natick MA The Mathwork
5. Show magnitude plots of w Select menu option or press ENTER to Exit 0 Help Please enter menu option or press ENTER to Exit piscrete time MIMO H infinity Control Design Discrete time MIMO H infinity Control Design MINO HrinPinity Control Design Disk Options Controller design Controller options 1 Save all variables 1 H infinity controller options moe ea h sin 2 Load all variables 2 Calculate H infinity controller en ee ae u 3 Contents disk 3 Controller reduction Salancinc ne Saar g gamma qe irecto 4 L riginal contr Sio SENDER EDS d i Ben IN Controller reduction method Minreal it Please enter menu option or pre it Please enter menu option or press ENTER to Exit P press ENTER to Exit Please enter menu option or press ER to Exit Program Structure The global program structure including all mhc functions is depicted in Fig B 1 Note that the standard MATLAB functions including the toolboxes which are used in the multivariable H control design toolbox are not mentioned in the overview The required toolboxes are described in Section 2 5 33 34 Program Structure mhc m mhc hm m mhc stm m mhc hl m mhc_stin m mhc im m mhc imf m mhc h2 m mhc rtf m mhc dtf m L mhc tfss m mhc c2o m L mhc mss m mhc ssr m mhc h21 m mhc_cm m mhc pzc m sbp m mhc_meta m mhc hcm m mhc h3 m mhc hco m
6. based on the 2 Riccati equation approach For controller reduction the options depicted in Fig 3 20 are available After selecting the control design options the actual controller calculation is started option 2 of he controller design menu In this case some information will scroll over the screen describing the minimal realization of the generalized plant bilinear transformation balancing and Fig 3 18b Help controller options Cont Discrete time MIMO H infinity Control Design H infinity type approach 1 SLC Safonov Limebeer Chiang loop shifting formulae 2 GD Glover Doyle all solution formulae 3 LK Limebeer Kasenally all solution formulae Please enter menu option or press ENTER to Exit Fig 3 19 Type of H approach Discrete time MIMO H infinity Control Design Controller Reduction Methods 1 Minimal state space realization 2 Optimal Hankel reduction 3 Schur reduction 4 Relative Schur reduction Please enter menu option or press ENTER to Exit Fig 3 20 Reduction methods Prepare Augmented Plant Size state matrix 12 Observability index 9 Controllability index 12 Working on state space minimization Please wait 3 states removed Working on inverse bilinear transformation Please wait Working on balancing augmented plant Please wait Fig 3 21 Preparing generalized plant 3 4 Controller Design 23 checki
7. inverse transformation to obtain the discrete time controller Tolerance Margin for Minimization Procedure To Solve the control design problem properly it is required that the constructed augmented plant has a minimal state space realization This might not be the case because of pole zero cancellations in the design filters When no minimal realization has been obtained this tolerance margin is used to detect and eliminate the unobservable uncontrollable modes Selected Input Signal The variable name of the input signal is defined which is used for the time simulation of the closed loop system Whenever the variable name does not exist in the workspace a time signal must be defined by the user Press any key to continue Fig 3 4a Help options menu Help Options Menu Generating META Files By selecting th s option amp filename will be requested after every plot to save the plot as a META file for later processing using GPP Lower Upper Frequency Bound and Number of Frequency Points These options define the frequency range for the magnitude plots Time For the continuous time this defines the length of the time simulation while for the discrete time the sample time is defined Press any key to continue Fig 3 4b Help options menu cont 3 2 Structure Initialization Discrete time MIMO H infinity Control Design The structure initialization Structure initialization menu 18 depicted in Fig 1
8. the standard solution method based on solving two Algebraic Riccati equations and implemented in the Robust Control Toolbox of MATLAB Chiang R C et al 1988 can be applied The methods available 1 Safonov Limebeer Chiang loop shifting formulae Safonov M G et ai 1989 2 Glover Doyle all solution formulae Glover K et al 1988 3 Limebeer Kasenally all solution formulae Limebeer D J N et al 1988 are only different in circumventing some of the numerical problems which generally arise when a design approaches its performance limits The solutions to the Riccati equations can be solved either by an eigenvalue or Schur decomposition The eigenvalue approach is the fastest but for design filters close to the H performance limits the Schur approach is numerically more reliable These routines calculate a controller if one exists only for a fixed value of y That is a controller is computed achieving F lt y However we are interested in Yopt for which a Stabilizing controller still exists Therefore the basic routine has been extended as follows with an iterative search procedure 2 2 Minimal Realization Generalized Plant 11 A start value y and a step size a 0 gt 1 are defined An interval Ymin Ymax is computed which contains the optimal solution 1 If a solution exists for Y Your Yo define Ymax Yo The lower bound of the interval be found by decreasing Y yon u
9. Boom A Damen A comparison of classical and modern controller design A case study Eindhoven Faculty of Electrical Engineering Eindhoven University of Technology The Netherlands 1990 EUT Report 90 E 244 Limebeer D J N and E M Kasenally I Jaimoukha M G Safonov All solutions to the four block general distance problem In Proc 27 IEEE Conf on Decision and Control Austin Texas December 7 9 1992 New York IEEE 1992 P 875 880 Maciejowski J M Multivariable feedback design Wokingham England Addison Wesley 1989 McFarlane D C and K Glover Robust controller design using normalized coprime plant descriptions Heidelberg Springer 1990 Lecture notes tn control and information sciences Vol 138 Morari M and E Zaftriou Robust process control Englewood Cliffs N J Prentice Hall 1989 Munro N and C Eng R S McLeod Minimal realization of transfer function matrices using the system matrix Proc IEE Vol 118 1971 No 9 p 1298 1301 Safonov M G and D J N Limebeer R Y Chiang Simplifying the H theory via loop shifting matrix pencil and descriptor concepts Int J Control Vol 50 1989 No 6 p 2467 2488 Stoorvogel A A and J H A Ludlage The discrete time minimum entropy H control problem Faculty of Mathematics and Computing Science Eindhoven University of Technology The Netherlands 1993 Internal report 46 References Dooren P M van T
10. Damen A A H et al 1994 the scaling of the filters has been adjusted such that H norm y becomes smaller than 1 3 6 Disk Functions 27 3 6 Disk Functions All information can be Discrete time MIMO H infinity Control Design saved and loaded using Disk Options l S 11 iabl disk options menn Load ail veriablem Contents disk Fig 3 34 Information change directory should be stored regularly Help Please enter menu option or press ENTER to Exit during the design because MATLAB errors due to Fig 3 34 Disk options menu numerical problems can terminate MHC Help Disk Options Menu The help screen for the This menu should be used to SAVE and LOAD variables correctly To avoid extensively checking of variables for existence dimensions etc in all disk options menu IS functions this menu has been included When starting up the design package all variables are initialized in a standard way Therefore variables can be entered either directly through the menus or by loading depicted In Fig 3 35 them from the workspace The available options are Save All Variables Load All Variables Contents Disk Change Directory Press any key to continue Fig 3 35 Help disk options The most important features of the H control design toolbox have been described together with the menus which will appear on the computer screen The exact screen input has not been described because the cont
