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Multivariable H-infinity ControlDesign ToolboxbyH.M. FalkusA.A.H. DamenEUT Report 94-E-282ISBN 90-6144-282-6April 1994

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Eindhoven University of Technology Research Reports

EINDHOVEN UNIVERSITY OF TECHNOLOGY

ISSN 0167-9708

Faculty of Electrical EngineeringEindhoven, The Netherlands

Coden: TEUEDE

Multivariable H-infinity ControlDesign Toolbox

User Manualby

H.M. FalkusA.A.H. Damen

EUT Report 94-E-282ISBN 90-6144-282-6

EindhovenApril 1994

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I I Multivariable Hw Control Design Toolbox

CIP-DATA KONINKLUKE BmLIOTHEEK, DEN HAAGFalkus, H.M.Multivariable H-infinity control design toolbox: usermanual / by H.M. Falkus, A.A.H. Darnen. - Eindhoven :Eindhoven University of Technology, Faculty of ElectricalEngineering. - Fig. - (EUT report, ISSN 0167 - 9708 ;94-E-282)With ref.ISBN 90-6144-282-6NUGI 832Subject headings: robust control I multi variable controlsystems I control simulation softwarc.

/

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Abstract

Multivariable H-infinity Control Design Toolbox: User manualH.M. Falkus and A.A.H. Darnen

A MA TLAB toolbox is presented for solving the multivariable control design problem.Algorithms are available (Robust control toolbox of MA TLAB) which solve the problem,once the control design configuration including process model and weighting functions hasbeen rewritten into a standard control problem. In this report a general package isdescribed that facilitates the controller design for various control configurations, the standard

control problem and the closed-loop system evaluation. Because no solution is known fortranslating design specifications such as desired behaviour, robustness, performance etc.directly into weighting functions in the frequency domain, the necessarily iterative designprocedure has been implemented in a flexible, menu driven way.Keywords: Robust control, Multivariable control systems, Control simulation software.- Falkus, H.M. and A.A.H. Damen

Multivariable H-infinity Control Design Toolbox: User manualEindhoven : 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

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iv Abstract

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Contents

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 General Control Design Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Structure Definition 72.2 Minimum Realization Generalized Plant 82.3 Controller Calculation2.4 Evaluation Controller Design2.5 Installation and Requirements

101212

3 Menu Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Options 143.2 Structure Initialization3.3 Input Matrix Functions3.4 Controller Design3.5 System Evaluation3.6 Disk Functions

1517202427

4 Conclusions . ............................................... 29A Menu Overview ............................................. 31B Program Structure ........................................... 33C Function Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35D List of Variables ............................................. 39

References ................................................. 43v

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VI Contents

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1Introduction

In the last few years, there has been much interest in the design of feedback controllers forlinear systems that minimize the norm of a specified closed-loop transfer function. Since1988, a state-space solution for general problems based on a "2-Riccati" approach, derivedby Glover, K. and J.e. Doyle (1988), has been available for the representation of allstabilizing controllers that satisfy an norm bound :

119"(G,K) L::; y (1.1)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 ofMATLAB) together with somenumerical variations and extensions of the basic solution are available now and can be appliedonce the original problem has been translated into the standard control problem.The generalized plant G contains what usually is called the plant in a control problem andincludes all weighting functions. The signal w E Rm! represents all external inputs, includingdisturbances, sensor noise and commands; the output z E RPI is the error vector; y E RP2 isthe observation vector; and u E Rm2 is the control input. The generalized plant G can bepartitioned according to the dimensions of the signals:

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2 Introduction

(1.2)

which results in the following closed-loop transfer function from w to z :(1.3)

w zGu y

KFig. 1.1 : Standard Problem.

The standard assumptions are : The triplet (A,B2,C2) - namely the plant transfer G22 - can be stabilized and detected,so that stabilizing controllers exist. rank(D I2) =m2 and rank(D21 ) = P2 in order to ensure realizability of the controllers. No zeros on the imaginary axis, PI m2 and m l P2 ensures that the solution to thecorresponding LQG problem is closed-loop asymptotically stable.

The main problem, however, is that every control problem has a different configurationbecause of different design constraints and control objectives. This implies that every newcontrol problem has to be rewritten again into the standard 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 control problem. Afterselecting the control setup, design constraints and objectives, the design configuration isdefined in a fairly simple way. Because no general solution is known for translating designspecifications such as desired behavior, robustness, performance etc. directly into weightingfunctions in the frequency domain, the control design is menu driven to ensure easy input

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3

of variables, controller calculation, and analysis of the results by computing both time andfrequency responses. In this way the necessary iterative design procedure for optimizing theH= control design problem becomes much easier. All tools in this toolbox are implementedin MATLAB by means of standard .m-files.In Section 2 the basic setup of the toolbox is presented using the process block diagram ofa floating platform laboratory process as an example. The floating platform with rotatingcrane has been built on laboratory scale to evaluate identification and control theories. Thisparticular process was chosen because it is an essentially linear MIMO system. It can be welldescribed by three decoupled, second order SISO systems. The model errors are then mainlydue to unmodeled waves, caused by the movement of the floats, which lead to linear transferswhich are however difficult to model. The fact that H= control is said to be particularly suitedfor robust control in cases of unmodeled linear dynamics, makes this laboratory process anexcellent example for testing the H= control synthesis procedure. On the platform, a cranehas been mounted rotating a load and thereby tilting the platform. The control to be designedshould prevent this tilting of the platform A detailed description of the process together withthe 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 inSection 3.

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4 Introduction

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2General Hoc ControlDesign Framework

In this section the most important parts of the general framework will be explained. Fig. 2.1depicts the control design configuration for the floating platform. The solid part illustratesthe basic control configuration, while the dashed part is added for the control design. Themain objectives in the design are disturbance attenuation (V dnd to prevent tilting of theplatform due to the rotating crane) and robustness (model errors represented by Vyny due tounmodeled waves). In addition saturation of the actuators C'NuUp) should be avoided.The transfers of Fig. 2.1 are described as follows :

P : Platform DynamicsR : Disturbance DynamicsVd : Shaping Crane DisturbanceV y : Shaping Model Disturbance

Wy : Weighting Process OutputW u : Weighting Control InputCtb : Feedback ControllerCff : Feedforward Controller

where we can define the following standard signals:

5

(2.1)

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6 General Hw Control Design Framework

nd r - - - ,- - .. .1 Vd l - - < > - - - < l ~ - - - + I ,1___ J

: nvr _t.._,'V 'v ,' _ _ _ J ,..--- ii, '

- - - - -1 - - - - - - - - - Wu - -v , ___ JYm r - - -, y+ } - ~ t - - o - ~ W y ~ - - ' "

, _ _ _ J

Fig. 2.1 : Multivariable Control Design Configuration Floating Platform

After deriving the various relations between the inputs and outputs, it follows that, in termsof the standard problem, the generalized plant G is defined by :

y WyVv WyRVd WyPnv( i i 0 0 Wu+ nd= =

Ym Vv RVd I P up (2.2)d 0 Vd 0

f' G" 1: 1G21 I G22The control acting on the outputs y is represented by :(2.3)

This yields the closed-loop system .'T:= .9\G,K) mapping w -t z which has norm:

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2.1 Structure Definition

11.rIL=

Wy(1 -pc fb t VvW u Cfb(1 -PC fbrl V v

Here the various transfer functions are named as follows :

.rl l : Sensitivity

.r21 : Control Sensitivity.r12 : Disturbance Attenuation.r22 : Saturation

7

(2.4)

Until now we have only described the standard approach to define the control problemfor the floating platform to achieve disturbance attenuation and avoiding saturation of thecontrol input. This results in the closed-loop system ~ G , K ) of Eq. 2.4. This configurationhowever is specific to the floating platform and can be significantly different for other controlproblems. To avoid this procedure of building (and implementing) different configurationsevery time when new control problems occur, we will generalize this setup.

2.1 Structure DefinitionBasicly, the design configuration (Fig. 2.1) can be built up of four major blocks (Fig. 2.2) :

1,2) Process models PI & P23) Shaping filters V for the input signals w4) Weighting filters W for the output signals z.

The extra process block P2 is sometimes necessary if there exists already a known feedback.Further, these four blocks are somehow connected. Rearranging of Fig. 2.1 into the blocksPI ' P2, V and W is carried out in Fig. 2.3 where IMI and 1M2 define the structure of thecontrol design configuration. 1M I and 1M2 reflect the interconnection structure of the variousblocks. These are constant matrices with entries 1 and 0, each entry corresponding to aspecific adding, subtracting or no connection of signals. The matrices 1\ to 14 define thefeed-through of signals which are necessary to build the state-space representation of thegeneralized plant G.We emphasize that this structure is general. That is, every control configuration of the formshown in Fig. 2.1 can be represented by a configuration of Fig. 2.2 resulting in Fig. 2.3.