11. General structure initialization 2 Change interconnection matrix IML 3 Change interconnection matrix IM2 3 5 The general structure as well as IM amp IM2 can be changed 0 Help Please enter menu option or press ENTER to Exit Fig 3 5 Structure initialization menu 16 Menu Description The generalized plant is described by defining the 11 signal dimensions of the blocks within the basic structure Fig 3 6 For the floating platform the transformation of the control configuration into Structure Initialization n2 x gt P2 gt n3 n4 gt 1 gt n6 n7 2 1 0 y gt M gt w n9 t 4 I gt gt 10 nil gt I gt MIMO Augmented Plant this structure is shown in Define the dimensions of the MIMO augmented plant ni 11 as shown above in MATLAB notation ni n2 n3 n4 n5 n n8 n9 n10 nil Section 2 1 The first step in the design is the defenition of the dimen Fig 3 6 General structure initialization sions Dimensions Dimensions 00212222312 The dimensions of the Build interconnection matrix interconnection matrices Actual matrix elements of IMl are fixed now and can be defined row by row After selecting a ro
12. REPRESENTATIONS OF DYNAMICAL SYSTEMS EUT Report 93 E 277 1993 ISBN 90 0144 277 1 Gorshkov Yu A and V I Vladimirov LINE REVERSAL GAS FLOW TEMPERATURE MEASUREMENTS Evaluations of the optical arrangements for the instrument EUT Report 93 E 278 1993 ISBN 90 5144 278 8 Crevghton Y L M and W R Rutgers van Veldhuizen IN SITU INVESTIGATION OF PULSED CORONA DISCHARGE EUT Report 93 E 274 1993 ISBN 90 6144 279 6 Li H Q and R P P Smeets LENGTH DEPENDENT PHENOMENA OF HIGH FREQUENCY VACUUM ARCS Report 3 E 280 1993 ISBN 90 6144 280 Di Chennian and Jochen A G Jess ON THE DEVELOPMENT OF A FAST AND ACCURATE BRIDGING FAULT SIMULATOR EUT Report 94 E 281 1994 ISBN 90 6144 281 6 K M and A A H Damen MULTIVARIABLE H INFINITY CONTROL DESIGN TOOLBOX User manual EUT Report 4 282 1994 ISBN 90 6144 282 6
13. case it should be mentioned here that the discrete time case is solved via bilinear transformation In Stoorvogel et al 1993 and Iglesias er al 1993 it is shown that designing a discrete time controller via a bilinear transformation to the continuous time domain might introduce an implicit and undesirable additional weighting function A simple free stable contraction map is added to eliminate this additional weighting In general the resulting H controllers are of high order because the order is equal to the order of the generalized plant process amp all design filters To reduce the order of the controller the following reduction techniques can be applied to the resulting controller 1 Minimal state space realization reduces within a predefined tolerance margin 2 Optimal Hankel reduction 3 Schur reduction 4 Relative Schur reduction 12 General Control Design Framework For the reduction methods 2 to 4 an additional option can be selected to reduce the controller with variable order and fixed error bound or fixed order and variable error bound A detailed description and more references for these reduction techniques can be found in Chiang R C et al 1988 2 4 Evaluation Controller Design After calculating the H controller the closed loop system is derived without shaping and weighting functions in order to evaluate the controller design For this purpose time as well as frequency re
14. 1529 1 0 1 0 n 9 0 Boa Boa Cpa L ny 0 20 3104 z 0 2z 222 z 1 9944z 0 9950 y 240 4172 2 0 5727 0 57251 z 0 5727 0 57251 Y 60 2 1 2 0 7022 0 53921 z 0 7022 0 53921 Design filters s 0 1 z 0 8 0 011 z 0 8 0 011 Y z 0 999 0 0011 z 0 999 0 0011 _ 40 z 0 995 0 0099i1 2 0 995 0 00991 2 0 9 2 20 Menu Description The numerator and denominator polynomials can be defined in several ways in the transfer function entry of Fig 3 12 For example Numerator V 0 3104 1 0 2 0 Denominator V 60 1 1 0 7022 0 5392 1 0 7022 0 5392 1 Note that MATLAB commands be used as well to define the transfer functions 3 4 Controller Design Discrete time MIMO H infinity Control Design Once the complete Controller design 1 1 H infinit troll ti generalized plant has 2 CAlculate H infini ty ecne oliot 3 Controller reduction been defi ned the 4 Load original controller controller design becomes 0 Help fairly simple Please enter menu option or press ENTER to Exit Fig 3 15 Controller design Help Controller Design Menu The help SCIEEN for the In this menu the actual controller design is performed All stabilizing controllers such that a specified closed loop transfer function has 1 1 H infinity norm less than a given scalar This characterization involves controller design menu 1 the solution to two algebraic Ricca
15. Design Fig 3 25 shows the menu System Evaluation to analyze the closed loop Show bode plots controlier Plot closed loop transfer function Show time simulation behavior by computing Help time and frequency Please enter menu option or press ENTER to Exit responses Fig 3 25 System evaluation menu Help System Evaluation Menu Help screen of the system evaluation menu In this menu the closed loop system behaviour can be evaluated by computing time and frequency responses Show Bode Plots Controller Shows the magnitude plot of the controller from a selected observation signal y t to a selected input signal uit for the frequency range defined in the Option menu Plot Closed loop Transfer Function The magnitude plot of a selected closed loop transfer function from the disturbance vector wit to the error vector yit without shaping weighting filters is shown for the specified frequency range Whenever the shaping and weighting blocks of the augmented plant are defined as diagonal functions the magnitude plot of the inverse shaping weighting filter scaled with the H infinity closed loop norm gamma is plotted as well This inverse function defines an upper bound of the corresponding closed loop transfer function over the whole frequency range This can be used to determine which transfer function is the limiting factor in the controller design and in which frequency range Press any key to continue
16. Fig 3 26a Help system evaluation Help System Evaluation Menu Show Time Simulation A closed loop time simulation is performed using the input signal defined in the Option menu The simulated outputs can be plotted by selecting the required output Press any key to continue Fig 3 26b Help system evaluation Cont 3 5 System Evaluation 25 The single transfers in the closed loop system are shown by selecting the corresponding input and output Working Show Discrete Closed loop Bode pilots Please enter number of input 1 2 or press Enter to Exit Please enter number of output 1 2 Please wait Fig 3 27 Plot closed loop transfer Only the magnitude plots for the closed loop evaluation will be shown here Discrete Clased loup PX IMP Magnitude dB 109 10 2 10 1 Frequency rad Fig 3 28 Closed loop transfer 1 1 Discrete Clased lvup P 21 ate Magnitude dB 10 10 10 Frequency rad Fig 3 30 Closed loop transfer 1 2 10 Discrete Closed toop P 12 Magnitude dB 10 2 10 101 Frequency rad Fig 3 29 Closed loop transfer 2 1 Discrete Closed loop P 22 gt Magnitude dB 10 40 10 10 103 10 2 10 Frequency rad Fig 3 31 Closed loop transfer 2 2 Before we can evaluate the time simulations an input signal must be defined in the options menu
17. Fig 3 3 The variable name of the input signal matrix can be entered If the variable 26 Menu Description name exists in the workspace for example generated before starting up mac and the number of columns correspond to the defined input dimensions the time simulations can be performed However if the variable name does not exist in the workspace the input matrix must be defined first Also the variable name of the output signal matrix must be defined For the floating platform example a disturbance signal can be generated corresponding with 3 rotations of the crane rotation frequency 0 04 Hz and a load of 1 kg After 10 seconds 100 samples sample time 0 1 sec the crane starts rotating and zeros have been added to create a time simulation of 100 seconds Name input dis Input signal zeros 100 1 9 81 s8in 2 pi 0 04 0 0 1 75 zeros 149 1 zeros 1000 1 Name output dis out Thiscrete Time Simulation of Closed loop System Discrete Time Simulation of Clased lanp System Output Signal dis out 1 Output Signal dis out 2 Time t 5 Time t s Fig 3 32 Simulation output signal Fig 3 33 Simulation control signal The corresponding time simulations are depicted in Fig 3 32 amp 3 33 The relative bad disturbance rejection in Fig 3 32 after 10 and 85 sec are caused by starting and stopping the rotation of the crane Note that compared to the designs described in Bouwels J P H M 1991 and
18. LAB The main mE A Please enter menu option or press ENTER to Exit menu which is depicted in Fig 3 1 will appear on the screen Fig 3 1 Main menu 13 14 Menu Description Help Main Menu H infinity Control Design Fig 32 depicts the help This menu structured H infinity control design package can be used to ensure easy definition of the design configuration input of variables Screen of the main menu controller calculation and analysis of the results by computing time and frequency responses Structure Initialization Defines the design configuration including process and weighting filters as a standard problem The structure is fixed by defining the dimensions and two interconnection matrices Input Matrix Functions V P and W Definition of process and weighting filters via SISO transfer functions or MIMO state space matrices Controller Design H infinity control design parametrizing all stabilizing controllers such that a specified closed loop transfer function has H infinity norm less than a given scalar This involves the solution to two algebraic Riccati equations each with the same order as the system and further gives feasible controllers also with this order Press any key to continue Fig 3 2a Help main menu Help Main Menu H infinity Control Design System Evaluation Analysis of closed loop system by computing time and frequency responses Options Definition of general op
19. R ROBUST CONTROL USING AN H infinity NORM BUT Report 92 E 261 1992 ISBN 90 0144 261 3 Groten M and W van Etten LASER LINEWIDTH MEASUREMENT IN THE PRESENCE OF RIN AND USING THE RECIRCULATING SELF HETERODYNE METHOD EUT Report 92 E 262 1992 ISBN 90 6144 262 1 Spolders RIGOROUS ANALYSIS OF THICK MICROSTRIP ANTENNAS AND WIRE ANTENNAS EMBEDDED IN A SUBSTRATE EUT Report 92 E 263 1992 ISBN 9 6144 263 Y Freriks L W and P J M Ciuitmans M J van Gils THE ADAPTIVE RESONANCE THEORY NETWORK Clustering behaviour in relation with brainsten auditory evoked potential patterns EUT Report 92 E 264 1992 ISBN 0 6144 264 8 AL ellen J 5 and F karouta A F C Schemmann E Smalbruace L M P Kaufmann MANUFACTURING AND CHARACTERIZATION OF G AS ALGAAS MULTIPLE QUANTUMWELL RIDGE WAVEGUIDE LASERS BUT Report 92 205 1992 ISBN 90 6144 265 6 Clustmans L J M USING GENETIC ALGORITHMS FOR SCHEDULING DATA FLOW GRAPHS BUT Report 92 E 266 1992 ISBN 90 6144 266 4 Jozwiak L and A P H van 011 A METHOD FOR GENERAL SIMULTANEOUS FULL DECOMPOSITION GF SEQUENTIAL MACHINES Algorithas and implementation BUT Report 92 b 267 1992 ISBN 90 6144 267 2 Boon van den and W van Etten W H C de Krom P van Bennekom F fiuiiskens L Niessen F de Leer AN OPTICAL ASK AND FSK PHASE DIVERSITY TRANSMISSION SYSTEM EUT Report 92 E 268 1992 ISBN 90 6144 268 6 Eindhoven Universit
20. additional viewing option has been included to verify the Fig 3 9 Input matrix functions Pl P2 V or W magnitude plots 0 Help Please enter menu option or press ENTER to Exit 18 Menu Description Help Input Matrix Functions P1 P2 Vor W Fig 3 10 shows the help As described in the Structure Initialization menu the augmented plant of the standard problem consists of three basic blocks screen for the input 1 Pl The process model part 1 2 P2 The process model part 2 matrix functions menu 3 V Shaping of the disturbance vector wit 4 W Weighing of the error vector zit These four blocks can be entered into the design package either as SISO transfer functions or as MIMO state space matrices For the SISO case the user must define the filters as numerator and denominator polynomials for every entry of the matrix function For the MIMO case the or D matrices must be defined The number of inputs and outputs of these blocks depends of course on the dimensions entered in the Structure Initialization menu Press any key to continue Fig 3 10a Help input matrix functions P1 P2 V or W Help input Matrix Functions P1 P2 V or W When entering the filters as SISO transfer functions the corresponding state space representation is derived automatically and vice versa A consequence of this representation of the blocks P1 P2 V and W in transfer function matrices and state space mat
21. bjectives the design configuration 1s defined in a fairly simple way Because no general solution is known for translating design specifications such as desired behavior robustness performance etc directly into weighting functions in the frequency domain the H control design is menu driven to ensure easy input of variables controller calculation and analysis of the results by computing both time and frequency responses In this way the necessary iterative design procedure for optimizing the H control design problem becomes much easier All tools in this toolbox are implemented in MATLAB by means of standard m files In Section 2 the basic setup of the toolbox is presented using the process block diagram of a floating platform laboratory process as an example The floating platform with rotating crane has been built on laboratory scale to evaluate identification and control theories This particular process was chosen because it is an essentially linear MIMO system It can be well described by three decoupled second order SISO systems The model errors are then mainly due to unmodeled waves caused by the movement of the floats which lead to linear transfers which are however difficult to model The fact that H control is said to be particularly suited for robust control in cases of unmodeled linear dynamics makes this laboratory process an excellent example for testing the H control synthesis procedure On the platform a crane has bee