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8 General Control Design Framework

r--------------------------------------------------,,,,,- II - - P, I--- PI I - -, I I zV - M M I - - WI 2- I, I--u YI , I .- - I - - :G-------------------------------------------------,

Fig. 2.2 : Basic structure generalized plant G.

V IMI PI 1M, W, - - - - - - - - - , - - - - - - - dl- - - - - - - - - r--------, , ---------V : 0 1'0,0 b ~ , ] '0 ' ] '0 W-:OJL-1d I I I _ I. _, _ I PI I - I t , I I I Y I II - - - - - - I I O ' O ' ] ~ , 1_"_1-1_ 1 1 -----.-y I : ,V I I I I 1 _________ I O'O'O'} ,UPI : I U0 I V" ] - ~ O - : - O :d,---------d: _ ~ _ : - - : 0 ,Wu :-

I I I I I !.......J ] '0 ' ] '0 I I - - ! - - ~ ---------1 I -r--I- I I I I I I ---------,o ] 0 v 'v - . . . . _. 'v, --------- : _:__ :_ I a.....! 0:1:0:0 ,Jm,---------up I I :UP! 0 I 0 11 'U I 3 IU I I I[: ____ ___ __ .'._J - ! ! ~ ________ ______ ____ ___K ,- - - - - - - - -, Ym,-----------!: c" iCrr [ ;..:- d - - - - ~ ~ , ~ - - - - ~ ______ ---

Fig. 2.3 : Floating platform configuration for MHC toolbox.

The control design configuration of the floating platform can now be described in a simpleway by defining the dimensions of V (Vd & Vv) ' PI (R & P), P2 (0), W (Wy & Wu) and IIto 14 together with the structure of the configuration using the interconnection blocks IMI and1M2, This is e:ffectively all input which is needed for the toolbox to convert it into a standardcontrol problem.

2.2 Minimal Realization Generalized PlantThe blocks V, PI ' P2 and W are represented in the toolbox either as MIMO state-spacerepresentations or, entry wise, as SISO transfer functions of a matrix. However, beforebuilding the generalized plant G, minimum state-space realizations of these blocks have tobe obtained.

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2.2 Minimal Realization Generalized Plant 9

Therefore we use the approach outlined in Munro, N. et. af. (1971). For every row in atransfer function matrix, the smallest common denominator is determined and the numeratorsare updated if necessary. The new MISO transfer functions can be transformed into anobserver canonical state-space representation which is minimal. Combining the state matricesof the MISO systems for every row in a block-diagonal form and adding the input, output andfeed-through matrices correctly results in an overall state-space representation which isobservable but not necessarily controllable. In Dooren, P.M. v. (1981) it is proven that if thecontrollability matrix of (A,B) has rank r ::; n, where n is the size of A, then there exists asimilarity transformation T such that :

B =TB t =CT T is =Dand the transformed system has a staircase form with the uncontrollable modes (being theeigenvalues of Auc), if any, in the upper left-hand comer.

(2.5)

where (Ac,Bc) is controllable, (Auc,Buc=O) is uncontrollable and Cc(sI-AcrIBc = C(sI-ArIB.If the process PI' P2 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. Inthat case a controllable but not necessarily observable state-space representation can bederived and all unobservable states have to be removed. So if the observability matrix of(A,C) has rank r ::; n, there exists again a similarity transformation such that the transformedsystem has a staircase form with the unobservable modes, if any, in the upper left-handcorner.

(2.6)

Because the blocks V, PI' P2 and W are now available as minimum state-space realizations,straightforward matrix computations for connecting state-space systems in series or parallelcan be used to build the generalized plant :

\) Build II' V and 12 parallel (System I).2) Build P I and 13 parallel (System 2).3) Build P2, Wand 14 parallel (System 3).4) Connect system \ in series with 1M I (System 4).

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10 General Hw Control Design Framework

5) Connect system 2 in series with 1M2 (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 P2 and II'

The s t a t e - s p a l ~ e system of the generalized plant might not be a minimum realization becauseof common modes in the various blocks. Removing again all uncontrollable (2.5) andunobservable (2.6) modes will yield a minimum state-space realization of the generalizedplant. This approach has been selected because obtaining the same minimal realization afterbuilding the generalized plant using the non-minimal state-space realizations of the variousblocks and applying Eq. 2.5 and 2.6 only once, might not be achievable due to numericalproblems (e.g. round-off errors).The constructed minimal state-space realization of the augmented plant might be badlyconditioned depending on the design filters and process behaviour. This can result innumerical problems when calculating the controller. Balancing of the augmented plant istherefore often desired to improve numerical reliability. The balancing approach described inWeiland, S. (1993) is used in order to handle unstable as well as stable systems.

2.3 Conttol/er CalculationBecause a minimum state-space representation is available, the standard solution methodbased on solving two Algebraic Riccati equations and implemented in the Robust ControlToolbox of MATLAB, Chiang, R.C. et. al. (1988), can be applied. The methods available:

1) SafonovlLimebeer/Chiang loop-shifting formulae; Safonov, M.G. et. al. (1989).2) Glover/Doyle all-solution formulae; Glover, K. et. al. (1988).3) LimebeerlKasenaily all-solution formulae; Limebeer, OJ.N. et. al. (1988).

are only different in circumventing some of the numerical problems which generally arisewhen a design approaches its performance limits.The solutions to the Riccati equations can be solved either by an eigenvalue or Schurdecomposition. The eigenvalue approach is the fastest but for design filters close to theperformance 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, acontroller is computed achieving II 7 y. However, we are interested in Yapt for whicha stabilizing controller still exists. Therefore the basic routine has been extended as followswith an iterative search procedure:

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2.2 Minimal Realization Generalized Plant 11

- A start value 'Yo and a step size 0: (0: > 1) are defined.An interval [ 'Ymin' Ymax 1 s computed which contains the optimal solution.I) If a solution exists for Yo (Yopt ::; Yo) define Ymax =Yo' The lower bound of the intervalcan be found by decreasing Y (Yk+\ = yo/o:k) until no solution exists defining

Ymin = Yk+\'2) If no solution exists for Yo (Yopt > Yo) define Ymin = Yo' The upper bound of theinterval can be found by increasing Y (Yk+\ = o:k.yk) until a solution exists definingYmax = Yk+\'Bisection search is used to find Yopt within a certain tolerance margin for which a

stabilizing solution exists.I) Define Yk = (Ymax + Ymin)122) If a solution exists for Yk adjust the upper bound Ymax = Yk' I f no solution exists forYk adjust the lower bound Ymin = Yk'

3) Repeat I & 2 until (Ymax - Ymin)/Ymin ::; tol.When starting the controller design, no information is available about Yopt which depends ofcourse on the design filters and the process. Because the final goal is to achieve Yi I , it isrecommended to start with 'Yo =1 and 0: =2 to reduce the number of iterations. Thisapproach has the advantage that it is reasonably fast (7 to 15 iterations depending on thetolerance margin) and that independent of the start value Yo 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 besolved properly anymore. Using the method proposed in Bruinsma, N.A. (1990), the normof the closed-loop system can be used to check the solution Yopt of the search procedure apostiori. Since the standard solution is only available for the continuous-time case, it shouldbe mentioned here that the discrete-time case is solved via bilinear transformation. InStoorvogel, A.A. et. al. (1993) and Iglesias, P.A. et. al. (1993), it is shown that designing adiscrete-time controller via a bilinear transformation to the continuous-time domain mightintroduce an implicit and undesirable additional weighting function. A simple free stablecontraction map is added to eliminate this additional weighting.In general, the reSUlting -controllers are of high order because the order is equal to theorder of the generalized plant (process & all design filters). To reduce the order of thecontroller, the following reduction techniques can be applied to the resulting controller:

I) Minimal state-space realization (reduces within a predefined tolerance margin).2) Optimal Hankel reduction.3) Schur reduction.4) Relative Schur reduction.

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12 General Control Design Framework

For the reduction methods 2 to 4 an additional option can be selected to reduce the controllerwith variable order and fixed error bound or fixed order and variable error bound. A detaileddescription and more references for these reduction techniques can be found in Chiang, R.C.et. al. (1988).