22. e interconnection matrices amp IM2 Fig 3 7 The help screen to define state space systems is shown in Fig 3 14 19 Define Discrete state space matrices of Pl Show Show change state matrix A Change input matrix B Show Change output matrix C Show Change feed through matrix D Change state dimension Help Please enter menu option or press ENTER to Exit Fig 3 13 Defining MIMO state space matrices Help Define State Space Matrices Menu The State Space matrices A B C D can be defined changed by selecting one matrix and then entering the values row by row The number of inputs and outputs has been defined in the Structure Initialization menu Only the number of states can be defined changed WARNING When changing the number of states all matrices are initialized to zero matrices and previously entered information will be lost Press any key to continue Fig 3 14 Help define state space matrices menu Before describing the actual H control design the following block information for the floating platform should be entered using the input menus for transfer functions and state space matrices Fig 3 11 3 12 amp 3 13 More detailed information about the modeling identification and H filter design can be found in Bouwels J P H M 1991 and Damen A A H et al 1994 0 1 0 0 0017 0 0045 A 1 0 1 B 0 0311 0 0007 Pioceses 3 794215 0 4044 0 1213 0
23. e copied e g copy mhc rn to the working directory of MATLAB If the files are copied to a directory different than the workspace of MATLAB this directory has to be added to the matlabpath The routines in this toolbox make use of standard MATLAB functions and the following MATLAB toolboxes Signal processing toolbox Control system toolbox Robust control toolbox Menu Description Before starting the H control design the specific control problem including design filters Fig 2 1 must be transformed into the standard configuration defined for this toolbox Fig 2 2 resulting in the required input information Fig 2 3 The Multivariable H Control design toolbox MHC is menu driven to ensure easy input of variables controller calculation and analysis of the results All menus of the toolbox will be described briefly and the controller design for a floating platform will be used as an example Any of the menu options can be selected by typing the correct number and pressing ENTER The previous menu will appear again by pressing just ENTER Every menu is provided with a help screen menu option 0 describing briefly the several menu options Startup MHC Continuous time MIMO H infinity Control Design Main menu St t inicializati To start the controller matris functions PI El Wick M Controller design design procedure execute System evaluation Options MHC from inside Disk functions MAT
24. e two equations can be solved either by eigenvalue or Schur decomposition The eigenvalue decomposition is faster but the Schur decomposition is numerically robuster for badly conditioned design problems TOlerance Margin Optimizing Gamma The H infinity controller will be designed in such a way that the H infinity norm of the closed loop system will be less than gamma Because the optimal gamma can only be approximated a tolerance margin must be defined indicating when the iterative design procedure optimizing gamma can be stopped Press any key to continue Fig 3 18a Help controller options 22 Menu Description Help Controller Options Menu Balancing Augmented Plant After building the augmented plant from the several blocks the overall system might be badly conditioned Balancing the augmented plant for badly conditioned problems can improve the final solution Controller Reduction Method The order of the feasible controllers will be the same as the order of the augmented plant containing the process model and all shaping weighting filters For more complex MIMO design problems the order will increase rapidly and therefore controlle reduction methods are often required to obtain lower order controllers Minimal State space realization Optimal Hankel Reduction Schur Reduction and Relative Schur Reduction Press any key to continue The following options are available to solve the H control design problem
25. fer function If the design filters V and W are diagonal matrices the inverse design function is also plotted Function to check pole zero cancellations in SISO transfer functions Returns the LQG or Riccati balanced state space representation of stable and unstable systems Function to replace an element in a transfer function matrix Routine to show Bode plot of a SISO transfer function Script file to generate system evaluation menu Function to calculate and show time simulation H loop shifting formulae derived by Safonov Limebeer and Chiang 38 Function name mhc_ssr m mhc stin m mhc stm m mhc tfss m Function Description Description Routine to show and define state space representation of a system Structure initialization function for H control design Script file to generate structure menu to define augmented plant MIMO transfer function matrix to state space conversion D List of Variables In this appendix a list of variables in alphabetical order with a short description is given which are used as input output arguments of the MHC functions described in Appendix C Variables Ac Bc Cc Dc Acl Bel Ccl Del Acon Bcon Ccon Dcon Acor Bcor Ccor Dcor Cpl Description State space matrices of controller in continuous time domain State space matrices of closed loop system without design functions State space matrices of final controller discrete c
26. for MHC toolbox The control design configuration of the floating platform can now be described tn a simple way by defining the dimensions of V V4 amp V P R amp P W W y amp Wp and 1 to I together with the structure of the configuration using the interconnection blocks IM and This is effectively all input which is needed for the toolbox to convert it into a standard H control problem 2 2 Minimal Realization Generalized Plant The blocks V P and W are represented in toolbox either as MIMO state space representations or entry wise as SISO transfer functions of a matrix However before building the generalized plant G minimum state space realizations of these blocks have to be obtained 2 2 Minimal Realization Generalized Plant 9 Therefore we use the approach outlined in Munro N et al 1971 For every row in a transfer function matrix the smallest common denominator is determined and the numerators are updated if necessary The new MISO transfer functions can be transformed into an observer canonical state space representation which is minimal Combining the state matrices of the MISO systems for every row in a block diagonal form and adding the input output and feed through matrices correctly results in an overall state space representation which is observable but not necessarily controllable In Dooren P M v 1981 it is proven that if the controllability matrix of A B has rank r S n w
27. g functions The signal w R represents all external inputs including disturbances sensor noise and commands the output z RP is the error vector yc RP is the observation vector and u R is the control input The generalized plant G can be partitioned according to the dimensions of the signals Introduction G Dj Pi 6 s1 A By B 1 2 Gi Gz P2 Pa C which results in the following closed loop transfer function from w to 2 F G K 6 6 K 1 Gy J Gy 1 3 G K Fig 1 1 Standard H Problem The standard assumptions are The triplet A B C namely the plant transfer G can be stabilized and detected so that stabilizing controllers exist rank D m and rank D p in order to ensure realizability of the controllers e No zeros on the imaginary axis p 2 m and m 2 p ensures that the solution to the corresponding LQG problem is closed loop asymptotically stable The main problem however is that every control problem has a different configuration because of different design constraints and control objectives This implies that every new control problem has to be rewritten again into the standard H control problem In this report a MATLAB toolbox is presented which enables us using computer routines to transform every multivariable control problem into the standard H control problem After selecting the control setup design constraints and o
28. he generalized eigenstructure problem in linear system theory IEEE Trans Aut Control Vol AC 26 1981 No 9 p 111 129 Weiland S A behavioral approach to balanced representations of dynamical systems Eindhoven Faculty of Electrical Engineering Eindhoven University of Technology The Netherlands 1993 EUT Report 93 E 277 Weinman W Uncertain models and robust control Heidelberg Springer 1991 47 Eindhoven University of Technology Research Reports ak 0167 9708 a Faculty of Electrical 1 296 257 298 259 260 261 262 263 2041 1265 266 2671 i268 Backx A C P M and A A H Damen IDENTIFICATION FOR THE CONTROL OF MIMO INDUSTRIAL PROCESSES BUT Report 91 E 256 1991 ISBN 96 6144 256 7 P J i de and ter Morsche J L M van den Broek i IM RECONSTRUCTION TECHNIQUE APPLICABLE TO MICROWAVE RADIOMETRY Report 92 257 1992 SBN 90 6144 257 5 E Vleeshouwers J N DERIVATION OF A MODEL OP THE EXCITER OF A BRUSHLESS SYNCHRONOUS MACHINE BUT Report 92 F 258 1992 ISBN 0 6144 258 3 Orlov V B DEFECT MOTION AS THE ORIGIN OF THE 1 F CONDUCTANCE NOISE IN SOLIDS EUT Report 92 E 259 1992 ISBN 9 6144 239 1 Rooijackers J B ALGORITHMS FOR SPEECH CODING SYSTEMS BASED GN LINEAR PREDICTION Report 92 E 260 1992 ISBN 90 6144 260 5 Boon T J 3 van den and A A H Damen Martin Kiompstra IDENTIFICATION PO
29. here n is the size of A then there exists a similarity transformation T such that B TB C cr and the transformed system has a staircase form with the uncontrollable modes being the eigenvalues of A if any in the upper left hand corner A 0 0 i uc AL 2 5 ZM NEU 2 5 21 C C where A B is controllable A B 0 is uncontrollable and C sI A B C sI A B If the process P P and the design blocks V and W are given in transfer function matrices this approach can be used to derive a minimum state space representation of every block The dual approach for realizing a minimum state space representation can also be used In that case a controllable but not necessarily observable state space representation can be derived and all unobservable states have to be removed So if the observability matrix of A C has rank r S n there exists again a similarity transformation such that the transformed system has a staircase form with the unobservable modes if any in the upper left hand corner A A B 5 ipu i Ji 0 C 2 6 0 A B Because the blocks and W are now available as minimum state space realizations straightforward matrix computations for connecting state space systems in series or parallel can be used to build the generalized plant 1 Build I V and I parallel System 1 2 Build P and I parallel System 2 3 Build W and I para
30. ion 0909 001 No stabilizing controller 5455 001 stabilizing controller 27727e 001 Solution OK 6591e 001 Solution OK 6023 001 stabilizing controller 6307 001 No stabilizing controller 6449e 001 Solution OK 6378e 001 No stabilizing controller 6449e 001 Solution OK 0 3 9 UI i Uu NN Hp YO MD AD MD AD MIDIIYtHP Optimal gamma 9 6449 001 Check H infinity norm 9 6458 001 closed loop system can be Gamma iteration has been terminated succesfully used to verify y and the Press any key to continue difference should be of the same order as the Fig 3 24 Gamma iteration tolerance margin Whenever the Riccati equations are not solved properly large residuals or other numerical problems the closed loop system might not be stable although the y iteration has been 24 Menu Description terminated successfully Redefining the design filters and reducing the constraints and objectives can often help to overcome this problem Because in general H controllers are of high order order generalized plant some controller reduction options option 3 of the controller design menu have been included to realize lower order controllers If this reduction results in an unstable closed loop system the original high order controller can be loaded again without new calculations option 4 of the controller design menu 3 5 System Evaluation Discrete time MIMO H infinity Control
31. ler reduction is often required to obtain lower order controllers Load Original Controller The controller reduction step can result in less accurate closed loop performance which makes it necessary to use the original high order controllers Press any key to continue Fig 3 16b Help controller design Cont Discrete time MIMO H infinity Control Design The options described in Controller options gt 1 Type of H infinity approach GD Section 2 3 can be 2 Type of Riccati solution approach eigen A 3 Tol nce i ptimizi amma 0 01 defined in the controller 1 Balancing sugdented oleae Yo NO 5 Controller reduction method Minreal options menu shown in Fig 3 17 0 Heip Please enter menu option or press ENTER to Exit Fig 3 17 Controller options menu Change for the floating platform design example the following options into 3 Tolerance margin optimizing gamma 0 001 4 Balancing augmented plant Yes Help Controller Options Menu The help screen for the Type of H infinity solution Several routines are available which solve the H infinity control problem in different ways according to controller options menu 1 Safonov Limebeer Chiang loop shifting formulae Glover Doyle ali solution formulae or Limebeer Kasenally all solution formulae depicted in Fig 3 18 Type of Riccati Equation Approach The calculation of the H infinity involves the solution to two algebraic Riccati equations Thes
32. llel System 3 4 Connect system 1 in series with IM System 4 10 General Control Design Framework 5 Connect system 2 in series with IM System 5 6 Connect system 4 in series with system 5 System 6 7 Connect system 6 in series with system 3 System 7 8 Partition system 7 according to the defined inputs outputs 9 Close the loop around P and 1 The state space system of the generalized plant might not be a minimum realization because of common modes in the various blocks Removing again all uncontrollable 2 5 and unobservable 2 6 modes will yield a minimum state space realization of the generalized plant This approach has been selected because obtaining the same minimal realization after building the generalized plant using the non minimal state space realizations of the various blocks and applying Eq 2 5 and 2 6 only once might not be achievable due to numerical problems e g round off errors The constructed minimal state space realization of the augmented plant might be badly conditioned depending on the design filters and process behaviour This can result in numerical problems when calculating the H controller Balancing of the augmented plant is therefore often desired to improve numerical reliability The balancing approach described in Weiland S 1993 is used in order to handle unstable as well as stable systems 2 3 Controller Calculation Because a minimum state space representation is available
33. n mounted rotating a load and thereby tilting the platform The control to be designed should prevent this tilting of the platform detailed description of the process together with the physical modeling identification and control design can be found in Bouwels J P H M 1991 and Damen A A H et al 1994 A detailed menu description of the toolbox together with several design options is given in Section 3 introduction General H_ Control Design Framework In this section the most important parts of the general framework will be explained Fig 2 1 depicts the H control design configuration for the floating platform The solid part illustrates the basic control configuration while the dashed part is added for the H control design The main objectives in the design are disturbance attenuation Vang to prevent tilting of the platform due to the rotating crane and robustness model errors represented by V n due to unmodeled waves In addition saturation of the actuators W up should be avoided The transfers of Fig 2 1 are described as follows P Platform Dynamics W Weighting Process Output R Disturbance Dynamics W Weighting Control Input Shaping Crane Disturbance C Feedback Controller V Shaping Model Disturbance C Feedforward Controller where we can define the following standard signals n TW 4 E usu 2 1 d ny gr wu 6 General Control Design Framework pu