2.4 Evaluation Controller DesignAfter calculating the controller, the closed-loop system is derived (without shaping andweighting functions) in order to evaluate the controller design. For this purpose time as wellas frequency responses can be calculated. Time simulations can be performed to check theclosed-loop behaviour with respect to design objectives and constraints in the time domainlike disturbance attenuation, reference tracking and/or input saturation. Frequency responseanalysis can be used to verify sensitivity and complimentary sensitivity functions. Wheneverthe design functions V and W are defined as diagonal blocks, which is recommended to keepthe design as simple as possible, the closed-loop behaviour from every input to every outputcan be compared with the corresponding inverse weighting functions (scaled with the H ~ norm y). This can simplify the iterative controller design because the Bode plots indicatewhich function in which frequency range is the limiting factor and where and how the designcan be improved.I f the controller design is not satisfactory, the menu driven structure of the toolbox ensuresthat the design filters can be changed fast and a new controller can be calculated easily inorder to optimize the controller design. At every stage of the design procedure, the controllerconfiguration 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 andthat correct data transfer between the various functions is guaranteed.

2.5 Instaflation and RequirementsAll names of the .m-files in the toolbox start with "rnhc" (App. B) and can be copied (e.g.copy rnhc*.m) to the working directory of MATLAB. I f the files are copied to a directorydifferent than the workspace of MATLAB, this directory has to be added to the matlabpath.The routines in this toolbox make use of standard MA TLAB functions and the followingMATLAB toolboxes:

- Signal processing toolboxControl system toolbox- Robust: control toolbox

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3

Menu Description

Before starting the 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 Controldesign toolbox (MHC) is menu driven to ensure easy input of variables, controller calculationand analysis of the results. All menus of the toolbox will be described briefly and thecontroller design for a floating platform will be used as an example. Any of the menu optionscan be selected by typing the correct number and pressing ENTER. The previous menu willappear again by pressing just ENTER. Every menu is provided with a help screen (menuoption 0) describing briefly the several menu options.Startup: MIICTo start the controllerdesign procedure, executeMHC from insideMATLAB. The mainmenu, which is depictedin Fig. 3.1 will appear onthe screen.

Cont inuous- t ime M1HO H - i n f i n i t y Con t ro l DesignMain menu

1) St ruc ture i n i t i a l i z a t i o n2) Inpu t matr ix fu n c t io n s P I, P2 , v , o r W3) Co n t r o l l e r d es ig n4) System e v a l u a t i o n5) Options6) Disk fu n c t io n s0) He l p

Please e n t e r menu o p t io n o r p r e s s ENTER t o E x i t

Fig. 3.1 : Main menu.13

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14

Fig. 3.2 depicts the helpscreen of the main menu.

3.1 Optil,ns

Before starting the actualcontroller design, severaloptions must be definedin the options menu. Thedefault menu is shown inFig. 3.3.

Menu Description

He!p Main Menu H-inf in i ty Co n tro l DesignTh is menu s t ruc tured H- inf in i ty cont ro l des ign package can be used toen su re easy d e f in i t i o n o f th e d es ig n conf igura t ion . input of v a r i ab l e s .co n t ro l l e r ca lcu la t ion and ana lys is of th e r e s u l t s by comput ing t imeand frequency responses .Struc ture I n i t i a l i z a t io n Defines th e d es ig n conf igura t ion inc ludingproces s an d weighting f i l t e r s as a s t andard problem. Th e s t ructure isf ixed by def in ing th e dimens ions and tw o i n t e rconnec t ion mat r i ces .Input Matr ix Functions v, P and W : Def in i t ion o f proces s and weig h t in gf i l t e r s v ia 61S0 t r an s f e r func t ions o r HIMO s t a t e - s p a c e mat r i ces .Contro l le r Design : H-inf in i ty cont ro l des ign pa ramet r i z ing a l ls t ab i l i z i n g c o n t r o l l e r s such t h a t a s p e c i f i e d c losed- loop t r an s f e rfunc t ion has H-inf in i ty norm l e ss than a g iv en s ca l a r . Th is i nvolvesth e s o lu t i o n to tw o a lgebra ic Ricca t i equa t ions , each with th e sameorder a s the system, an d fur the r gives f eas ib le co n t ro l l e r s a lso wi tht h i s o r d e r .

Press any key t o cont inue

Fig. 3.2a : Help main menu.Help Main Menu H-inf in i ty Co n tro l Design

System EValuat ion : AnalySis of c losed- loop system by comput ing t imeand frequency responses .Options : Def in i t ion o f genera l op t ions for des ign package .Disk Fu n c t i o n s : Menu to lo ad and save v a r i a b l e s . WARNING a l l v a r i a b l e sa re i n i t i a l i z e d in a s t andard form when s t a r t i n g up th e design package.

Press any key to co n t in u e

Fig. 3.2b Help main menu (cont.).

Continuous-t ime MIMO H - in f in i t y Co n tro l DeSignOptions

1) Mode2) Tolerance margin fo r minimization procedure3) Selec ted input s ig n a l4) Generating META f i l e s5) Lower frequency bound6) Upper frequency bound7) Number of frequency p o in t s8) En d o f t ime i n t e rv a l0 ) Help

Se lec t menu opt ion or p r e s s ENTER to E x i t

Fig. 3.3 Options menu.

Continuousle-010No-2 rad2 rad501 sec

Change for the floating platform design example the following options into1) Mode Discre te

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3.2 Structure Initialization 15

2) Tolerance margin: 1 e - 65) Lower frequency bound -4 (10-4)7) Frequency points8) Sample time :

The help screen providesa brief explanation of theseveral options.

1000 .1

Help Options MenuMode : The H-inf inicy cont ro l des ign be done e i the r in thec o n t in u o u s t ime or d i s c r e t e t ime . Because th e a lg o r i th ms a re o n lyv a l i d for the cont inuous t ime problem, th e d i s c r e t e t ime i s SOlved byt ransforming th e d i s c r e t e augmented p la n t v i a b i l i n ea r t r ans format ionto the cont inuous t ime, c a l c u l a t i n g th e c o n t r o l l e r an d apply ing th einverse t r an s fo rmat io n t o ob ta in th e d i s c r e t e t ime c o n t r o l l e r .To le ra n c e Margin fo r Minimiza t ion ProcedUre : To so lve t h e c o n t r o ldesign problem p ro p er ly , it i s requi red t h a t th e cons t ruc ted augmentedp la n t has a minimal s t a t e - s p a c e r e a l i z a t i o n . This might n o t be th e casebecause o f p o le - ze ro c a n c e l l a t i o n s in th e d es ig n f i l t e r s . When no min imalr e a l i z a t i o n has been o b ta in ed , t h i s t o l e rance margin i s u se d t o d e t e c tan d e l imina te th e u n o b serv ab le /u n co n t ro l lab le modes.Se lec ted Input Sig n a l , Th e var iab le name o f th e in p u t s i g n a l i s d e f i n ewhich i s u se d fo r th e t ime s imu la t io n o f th e c lo sed - lo o p system.Whenever th e v a r i a b l e name does n o t ex i s t in th e workspace, a t imes i g n a l must be def ined by th e u s e r .

Press an y key to co n t in u e

Fig. 3.4a Help options menu.Help Options Menu

G e n e ra t in g META F i l e s By s e l e c t i n g t h i s o p t io n , a f i lename w i l l ber eq u es ted a f t e r every p lo t , to save th e p l o t as a META f i l e fo r l a t e rp ro cess in g u s in g GPP.Lower/Upper Frequency Bound an d Number o f Frequency Po in ts : Theseo p t io n s d ef in e th e f requency range for the magni tude p l o t s .Time : For th e cont inuous t ime t h i s def ines th e l ength of the t imes imu la t io n w h i l e for the d i s c r e t e t ime th e sample t ime i s d e f i n e d .

Press an y ke y to co n t in u e

Fig. 3.4b Help options menu (cont.).

3.2 Structure Initialization

The structure initializationmenu is depicted in Fig.3.5. The general structureas well as 1M 1 & 1M2can be changed.

Discre te - t ime MIMO H-inf in i ty Co n tro l DesignS t r u c t u r e i n i t i a l i z a t i o n

1) Gener a l s t r u c tu r e i n i t i a l i z a t i o n2) Change in te r co n n ec t io n mat r ix IM l3) Change i n t e rconnec t ion m a t r i x 1M20) Help

Please e n t e r menu o p t io n o r p r e s s ENTER to E x i t

Fig. 3.5 Structure initialization menu.

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16

The generalized plant isdescribed by defining the11 signal dimensions ofthe blocks within thebasic structure (Fig. 3.6).For the floating platformthe transformation of thecontrol configuration intothis structure is shown inSection 2.1. The first stepin the design is thedefenition of the dimen-SlOns.