34. nfinity standard problem with an augmented plant that contains what usually is called the plant in a control problem plus all weighting functions The augmented plant consists of four basic blocks 1 2 Pl amp P2 Process models 3 V Shaping the disturbance vector w t and 4 W Weighting the error vector zi t Further these four blocks are somehow connected through the interconnection matrices amp IM2 containing only the elements 1 and 0 which corresponds with adding subtracting or no connection Every control configuration can now be described in a simple way by defining the dimensions nl nil together with the structure of the configuration using the interconnection matrices Press any key to continue Fig 3 8b Help structure initialization menu cont 3 3 Input Matrix Functions After initializing the Discrete time MIMO H infinity Control Design structure the blocks Input Matrix Functions Pl 2 V or W P2 V and W must be Enter Pl as SISO transfer functions Enter Pl as MIMO st te space matrices defined This menu is Show magnitude plots of Pl Enter P2 as SISO transfer functions depicted in Fig 3 9 The re matrices blocks can be defined Enter V as SISO transfer functions Enter V as MIMO state space matrices either as SISO transfer Show magnitude plots of V functions per entry or as Enter W a NINO state space matrices MIMO state space Show magnitude plots of W matrices An
35. ng the conditions to solve the H control Transfer G22 from the control input vector u t to the problem observation vector y t is stabilizable and detectable Go A B stabilizable Transfer G12 from the control input vector u t to the error and A C detectable vector z t has full column rank at infinity Dj full column rank Transfer G21 from the disturbance vector w t to the observation D full TOW rank vector has full row rank at infinity No transmission zeros have been detected in the transfer G12 column rank from the control input vector u t to the error vector zit No transmission zeros have been detected in the transfer G21 row rank from the disturbance vector wit to the observation vector y t Only if a condition is violated an error message Fig 3 22 Check solution conditions will appear For y minimization a new Calculate H infinity Controller start value and step size value of gamma 1 must be defined Please enter new start value of gamma or press ENTER Old step size 2 Please enter new step size or press ENTER Start Value 1 Step size 1 1 Fig 3 23 Initializing Y The tolerance margin defined in the controller options determines the number of iterations Fig 3 24 For an optimal design y should be slightly smaller than 1 The H norm of the End of iteration H H Gamma Comment 0000 000 Solut
36. ntil no solution exists defining Ymin 7 Tk 1 2 If no solution exists for Yo Yop gt Yo define min Yo The upper bound of the interval can be found by increasing Y y until a solution exists defining Ymax 7 Bisection search is used to find within a certain tolerance margin for which a stabilizing solution exists 1 Define Yk Ymax Ymin 2 2 If a solution exists for adjust the upper bound Ymax If no solution exists for Yx adjust the lower bound Ymin 3 Repeat 1 amp 2 until Ymax Ymin Ymin tol When starting the controller design no information is available about Yopt which depends of course on the design filters and the process Because the final goal is to achieve y T 1 it is recommended to start with 1 and a 2 to reduce the number of iterations This approach has the advantage that it is reasonably fast 7 to 15 iterations depending on the tolerance margin and that independent of the start value y a sub optimal solution is found The variable tolerance margin has been introduced to speed up the design fewer iterations and because of the fact that if this margin becomes too small the Riccati equations cannot be solved properly anymore Using the method proposed in Bruinsma 1990 the norm of the closed loop system can be used to check the solution Yopt of the search procedure a postiori Since the standard solution is only available for the continuous time
37. on Help screen for structure initialization menu Help screen for input matrix functions menu Help screen for define state space matrices menu Help screen for controller design menu Help screen for controller options menu Help screen for system evaluation menu Help screen for options menu Help screen for disk options menu H controller basic function which prepares the variables for the general MIMO configuration and minimizes Y to calculate the optimal controller Script file to generate H control menu Shows H controller options menu Routine to calculate norm of a state space system which is the maximum over all frequencies of the maximum singular value Function name mhc_hm m mhc_im m mhc_imf m mhc_kgjd m mhc_map m mhc_meta m mhc_mss m mhc_opt m mhc pcl m mhc pzc m rbal m mhc rtf m mhc sbp m mhc sem m mhc sim m mhc slrc m 37 Description Help screen for main menu Function to build interconnection matrices Script file to generate input matrix functions menu H all solution formulae derived by Glover and Doyle Function to construct minimal realization of the augmented plant for the basic structure This function file generates a meta file using a filename defined by the user and writes the current graph to for late processing Routine to calculate minimal state space realization Script file to generate options menu Function to plot closed loop trans
38. ontinuous time high low order depending on design options State space matrices of original controller in continuous time domain backup of Ac Bc Cc Dc if controller reduction fails State space representation of process block P1 39 40 Variables Ap2 Bp2 Cp2 Dp2 Av Bv Cv Dv Aw Bw Cw Dw IM1 IM2 alpha dim flag List of Variables Description State space representation of process block P2 State space representation of design block V State space representation of design block W Interconnection matrices for basic MHC structure Step size Dimension array for basic MHC structure Information array about current status D 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode 1 Continuous 2 Discrete Configuration structure 1 Known 0 Unknown Process block Process block P2 0 Unknown 1 Transfer function Shaping block V 2 State space matrices Weighing block W Generating META files 1 Yes 0 No Valid controller design 1 Yes 0 No Valid controller reduction 1 Yes 0 No H type approach 1 Safonov Limebeer Chiang loop shifting formulae 2 Glover Doyle all solution formulae 3 Limebeer Kasenally all solution formulae Type of Riccati solution approach 1 Eigen 2 Schur Balancing augmented plant 1 Yes No Controller reduction method 1 Minimal realization 2 Optimal Hankel method 3 Schur reduction method
39. rewritten into a standard H control problem In this report a general package is described that facilitates the controller design for various control configurations the standard H control problem and the closed loop system evaluation Because no solution is known for translating design specifications such as desired behaviour robustness performance etc directly into weighting functions in the frequency domain the necessarily iterative design procedure has been implemented in a flexible menu driven way Keywords Robust control Multivariable contro systems Control simulation software Falkus H M and A A H Damen Multivariable H infinity Control Design Toolbox User manual Eindhoven Faculty of Electrical Engineering Eindhoven University of Technology The Netherlands 1994 EUT Report 94 E 282 Address of the Authors Measurement and Control Section Faculty of Electrical Engineering Eindhoven University of Technology P O Box 513 5600 MB Eindhoven The Netherlands 111 Abstract der cow Introduction General H Control Design Framework 2 Structure Definition 2 2 Minimum Realization Generalized Plant 2 3 Controller Calculation 2 4 Evaluation Controller Design 2 5 Installation and Requirements Menu Description 3 1 Options 3 2 Structure Initialization 3 3 Input Matrix Functions 3 4 Controller Design 3 5 System Evaluation 3 6 Disk Functions Conclusions Menu Ove
40. rices is that all the process and design blocks must be proper In addition the magnitude of the designed filters can be plotted REMARK To simplify the H infinity control design it is recommended to define the shaping and weighting blocks V and W respectively as square functions equal number of inputs and outputs with elements only on the diagonal Press any key to continue Fig 3 10b Help input matrix functions P1 P2 V or W cont When defining a block as SISO transfer function Enter input number 1 1 or press ENTER to Exit the correct element of the Enter output number 1 1 Discrete Transfer Matrix Pl matrix must be selected first Fig 3 11 Selecting element of transfer matrix Every transfer function is Define Discrete Transfer Function of Pi ll defined by its numerator Please define the following polynomials in MATLAB notation Example 2 3 2272 32 1 gt 12 3 1 and denominator Both polynomials can be ora numerator 0 entered as arrays in New numerator MATLAB notation The old denominator New denominator polynomials must be defined in powers of Z or 8 for the discrete continuous time respectively Fig 3 12 Defining SISO transfer function 3 3 Input Matrix Functions The menu to define the state space matrices A B C amp D is depicted in Fig 3 13 These matrices can be entered exactly the same way as th
41. rol design is rather straightforward and the required user input is fairly simple The example of the floating platform should be sufficient to guide the user through all menus of the design procedure It is not the intention to show with this example a complete H control design procedure for all shaping and weighting filters This is described in more detail in Bouwels J P H M 1991 28 Menu Description Conclusions Any control configuration can be rewritten in the presented basic structure which is automatically transformed into a standard control problem The menu driven structure of the toolbox makes the necessarily iterative design procedure fast due to easy input of variables and simple analysis of the results by calculating time and frequency responses The H control design of a laboratory process has been used to show the user how to define the basic control structure An extensive description of all menus and heip facilities should guide the user through the design and explain all options 29 30 Conclusions A Menu Overview The menus of the multivariable H control design MHC toolbox are presented in one scheme to provide an overview of the most important functions This overview can be used as quick reference guide by the user during the H control design 31 Discrete time MIMO H infinity Control Design Discrete time MIMO H infinity Control Design Structure initialization Sy
42. rview Program Structure Function Description List of Variables References w s E 4 w 4 ss gt vw a 4 gt kk gt w 4 4 w a n 4 4 4 n gt n w w 4 w n n a a W gt w s gt an 4 4 ns 9 w 4 a 4 9 w ns 83 n 4 o n 9 4 o o a Contents vi Contents Introduction In the last few years there has been much interest in the design of feedback controllers for linear systems that minimize the norm of a specified closed loop transfer function Since 1988 a state space solution for general H problems based on a 2 Riccati approach derived by Glover K and J C Doyle 1988 has been available for the representation of all stabilizing controllers that satisfy an H norm bound sic 1 aa A more detailed explanation and a proof of its validity is outlined in Doyle J C et al 1989 Standard program packages Robust control toolbox of MATLAB together with some numerical variations and extensions of the basic solution are available now and can be applied once the original problem has been translated into the standard control H problem The generalized plant G contains what usually is called the plant in a control problem and includes all weightin
43. s 1988 Damen A A H and H M Falkus J P H M Bouwels Modeling and control of a floating platform To be published in IEEE Trans on Aut Control 1994 43 44 References Doyle J and B Francis A Tannenbaum Feedback control theory New York McMillan 1990 Doyle J C and K Glover P P Khargonekar B A Francis State space solutions to standard H and H control problems IEEE Trans Aut Control Vol 34 1989 No 8 p 831 847 Falkus H M and A A H Damen J Bouwels General MIMO H control design framework In Proc 31 IEEE Conf on Decision and Control Tucson Arizona December 16 18 1992 New York IEEE 1992 P 2181 2186 Francis B A A course in H control theory Heidelberg Springer 1987 Lecture notes in control and information sciences Vol 88 Glover K and J C Doyle State space formulae for all stabilizing controllers that satisfy an H norm bound and relations to risk sensitivity Systems amp Control Letters Vol 11 1988 p 167 172 Grace A and A J Laub J N Little C Thompson Control system toolbox User s guide Natick MA The Mathworks 1990 Iglesias and D Mustafa State space solution of the discrete time minimum entropy control problem via separation IEEE Trans on Aut Control Vol AC 38 1993 No 10 p 1525 1530 Kailath T Linear systems Englewood Cliffs N J Prentice Hall 1980 45 Klompstra M and T van den
44. sign configuration Fig 2 1 can be built up of four major blocks Fig 2 2 1 2 Process models P amp P 3 Shaping filters V for the input signals w 4 Weighting filters W for the output signals z The extra process block P is sometimes necessary if there exists already a known feedback Further these four blocks are somehow connected Rearranging of Fig 2 1 into the blocks P Pa V and W is carried out in Fig 2 3 where IM and IM define the structure of the control design configuration IM and IM reflect the interconnection structure of the various blocks These are constant matrices with entries 1 and 0 each entry corresponding to a specific adding subtracting or no connection of signals The matrices I to I define the feed through of signals which are necessary to build the state space representation of the generalized plant G We emphasize that this structure is general That is every control configuration of the form shown in Fig 2 1 can be represented by a configuration of Fig 2 2 resulting in Fig 2 3 8 General Control Design Framework 7 V M M W pU u MEUM G ba bs Fig 2 2 Basic structure generalized plant G 3 P IM W um 7777700 va Qa eu nn a ay Be 2 a Wyp a AR ANA 1 51 aw mw w oak m Fig 2 3 Floating platform configuration