Menu Description

- - - Struc ture I n i t i a l i z a t io n - - -+-------------------------------------------------+I II + - - - + n1 + - - - + + - - - + n2 + - - - + I+->1 I 1--/->1 I I 1--/->1 P21->+

+- - -+ I I nJ +- - -+ n4 I I +- - -+I - - / -> I PI1- - I -> I Iw n5 +- - -+ n6 I I I +- - -+ I I I n7 +- - -+ n S " z-----"--1->1 V 1--/->1 M I I M 1--/->1 w 1--/--*----> I I n9 I 2 I 1 ' - - -+ "1--/->1 I 1---->1 Iu " + - - - + n1 0 I I I I n I l +- - -+ " Y-----"---->1 I 1--/-> --/->1 I 1-----*---->+ - - - + + - - - + + - - - + + - - - +

MIMO Augmented Plant............................................Define th e dimens ions of th e MIMO augmented p la n t n1 .. . n I l as shownabove in MATLAB nota t ion ( - I n1 n2 n3 n4 n5 n6 n7 n8 n9 n1 0 nIl 1 I .Dimensions :

Fig. 3.6 General structure initialization.

Dimensions: [ 0 0 2 1 2 2 2 2 3 121

The dimensions of theinterconnection matricesare fixed now and can bedefined row by row. Afterselecting a row number,this row can be definedas an array in matlabnotation. Fig. 3.7 depictsthe screen for 1M 1.

Interconnection matrices :

- - - B ui ld i n t e rconnec t ion mat r ix IH lActual mat r ix elements of 1Ml :

1o1ooooo1o

o1oo1Enter ro w number ( 1 - - 4 ) o r p r e s s ENTER to E x i t : 3Enter elements ro w 3 in MATLAB nota t ion : [ 1 0 0 J

Fig. 3.7 : Interconnection matrix IMl

1 0 0 1 0 1 00 0 1 0 0 0 1:I:M1 = 1 0 0 I :I:M2 = 1 0 1 00 1 0 0 1 0 00 0 1

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3.3 Input Matrix Functions

Fig. 3.8 shows the helpscreen of the structureinitialization menu.

17

Help Struc ture I n i t i 4 1 i ~ a t i o n MenuH-inf in i ty s t andard ~ r o b l e m :

+-------------------------------------------------+I + - - -+ n1 + - - - + + - - - + n2 + - - - + I+->1 I 1--1->1 1 1--I->.I_p_2_.I--++ - - -+ I n3 + - - - + n41 1--1->1 Pl l - - / -> [ 1w nS +---+ n6 I +- - -+ I I n7 +- - -+ n8 z-----*--1--1 V 1--/->1Mil 1--1->1 W 1--1------->.. +---+ I 1 n9 +- - -+ 2 +-- -+ *1 1--1->1' 1---->1 1 u +---+ n1 0 I +- - -+ I I n I l +---+ y-----:-----1-=_1-- 1 ->1 ___1 1---1-- /->1-=-1-----:---->

MIMO Augmented Plant* . ~ * . * * * * * * .*

Press an y key to co n t in u e

Fig. 3.8a Help structure initialization menu.Help S t r u c t u r e I n i t i a l i z a t io n Menu

Before des iQning an H-inf in i ty cont ro l l er , th e cont ro l se tup has to bedefined toge the r with th e design c o n s t r a in t s an d objec t ives . ( e . g .t r ack ing , dis turbance re jec t ion , i n p u t s a t u r ~ t i o n e t c . ) This des ignconf igura t ion must be t ranSformed in to the H-inf in i ty s t andardproblem with an augmented p la n t t h a t conta ins what usua l ly i s cal l edth e p la n t in a cont ro l problem plus a l l weighting func t ions .Th e augmented plan t cons is t s o f four bas ic b l o c k s : 1,2) PI&P2 -Process models, 3) V - Shaping th e di s turbance vec tor wIt) an d 4) W -Weighting th e e r ro r vec tor z I t ) . Fur the r these four b locks a re somehowconnected through t he in t e rconnec t ion mat r i ces IMl & 1M2 conta in ingonly the elements + / - l a n d 0 which corresponds wi th ~ d d i n a , s u b t r a c t i n ~ o r no connection .Every cont ro l conf igura t ion can now be desc r ibed in a s imple wa y b ydef in ing th e dimensions nl .. . n I l t oge the r wi th the s t ru c t u re o f th econf igura t ion using t he in t e rconnec t ion mat r i ces .

Press an y key to cont inue

Fig. 3.8b Help structure initialization menu (cont.).

3.3 Input Matrix FunctionsAfter initializing thestructure, the blocks PI ,P2, V and W must bedefined. This menu ISdepicted in Fig. 3.9. Theblocks can be definedeither as SISO transferfunctions per entry or asMIMOmatrices.

state-spaceAn additional

viewing option has beenincluded to verify themagnitude plots.

Discre te - t ime MIMO H-inf in i ty C o n tro l DesignInput Matr ix FUnctions PI , P2 , V or W

1) En ter PI a s SISO t r an s f e r func t ions2) En ter PI a s MIMO s ta te - space mat r i ces3) Show magnitude p l o t s o f PI4) En ter P2 as SISO t r an s f e r func t ions5) En ter P2 as MIMO s ta te - space mat r i ces6) Show magnitude p l o t s o f P2i ) En ter V as 5ISO t r an s f e r func t ions8) En ter v as MIMO s t a t e - s p a c e matrices9) Show magnitude p l o t s o f V

10 ) En ter W 5150 t r ~ n s f e r func t ions11) En ter W as MIMO s ta te - space mat r i ces12) Show magnitude p l o t s o f W0) Help

Please ente r menu opt ion or press ENTER to Exi t

Fig. 3.9 Input matrix functions PI , P2, V or W

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18

Fig. 3.10 shows the helpscreen for the inputmatrix functions menu.

Menu Description

Help Input Matr ix Functions PI , P2 , V o r WAs desc r ibed in th e Struc ture I n i t i a l i z a t io n menu. th e augmented p l a n to f t he s t andard problem c o n s i s t s o f th ree bas ic b lo ck s

1) P l2) POl3 ) V4) W

Th e p ro cess model (pa r t 1 ) .Th e p ro cess mOdel ( p a r t 2 ) .Shaping o f th e di s turbance vec tor wIt) .Weighing of th e e r r o r vec tor z I t ) .These four b locks ca n be ente red in to the des ign package e i t h e r a sSI$O t r an s f e r funct ions or as MIMO s ta te - space mat r i ces . For th e 5150case th e use r must def ine the f i l t e r s as numerator and denominatorpolynomials fo r every ent ry o f t he mat r ix func t ion . Fo r th e MIMO casethe A,B.C o r D matrices must be def ined . Th e number of i nput s andoutput s of these blocks depends o f course on th e dimens ions ente red inthe Struc ture I n i t i a l i z a t io n menu.

Press an y ke y to cont inue

Fig. 3.10a Help input matrix functions PI, n, V or W.Help Input Matr ix Functions P l. P2 . V o r W

When ente r ing th e f i l t e r s as SISO t r an s f e r func t ions . t he co r r esp o n d in gs ta te - space r epresen ta t ion i s der ived automat i ca l ly and vice versa .A consequence of t h i s r epresen ta t ion of the blocks P l. P2 . V and W int r an s f e r func t ion m at r i ces and s ta te - space mat r i ces i s t h a t a l l th eproces s and des ign blocks must be proper .In addi t ion th e magni tude o f th e designed f i l t e r s can be plo t t ed .REMARK : To s impl i fy th e H - in f in i t y cont ro l des ign it i s recommendedt o de f ine the shaping an d weight ing blocks, V an d W r e spec t ive ly , a ssq u are func t ions (equal number o f i nput s and outputs) wi th elementsonly on t he d iagona l .

Press any ke y to co n t in u e

Fig. 3.10b Help input matrix functions PI, n, V or W (cont.).

When defining a block asSISO transfer function,the correct eit:ment of thematrix must be selectedfirst.

Every transfer function isdefined by it.s numeratorand denominator. Bothpolynomials can beentered as arrays InMATLAB nDtation. Thepolynomials must bedefined in powers of 'z' or's' for the discrete orcontinuous time respectively

Discre te Trans fe r Matr ix PIEn ter i n p u t number 1 I ) or press ENTER to Exi t ;En ter o u tp u t number ( I - - 1 )

Fig. 3.11 Selecting element of transfer matrix PI.

Define Discre te Trans fe r Function of PI ( I I )Please de f ine the fo llowing polynomials in MATLAB nota t ion .Ex amp le ; z"3 2z"2 - 3z + I H [ 1 2 -3 I 1press ENTER if a polynomial should no t be c h a n g e d ! ! ! ! !Old numerator , 0 INew numeratorOld denominator , 1 INew denominator

Fig. 3.12 Defining SISO transfer function.