45. sponses can be calculated Time simulations can be performed to check the closed loop behaviour with respect to design objectives and constraints in the time domain like disturbance attenuation reference tracking and or input saturation Frequency response analysis can be used to verify sensitivity and complimentary sensitivity functions Whenever the design functions V and W are defined as diagonal blocks which is recommended to keep the design as simple as possible the closed loop behaviour from every input to every output can be compared with the corresponding inverse weighting functions scaled with the norm This can simplify the iterative controller design because the Bode plots indicate which function in which frequency range is the limiting factor and where and how the design can be improved If the controller design is not satisfactory the menu driven structure of the toolbox ensures that the design filters can be changed fast and a new controller can be calculated easily in order to optimize the controller design At every stage of the design procedure the controller configuration as well as the actual results can be saved to ensure continuation if necessary The toolbox is built up in such a way that the input required from the user is minimized and that correct data transfer between the various functions is guaranteed 2 5 Installation and Requirements All names of the m files in the toolbox start with mhc App B and can b
46. stem Evaluation Men us 1 General Structure initialization Show bode plots controller Plot closed loop transfer function 2 Change interconnection matrix a s 3 Change interconnection matrix IM2 Show time simulation Multi variable 01 Help Help n ABER an B m Please enter menu option or prass ENTER to Exit Please enter menu option or press ENTER to Exit Fi Design OO Discrete time MIMO H infinity Control Design Input Matrix Functions P1 P2 V or W Discrete time MIMO H infinity Contro Design Enter Pl as SISO transfer functions Discrete time MIMO H infinity Control Design Enter as MIMO state space matrices Show magnitude plots of PI Options Main menu Mode Discrete Enter P2 as SISO transfer functions 1 Structure initialization Tolerance margin for minimization procedure 1e 06 Enter P2 as MIMO state space matrices 2 Input matrix functions P1 P2 V or W Selected input signal Show magnitude plots of 2 3 Controller design Generating META files No 4 System evaluation Lower frequency bound 4 rad Enter V as SISO transfer functions 5 Options Upper frequency bound 3 14 rad Enter V as MIMO state space matrices 6 Disk functions Number of frequency points 100 Show magnitude plots of V End of time interval 0 1 sec 0 Help Enter W as SISO transfer functions Help Enter W as MIMO state space matrices Please enter menu option or press ENTER to Exit
47. ti equations each with the same order as the system and further gives feasible controllers also with depicted in Fig 3 16 this order H infinity Controller Options In this menu some controller relevant settings are defined like type of H infinity solution type of Riccati equation solution and a tolerance margin indicating the accuracy of the closed loop H infinity norm with respect to the optimal gamma In addition an option for balancing of the augmented plant can be selected This can improve the numerical stability of the controller design Also a reduction technique can be selected to reduce the order of the H infinity controller Press any key to continue Fig 3 16a Help controller design 3 4 Controller Design 21 Help Controller Design Menu Calculate H infinity controller After checking the conditions for the existence of a stabilizing controller satisfying an H infinity norm of the closed loop system like whether the system is stabilizable and detectable rank conditions to ensure that the controllers are proper and no zeros on the imaginary axis of some transfers a start value and step size for gamma must be defined After deriving an interval containing the optimal solution the nearly optimal gamma will be derived using bisection search until the required accuracy has been obtained Controller Reduction Because the order of the controllers will be the same as the order of the system control
48. tions for design package Disk Functions Menu to load and save variables WARNING all variables are initialized in a standard form when Starting up the design package Press any key to continue Fig 3 2b Help main menu cont 3 1 Options Continuous time MIMO H infinity Control Design Before starting the actual Options 1 Mod Conti controller design several ko lot R G margin for minimization procedure ag Selected i i 1 options must be defined Generating META files No Lower frequency bound 2 rad in the options menu The Upper frequency bound 2 rad Number of frequency points 50 default menu is shown in 1 sec Fig 3 3 di Select menu option or press ENTER to Exit Fig 3 3 Options menu Change for the floating platform design example the following options into 1 Mode Discrete 3 2 Structure initialization 15 2 Tolerance margin 1e 6 5 Lower frequency bound a 10 7 Frequency points 100 8 Sample time 0 1 Help Options Menu The help screen provides Mode The H infinity control design can be done either in the continuous time or discrete time Because the algorithms are only 1 1 valid for the continuous time problem the discrete time is solved by a brief explanation of the transforming the discrete augmented plant via bilinear transformation to the continuous time calculating the controller and applying the several options
49. w number Enter row number 1 4 or press ENTER to Exit this row be defined Enter elements row 3 in MATLAB notation 100 as an array in matlab notation Fig 3 7 depicts the screen for Fig 3 7 Interconnection matrix IMI Interconnection matrices IMi IM2 oOo O F o Hd H oo HO O PM H o O H o 3 3 Input Matrix Functions 17 Help Structure Initialization Menu Fig 3 8 shows the help H infinity standard problem screen of the structure BEER er td ee me M diui mom m EK Ee ee m mm eee m mmc mco Ao ooo o initialization menu DL eee gt I gt n a pi n6 I v gt gt W 7 1 a z 4 n Bod RR 1 10 nii gt gt 1 MIMO Augmented Plant SPEEA A x Press any key to continue Fig 3 8a Help structure initialization menu Help Structure Initialization Menu Before designing an H infinity controller che control setup has to be defined together with the design constraints and objectives e g tracking disturbance rejection input saturation etc This design configuration must be transformed into the H i
50. y of Technology Research Reparts ISSN 0167 9708 Facuitwv 2691 2703 1424 273 12741 2763 278 279 280 2811 Coden TEUEDE of Electricai Engineering mman Vie Putten K A van der MULTIDISCIPLINAIR SPECIFICEREN EN ONTWERPEN VAN MICRGELEKTRONICA IN PRODUKTEN in Dutch EUT Report 93 E 269 1993 ISBN 90 6144 269 Blows R H J FROGRIL A ianquage for the definition of protocol grammars BUT Report 93 E 270 1993 ISDN 90 6144 270 2 loks KAJ CODE GENERATION FOR THE ATTRIBUTE EVALUATOR OF THE PROTOCOL ENGINE GRAMMAR PROCESSOR UNIT BUT Report 95 E 271 1993 ISBN 90 6144 271 6 ca Yan Keping and E M van Veldhuizen FLUE GAS CLEANING BY PULSE CORONA STREAMER Report 93 E 272 1993 ISBN 90 6144 272 9 Smolders A b FINITE STACKED MICROSTRIP ARRAYS WITH THICK SUBSTRATES EUT Report 95 E 273 1993 ISBN 0 6144 273 7 tax plien M H w and A A van Houten GN INSULAR POWER SYSTEMS Drawing up an inventory of phenomena and research possibilities EUT Report 93 E 274 1993 ISBN 90 6144 274 5 Deursen van ELECTROMAGNETIC COMPATIBILITY Part 3 installation and mitigation guidelines section 3 cabling and wiring EUT Report 93 E 275 1993 ISBN 90 6144 275 3 bolien M H LITERATURE SEARCH FOR RELIABILITY DATA OF COMPONENTS IN ELECTRIC DISTRIBUTION NETWORKS BUT Report 93 E 276 1993 ISBN 90 6144 276 i inani Sien A BEHAVIORAL APPROACH TO BALANCED
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