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3.3 Input Matrix Functions

The menu to define thestate-space matrices A, B,C & D is depicted in Fig.3.13. These matrices canbe entered exactly thesame way as theinterconnection matricesIMI & 1M2 (Fig. 3.7).

19

Define Disc re t e s t a t e - s p ace mat r i ces of P l - - -1) Show / change s t a t e matrix A2) Show / change i npu t ma t r ix B3) Show I c h ~ g e output matr ix C4) Show I Change f eed- thr ough mat r ix D5) Change s t a t e dimension0) Help

Please en t e r menu opt ion o r p r e s s ENTER to EXit

Fig. 3.13 Defining MIMO state-space matrices.Help Define State-Space Matr ices Menu

The help screen to definestate-space systems isshown in Fig. 3.14.

Th e s t a t e - s p ace matr ices [A,a,C,D) c an be def ined /changed by s e l e c t i n gon e mat r ix an d then en te r ing the va lues ro w by row. The number o f i n p u tand outpu t s has been def ined in th e Structu re I n i t i a l i z a t i o n menu. Onlyth e number o f s t a t e s can be def ined /changed .WARNING When changing th e number o f s t a t e s . a l l mat r i ces a rei n i t i a l i z e d to zero mat r i ces and prev ious ly en te red

in f or mat ion w i l l be l o s t 1! 111Pr es s any key to cont inue

Fig. 3.14 Help define state-space matrices menu.

Before describing the actual control design, the following block information for thefloating platform should be entered using the input menus for transfer functions and statespace matrices (Fig. 3.11, 3.12 & 3.13). More detailed information about the modeling,identification and filter design can be found in Bouwels, J.P.H.M (1991) and Damen,A.A.H. et. al. (1994).

Process:C p1

Design filters:

.[: 0 .1 01 0 .1- 3 . 7 9 4 2 1 5 0 . 4 0 4 4==

[ 1 0 . 1 0 ]Bp, = Cp = [ ]

- 0 . 3 1 0 4 (z'-0.2z)z' - 1 . 9 9 4 4 z + 0 . 9 9 5 0

1 [ ..'" 0 . 0 0 4 5 1p l = 0 . 0 3 1 1 - 0 . 0 0 0 7- 0 . 1 2 1 3 0 . 1 5 2 9Dpl = [ 0 0 ]Dp,=[O]

( z + 0 . 4 1 7 2 ) ( z - 0 . 5 7 2 7 - 0 . 5 7 2 5 1 . ) ( z - 0 . 5 7 2 7 +0 . 57251 . )60 ( z 1 ) (z 0 . 7 0 2 2 0 .5 3 9 2 1 . ) ( z 0 . 7 0 2 2 + 0 . 5 3 9 2 1 . )0 .1 (z-0.8-0.011.) ( z - 0 . 8 + 0 . 0 1 1 . )

( z 0 . 9 9 9 0 . 0 0 1 1 . ) ( z - 0 . 9 9 9 + 0 . 0 0 1 1 . )= 4 0 ( z - 0 . 9 9 5 - 0 . 0 0 9 9 1 . ) ( z - 0 . 9 9 5 + 0 . 0 0 9 9 1 . )

(z+0.9)'

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20 Menu Description

The numerator and denominator polynomials can be defined In several ways in the transferfunction entry of Fig. 3.12. For example:

Numerator Vo' - 0 . 3 1 0 4 * [ 1 -0.2 0 ]

Denominator Vv 6 0 * p o 1 y ( [1; 0 . 7 0 2 2 + 0 . 5 3 9 2 *i; 0 . 7 0 2 2 - 0 . 5 3 9 2 *i] )

Note that MA TLAB commands can be used as well to define the transfer functions.

3.4 Controller Design

Once the completeplant haseneralized

defined,een thecontroller design becomesfairly simple.

The help screen for thecontroller design menu isdepicted in Fig. 3.16.

Disc re t e - t ime MIMO H - i n f i n i t y Cont r o l DesignCont ro l l e r des ign

1) H - i n f i n i t y c o n t r O l l e r op t ions2) Calcu la t e H - i n f i n i t y co n t r o l l e r3) Controller reduction4) Load o r i g i n a l co n t r o l l e r0) Help

Please en t e r menu o p t i o n o r p r e s s ENTER to Exi t

Fig. 3.15 Controller design.

Help C o n t r o l l e r Design MenuIn t h i s menu th e ac tua l co n t r o l l e r design i s performed. A ll s t a b i l i z ingco n t r o l l e r s such t h a t a s p ec i f i ed c losed- loop t r a n s f e r func t ion hasH - i n f i n i t y norm l e s s than a given s c a l a r . This ch a r ac t e r i za t i o n i nvo lveth e s o l u t i o n to tw o a lgebra i c Ricca t i e q u a t i o n s , each wi th th e sameo r d e r a s th e system, an d fu r the r g ives f ea s i b l e co n t r o l l e r s a l s o wi tbt h i s o r d e r .H - i n f i n i t y C o n t r o l l e r Options : In t h i s menu some c o n t r o l l e r r e l e v a n ~ s e t t i n g s a re d e f i n e d , l i k e type o f H - i n f i n i t y s o l u t i o n , type of R i cca t iequa t ion so lu t ion and a t o l e rance margin i n d i ca t i n g the accur acy o fth e c losed- loop H-inf in i ty norm with r e s p e c t t o the op t ima l gamma. Ina d d i t i o n an o p t i o n f or ba lanc ing o f th e augmented p l an t can be s e l ec t ed .This can improve th e n u m er i ca l s t a b i l i t y o f th e c o n t r o l l e r des ign . Alsoa reduc t ion t echnique can be s e l ec t ed to reduce th e o r d e r o f th eH - i n f i n i t y co n t r o l l e r .

Press an y key to c o n t i n u e

Fig. 3.16a Help controller design.

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3.4 Controller Design

The options described inSection 2.3 can bedefined in the controlleroptions menu shown inFig. 3.17.

21

Help Cont ro l l e r Design MenuCalcula te H- inf in i ty c o n t r o l l e r : A f t e r c he c k ing t he condi t ions for th eexi s t ence o f a s t ab i l i z i n g c o n t r o l l e r s a t i s f y in g an H - in f in i t y norm o fth e c lo sed - lo o p sys tem, l i k e whether th e sys tem i s s t ab i l i zab l e andd e t e c t a b l e , rank condi t ions to ensure t h a t th e c o n t r o l l e r s a r e proper ,an d no zero s on th e imaginary a x i s o f some t r a n s f e r s , a s t a r t va luean d s t ep s ize fo r gamma must be d ef in ed . Af te r der iv ing an i n t e r v a lco n ta in in g th e opt imal s o l u t i o n , th e near ly o p t imal gamma w i l l b e d er iv edusing bisec t ion search u n t i l th e requi red accu racy h as been obta ined .Cont ro l l e r Reduct ion : Because th e o rd er o f th e c o n t r o l l e r s w i l l be th esame as th e o rd er o f th e sys t em, c o n t r o l l e r r ed u c t io n i s o f t e n requi redto o b ta in lower o rd er c o n t r o l l e r s .Load Orig ina l Cont ro l l e r Th e c o n t r o l l e r r ed u c t io n s t e p can r e s u l t i nl e ss accu ra te c lo sed - lo o p performance , which makes it n e c e s s a r y tou se th e o r ig in a l h igh o rd er c o n t r o l l e r s .

Press any key to co n t in u e

Fig. 3.16b Help controller design (Cont.).

Discre te - t ime MIMO H-inf in i ty c o n t r o l Designc o n t r o l l e r OPtions

1) Type of H-inf in i ty approach2) Type of Ri c c a t i s o l u t i o n approach3) Tolerance margin O P t i m i ~ i n g gamma4) Balanc ing augmented p la n t5) Cont ro l l e r r ed u c t io n method0) Help

GOe ig en0.01NoMinre a l

Please e n t e r menu o p t io n o r p r e s s ENTER to E x i t

Fig. 3.17 Controller options menu

Change for the floating platform design example the following options into3)4)

Tolerance margin optimizing gammaBalancing augmented plant:

0 . 0 0 1Yes

The help screen for thecontroller options menu isdepicted in Fig. 3.18.

Help Cont ro l l e r Opt ions MenuType o f H - i n f i n i t y s o l u t i o n Sev era l rout ines a r e a v a i l a b l e whiChso lv e th e H-inf in i ty c o n t r o l problem i n d i f f e r en t ways acco rd in g toSafonov/Limebeer /Chiang l o o p - s h i f t i n g formulae , Glover/Doyle all-s o l u t i o n formulae o r Limebeer/Kasena l ly a l l - s o l u t i o n formulaeType of Ri c c a t i Equat ion Ap p ro ach : Th e ca lcu la t ion of th e H - in f in i t yinvolves th e so lu t ion to tw o a lgebra ic Ricca t i e q u a t i o n s . These tw oeq u a t io n s can be so lve d e i t h e r by e ig en v a lu e o r Schur de c ompos i t i on .The e ige nva lue decomposi t ion i s f a s t e r b u t th e Schur de c ompos i t i on i sn u mer ica l ly robus te r for ba d ly co n d i t io n ed design problems.Tolerance Margin O p t i m i ~ i n g Gamma Th e H - in f in i t y c o n t r o l l e r w i l l bede s igne d in such a wa y t h a t th e H - in f in i t y norm o f th e c losed- loopSystem w i l l be l e ss than gamma. Because th e o p t imal gamma can o n ly beapproximated, a to le r an ce margin must be d ef in ed i n d i c a t i n g when th ei t e r a t i v e design procedure o p t i m i ~ i n g gamma can be s topped.

P ress any key to co n t in u e

Fig. 3.18a : Help controller options.

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22

The following options areavailable to solve thecontrol design problembased on the "2-Riccati"equation approach.

For controller reductionthe options depicted 10Fig. 3.20 are available.

After selecting the controldesign options, the actualcontroller calculation isstarted (option 2 of hecontrollerIn this

design menu).case some

information will scrollover the screen describingthe minimal realization ofthe generalized plant,bilinear transformation,balancing and ...

Menu Description

Help Cont ro l l e r Options MenuBalancing Augmented Plant : Afte r bu i ld ing the augmented p la n t fromth e s e v e r a l b lo ck s , th e overa l l system mi g h t be badly condi t ioned .Balanc ing th e augmented p la n t fo r b ad ly co n d i t io n ed prob lems canimprove th e f i n a l so lu t ion .Cont ro l l er Reduction Method The orde r of the feas ib le con tro l le r sw i l l be th e sarne as th e order o f th e augmented p la n t conta in ing theproces s model and all sh ap in g /weig h t in g f i l t e r s . For more complex MIMOd es ig n prob lems th e o rd er w i l l i nc rease rap id ly and t h e r e f o r e c a n t r a l l ereduc t ion methods a re of ten requi red to ob ta in lower order co n t ro l l e r s(Minimal Sta te -space r e a l i z a t i o n , Optimal Hankel Reduct ion, SchurReduct ion an d Rela t ive Schur Reduct ion) .

Pre ss an y ke y to co n t in u e

Fig. 3.1Sb : Help controller options (Cont.).

Discre te - t ime MIMO H - in f in i t y Co n tro l DesignH - in f in i t y type approach

1) SLC (Safonov/Limebeer /Chiangl l o o p - s h i f t i n g fo rmulae2) GD (Glover /Doyle) a l l - s o l u t i o n fo rmulae3) LK (Limebeer /Kasenal ly ) a l l - s o l u t i o n fo rmulaePlease ente r menu o p t io n o r p r e s s ENTER to Exi t

Fig. 3.19 : Type of approach.Discre te - t ime MIMO H - in f in i t y Co n tro l Design

Co n t r o l l e r Reduct ion Methods1) Minimal s t a t e sp ace r ea l i za t i o n2) Optimal Hankel reduc t ion3) Schur reduc t ion4) Rela t ive Schur reduc t ion

Please ente r menu o p t i o n o r p r e s s ENTER to E x i t :

Fig. 3.20 : Reduction methods.

Prepare Augmented P l a n t

S ize s t a t e mat r ix : 12

Observabi l i ty index : 9

Co n t r o l l a b i l i t y index : 12

.. . Working on s ta te - space min imiza t io n .. . Please wai t

s t a t e s removed.

Working on i nverse b i l i n ea r t r ans format ion .. . please w a i t

Working on b a lan c in g augmented p la n t .. . Please w a i t

Fig. 3.21 Preparing generalized plant.

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3.4 Controller Design

checking the conditions tosolve the controlproblem:- G22 : (A,B2) stabilizableand (A,C2) detectable- 012 full column rank.- 02 1 full row rank.- G 12 : (A,B2,C1,012) fullcolumn rank.- G21 : (A,B 1,C 2,021) full

row rank.Only if a condition ISviolated an error messagewill appear.For y minimization a newstart value and step sizemust be defined :Start Value 1Step size: 1.1

The tolerance margindefined in the controlleroptions determines thenumber of y(Fig. 3.24).

iterationsFor an

optimal design, y shouldbe slightly smaller than I.The norm of theclosed-loop system can beused to verify y and thedifference should be ofthe same order as thetolerance margin.

- - - check so luc ion condi t ions

Tran s fe r G22 from th e cont ro l in p u t vec tor u( t ) to th eobserva t ion vec tor y( t ) i s s t ab i l i zab l e an d de tec tab le .

Trans fe r GI 2 from th e c o n t r o l input v e c t o r u ( t l t o the e r r o rvec tor z I t ) has fu l l column rank a t i n f in i ty .

Tran s fe r G21 from t he d i s turbance vec tor w It ) to th e observa t ionv e c t o r y ( t ) has f u l l ro w rank a t i n f i n i t y .

No t r an smiss io n ze ro s have been de tec ted in the t r an s f e r Gl 2from t he cont ro l input vec tor u( t ) t o the e r ro r v e c t o r z ( t ) .

No t r an smiss io n ze ro s have been de tec ted in th e t r an s f e r G21from th e di s turbance vec tor w(tl to the observa t ion vec tor y( t ) .

Fig. 3.22 : Check solution conditions.

Calcula te H - in f in i t y Co n t r o l l e rDIu v a lu e o f gamma : IPlease ente r new s t a r t v a lu e o f gamma or p r e s s ENTERDIu s tep s i ~ e : 2Please ente r new s tep s i ~ e o r pres s ENTER

Fig. 3.23 : Initializing y.

End o f i t e r a t i o n t e r Comment1 1 OOOOe+OOO Solu t ion OK2 9 090ge-001 NO s t a b i l i ~ i n g c o n t r o l l e r ! !!3 9 5455e-001 NO s t a b i l i ~ i n Q c o n t r o l l e r ! ! !4 9.7727e-001 so lu t ion OK5 9.6591e-001 so lu t ion OK6 9.6023e-001 NO s t a b i l i ~ i n g c o n t r o l l e r ! ! !7 9.6307e-001 No s t ab i l i z i n g c o n t r o l l e r !! !8 9. 644ge-001 Solu t ion OK9 9 6378e-001 No s t ab i l i z i n g c o n t r O l l e r " ,10 9 644ge-001 Solu t ion OK

Opt imal gamma' 9 .644ge-001Check H_inf in i ty norm 9.6458e-001Gamma i t e r a t i o n has been t e rmin a ted su cces fu 1 ly i i!

Press any key to continue

Fig. 3.24 Gamma iteration.

23

Whenever the Riccati equations are not solved properly (large residuals or other numericalproblems), the closed-loop system might not be stable although the y iteration has been

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24 Menu Description

terminated successfully. Redefining the design filters and reducing the constraints andobjectives can often help to overcome this problem. Because in general controllers are ofhigh order (order generalized plant), some controller reduction options (option 3 of thecontroller design menu) have been included to realize lower order controllers. I f his reductionresults in an unstable closed-loop system, the original high-order controller can be loadedagain without new calculations (option 4 of the controller design menu).

3.5 System Evaluation

Fig. 3.25 shows the menuto analyze the closed-loopbehavior by computingtime and frequencyresponses.

Help screen of the systemevaluation menu.

D i s c r e t e - t i m e MIMO H - i n f i n i t y C o n t r o l DesignSystem Ev alu a t io n

1) Show bode p lo t s c o n t r o l l e r2) Plo t c losed- loop t r an s f e r func t ion3) Show t ime s imula t ion0) Help

Please e n t e r menu o p t io n o r p r e s s ENTER to E x i t

Fig. 3.25 System evaluation menu.

Help System Ev alu a t io n MenuIn t h i s menu th e c lo sed - lo o p sys tem behaviour can be e v a l u a t e d bycomput ing t ime and f r e que nc y r e sponse sShow Bode Plot s C o n t r o l l e r : Shows th e magni tude p l o t o f th e c o n t r o l l e rfrom a s e l e c t e d o b serv a t io n s i g n a l y ( t ) to a s e l e c t e d in p u t s i g n a l u ( t )fo r th e f r e que nc y range d ef in ed in th e Opt ion menu.P l o t Close d - loop Tran s fe r Func t ion : Th e magni tude p l o t o f a s e l e c t e dc lo sed - lo o p t r a n s f e r fu n c t io n from th e d i s t u r b a n c e v e c t o r w It ) to th ee r r o r v e c t o r y ( t ) w i thou t sha p ing /w e igh t ing filters is shown fo r th es p e c i f i e d f requency range . Whenever th e sha p ing and w e igh t ing b lo ck so f th e augmented p l a n t a r e d ef in ed as d iag o n a l fu n c t io n s . th e m a gn i t udep l o t of th e in v er se sha p ing /w e igh t ing filter s c a l e d w i t h t h e H - in f in i t yc lo sed - lo o p norm gamma is p lo t t e d a s w e l l . This in v er se fu n c t io n d e f i n ean uppe r bound o f th e c o r re spond ing c lo sed - lo o p t r a n s f e r fu n c t io n o v erth e whole f r e que nc y range . This can be used to de te rmine whiCh t r a n s f e rfu n c t io n i s th e l i m i t i n g f a c t o r i n t h e c o n t r o l l e r de s ign an d in whichf requency range .

P re ss any key to co n t in u e

Fig. 3.26a : Help system evaluation.Help System Ev alu a t io n Menu

Show Time Simu la t ion : A c lo sed - lo o p t ime s i m u l a t i o n is performed u s i n gth e in p u t s i g n a l d ef in ed i n th e Opt ion menu. Th e s imu la ted o u tp u ts canbe p lo t t e d by s e l e c t i n g t h e r eq u i r ed o u tp u t .

p r es s an y key to co n t in u e

Fig. 3.26b Help system evaluation (Cont.).

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3.5 System Evaluation

The single transfers in theclosed-loop system areshown by selecting thecorresponding input andoutput.

- - - Show Discre te C losed - loop Bode p l o t s - -Please ente r number of input ( 1 - - 2 ) o r p r e s s Enter to E x i tPlease ente r number of ou tpu t ( 1 - - 2 1 :

Working . . Please wai t .. .Fig. 3.27 : Plot closed-loop transfer.

Only the magnitude plots for the closed-loop evaluation will be shown here.DiSC['de C I " ~ a J - l " u p Pi II ) Disatte Closed-loop P( 12)., 80

., ,,"- 60."20

..... 4

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26 Menu Description

name exists in the workspace (for example generated before starting up MHc) and the numberof columns correspond to the defined input dimensions, the time simulations can beperformed. However, if the variable name does not exist in the workspace, the input matrixmust 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 with3 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 tocreate a time simulation of 100 seconds.

Name input: d:l.sInput signal: [ [ z e r o s ( 1 0 0 . 1 ) ;-9. 81*s : l .n(2*p: l.*0 . 0 4* ( 0 . 0 . 1 . 7 5 ';

z e r o s ( 1 4 9 . 1 ) ] z e r o s ( 1 0 0 0 . 1 ) ]Name output: d : l . s_ou t

tliscrete Time Simula!ion of Closed-loop Symm0.01 ; - ~ - , - - - - - , - - - __ - - , - - - - - - ' , - ' - - - - _ ~ - - - - , (1.0080.006

::::: 0.0040.002

Of--- \ j . \ ' t " ~ ~ ~ - - L - ~ - + ~ ~ . -0.002L.OO4

TIl!IeI(s)

Fig. 3.32 : Simulation output signal.

Discrete Time Simulatiun of Clod-loop S)'lItem0.';-_-.:=::::;.:=:;=::::;:.=:;:::.::::::;:=:.-_----,0.2

0.1,;l 0I -0.1

~ . 2

Time I (. )Fig. 3.33 : Simulation control signal.

The corresponding time simulations ate depicted in Fig. 3.32 & 3.33. The relative baddisturbance r , ~ j e c t i o n in Fig. 3.32 after 10 and 85 sec. are caused by starting and stopping therotation of the crane. Note that compared to the designs described in Bouwels, l.P.H.M.(1991) and Damen, A.A.H. et. al. (1994) the scaling of the filters has been adjusted such that

-norm y becomes smaller than I .

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3.6 Disk Functions

3.6 Disk FunctionsAll information can besaved and loaded usingthe disk options menu(Fig. 3.34). Informationshould be stored regularlyduring the design becauseMATLAB errors due tonumerical problems canterminate Mlle.

The help screen for thedisk options menu ISdepicted in Fig. 3.35.

27

Discre te - t ime MIMO H - in f in i t y Co n tro l DesignDisk Options

1) Save a l l var iab les2) Load a l l var iab les3) Contents disk4) Change d i r e c t o r y0) Help

Please e n t e r menu o p t io n o r p r e s s ENTER to E x i t

Fig. 3.34 : Disk options menu.

Help Disk Options MenuThi s menU should be used to SAVE and LOAD v a r i a b l e s c o r r e c t l y . To av o idextens ive ly checking o f v a r i a b l e s for ex i s t ence . dimens ions e t c . in a l lfu n c t io n s , t h i s menu has been in c lu d ed . When s t a r t i n g up th e d es ig npackage, a l l var iab le s a re i n i t i a l i z e d in a s t andard way. There forev a r i a b l e s can be e n t e r e d e i t h e r d i r ec t l y through th e menus o r by 10a d inthem from the workspace. Th e ava i lab le opt ions a re

Save All Var iab les .Load A ll Var iab les .Contents Disk .Change Direc to ry .

P res s any key to co n t in u e

Fig. 3.35 Help disk options.

The most important features of the control design toolbox have been described togetherwith the menus which will appear on the computer screen. The exact screen input has notbeen described because the control design is rather straightfOlward and the required user inputis fairly simple. The example of the floating platform should be sufficient to guide the userthrough all menus of the design procedure. It is not the intention to show with this examplea complete control design procedure for all shaping and weighting filters. This isdescribed in more detail in Bouwels, l.P.H.M. (1991).

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28 Menu Description

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4Conclusions

Any control configuration can be rewritten in the presented basic structure which isautomatically transformed into a standard control problem. The menu driven structure ofthe toolbox makes the necessarily iterative design procedure fast, due to easy input ofvariables and simple analysis of the results by calculating time and frequency responses.The control design of a laboratory process has been used to show the user how to definethe basic control structure. An extensive description of all menus and help-facilities shouldguide the user through the design and explain all options.

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30 Conclusions

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AMenu Overview

The menus of the multivariable control design (MHC) toolbox are presented in onescheme to provide an overview of the most important functions. This overview can be usedas quick reference guide by the user during the control design.

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-

-

Discre te - t ime MIMO H-inf in i ty Control Design - - -Structure i n i t i a l i z a t i o n

1) General s t ruc ture in i t ia l iza t ion2) Change i n t e rconnec t ion mat r ix IMl3) Change i n t e rconnec t ion mat r ix 1M201 Help

please ente r menu OPtion or pre ss ENTER to Exi t

1

- - Discre te - t ime MIMO H-inf in i ty Control Design - - -Input Matrix. Funct ions P1. P2 , V o r W

1) Enter P1 as SISO t rans fer func t ions -21 En t er P1 as KIKO s ta te - space mat r i ces3J Show magni tude p l o t s of P141 En t er P2 as SISO t rans fer func t ions5) En t er P2 ss MIMO s ta te - space mat r i ces6) Show magnitude p l o t s o f p27) En t er V as SISO t r ansfe r func t ionsB) En t er V s . MIMO state-spacl II mat r i ces" Show magnitude plo t s of V10) En t er w as SISO t rans fer func t ions11 ) En t er w a s MIKO s ta te - space mat r i ces12) Show magnitude plo t s o f W0) Help

Please ente r menu opt ion or pre ss ENTER to Elcit ,

6- - Discre te - t ime MIKO H-inf in i ty Control Design - - - -

Disk Opt ions1) Save a ll var iab le s2) Load a ll var iab le s3J Contents disk41 Change direc tory0) Help

Please ente r menu opt ion or pre ss ENTER to Exi t ,

Discre te - t ime MIMO H-inf in i ty C ont ro l Design - - -System Evalua t ion

1) Show bode p l o t s co n t r o l l e r2) Plot c losed- loop t r an s f e r function3) Show time s imula t ion0) Help

Please ente r menu opt ion o r press ENTER to Exi t

4

- - Discre te - t ime MIMO H-inf in i ty C ont ro l Design - - -Main menu

1) Struc ture i n i t i a l i z a t i o n21 Input matrix. func t ions P1 , P2 , V, or W3J Contro l le r des ign41 System eva lUa t ion5) Options6) Disk func t ions0) Help

Please ente r menu opt ion or pre ss ITER to Elcit ,

3- - Discre te - t ime MIMO H-inf in i ty C ont ro l Design - - -Contro l le r des ign

1) H - i n f i n i t y co n t r o l l e r opt ions2) Calcula te H-inf in i ty cont ro l l er3J Contro l le r r educ t ion4) Load orig inal cont ro l l er0) Help

Please ente r menu opt ion or pre ss ENTER to Exi t ,

MenusMultivariableHoo .... . . -. .(;onrrOI ueslgnDisc re te - t ime MIMO H-inf in i ty C ont rol Design _ _ _

Options1) Mode2) Tole rance margin fo r minimiZation procedure3) Se lec ted input s igna l4) Genera t ing META f i l e s5) Lower frequency bound6) Upper frequency bound7) Number o f frequency poin ts8) End of t ime in terval0) Help

Se lec t menu opt ion or press ENTER to EXit :

Disc re te1 e -0 6No-4 ra d3 .1 4 ra d10 00. 1 se c

Disc re te - t ime MIMO H - i n f i n i t y C ont rol Design - - -Contro l le r op t ions

1) Type o f H-inf in i ty approach2) Type of R icca t i Solu t ion approach3) Tolerance margin opt imiz ing gamma4) Balancing augmented plan t5) Contro l le r reduction method0) Help

GOe igen0 . 0 0 1YesMinrea1

Please e n t e r menu opt ion or p r e s s ENTER to Exi t

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B

Program Structure

The global program structure including all rnhc-functions is depicted in Fig. B.l. Note thatthe standard MATLAB functions (including the toolboxes) which are used in the multivariable

control design toolbox are not mentioned in the overview. The required toolboxes aredescribed in Section 2.5.

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34 Program Structure

mhe.m mhc_hm.rnmhc_stm.m 1 he_hl.mmhc_st in.mmhc_im.mmhc_imf.m mhc_h2.m

mhc_rtf.m 1 mhc_dtf.mmhc_tfss.m 1 rnhc_c2o.mmhc_mss.mmhc_ssr.m

1hc_h21.m

robe_cm.mmhc-pzc.m

mhc_sbp.m mhc_rneta.mrohc:_hcm.m mhc_h3.m

mhe_hco.m mhc_h31.mmhc_hcb.m mhC-"lap.m - mhe_Inss.m

mhc_rbal.rnmhe_esc.mrnhc_slrc.rn 1 mhc_are.mmhc_d2ss.mrnhc_kgjd.m 1 mhe_are.Inmhc_d2ss.mmhc_djnl .m 1 mhe_are.mmhc_d2ss.mrnhe_hin.m

mhe_crm.m mhe_ffiss.mmhc cel .m

mh::_sem.m1hc_h4.mmhc_sbp.m mhc_meta.mrnhc-pc1 . m mhc_meta.mmhc_sim.m m h c ~ e t a . m mhc_opt.m mhc_h5.mmhc_disk.m mhc_h6.m

Fig. B.I Global program structure.

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cFunction Description

A brief description of all functions (in alphabetical order) presented in the overview ofAppendix A will be given.

Function name

mhc.mDescription

Initialization script file showing the main menu.Computes the algebraic Riccati equation for the control problemState-space transformation from controller canonical form to observercanonical form.Calculates the closed-loop system consisting of the process blocks PI& P2 and the controller.

Function to change rows in a matrix.35

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36

Function name

Function Description

Description

Computing controller reduction according to several methods.Checks the conditions to solve the control problem.Disk options menu.

all solution formulae derived by Limebeer and Kasenally.

Define transfer function.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.

controller basic function which prepares the variables for thegeneral MIMO configuration and minimizes y to calculate the optimalcontroller.Script file to generate control menu.Shows controller options menu.

Routine to calculate -norm of a state-space system which is themaximum over all frequencies of the maximum singular value.

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Function name

37

Description

Help screen for main menu.Function to build interconnection matrices.Script file to generate input matrix functions menu.

all solution formulae derived by Glover and Doyle.

Function to construct minimal realization of the augmented plant forthe basic structure.This function file generates a meta file using a filename defined bythe 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 transfer function. If the design filters Vand W are diagonal matrices the inverse design function is alsoplotted.Function to check pole-zero cancellations in SISO transfer functions.Returns the LQG or Riccati balanced state-space representation ofstable 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.

loop-shifting formulae derived by Safonov, Limebeer and Chiang.

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38

Function name

mhc_stm.mmhc_tfss.m

Function Description

Description

Routine to show and define state-space representation of a system.Structure initialization function for control design.Script file to generate structure menu to define augmented plant.MIMO transfer function matrix to state-space conversion.

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D

List of Variables

In this appendix a list of variables in alphabetical order with a short description is givenwhich are used as input/output arguments of the MHC functions described in Appendix C.

VariablesAc, Bc, Cc, DcAcl,BcI,CcI,DcI

DescriptionState-space matrices of controller in continuous time domain.State-space matrices of closed-loop system without designfunctions.

Acon, Bcon, Ccon, Dcon State-space matrices of final controller (discrete/continuous time,high/low order depending on design options).

Acor, Bcor, Ccor, Dcor State-space matrices of original controller in continuous timedomain (backup of Ac, Bc, Cc, Dc if controller reduction fails).

Ap I, Bp I, Cp I, Dp I State-space representation of process block PI.39

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40

VariablesAp2, Bp2, Cp2, Dp2Av, Bv, Cv, DvAw, Bw, Cw, DwIM1,IM2

alphadimflag

List of Variables

DescriptionState-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 sizeDimension array for basic MHC structure.Information array about current status :1) Mode; 1 =Continuous, 2 =Discrete2) Configuration structure; 1 = Known, 0 = Unknown3) Process block PI4) Process block P2 ; 0 = Unknown, I = Transfer function5) Shaping block V ; 2 = State-space matrices6) Weighing block W7) Generating META files; 1 = Yes, 0 = No8) Valid controller design ; 1 =Yes, 0 = No9) Valid controller reduction ; 1 =Yes, 0 =No10) type approach :

1 = Safonov/Limebeer/Chiang loop-shifting formulae2 = GloverlDoyle all-solution formulae3 = Limebeer/Kasenally all-solution formulae11) Type of Riccati solution approach ; 1 = Eigen, 2 = Schur

12) Balancing augmented plant; I =Yes, 0 =No13) Controller reduction method:

1 =Minimal realization2 =Optimal Hankel method,3 = Schur reduction method4 = Relative Schur reduction

14) Type of controller reduction for method 2, 3 & 4 :1 =Variable order & Fixed error bound2 = Fixed order & Variable error bound

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Variablesfreq

gamtol

time

tol

41

DescriptionArray defining frequency information :1) Lower bound2) Upper bound3) Number of frequency pointsTolerance margin for r minimizationNumerator/denominator transfer matrix of process block PINumerator/denominator transfer matrix of process block P2Numerator/denominator transfer matrix of design block VNumerator/denominator transfer matrix of design block WTime information : End of time interval (continuous mode) orSample time (discrete mode)Tolerance margin for minimal state-space realization

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42 List of Variables

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References

Bouwels, J.P.H.M.Ontwikkeling en beproeving van verschillende regelaars (LQG, H.J voor de horizontaleafregeling 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. SteinbuchA fast algorithm to compute the -norm of a transfer function matrix.Systems & Control Letters, Vol. 14 (1990), p. 287-293.Chiang, R.C. and M.G. SafonovRobust control toolbox: User's guide.Natick, MA : The Mathworks, 1988.Damen, A.A.H. and H.M. Falkus, J.P.H.M. BouwelsModeling and control of a floating platform.To be published in IEEE Trans. on Aut. Control (1994).

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44

J. and B. Francis, A. TannenbaumFeedback control theory.New York : McMillan, 1990.

J.e. and K. Glover, P.P. Kbargonekar, B.A. FrancisState-space solutions to standard H2 and control problems.IEEE Trans. Aut. Control, Vol. 34 (1989), No.8, p. 831-847.Falkus, H.M. and A.A.H. Damen, J. BouwelsGeneral MIMO control design framework.

References

In : Proc. 31 51 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 control theory.Heidelberg : Springer, 1987.Lecture notes in control and information sciences, Vol. 88.Glover, K. and J.e. DoyleState-space formulae for all stabilizing controllers that satisfy an H ~ - n o r m bound andrelations to risk sensitivity.Systems & Control Letters, Vol. 11 (1988), p. 167-172.Grace, A. and A.J. Laub, J.N. Little, C. ThompsonControl system toolbox: User's guide.Natick, MA : The Mathworks, 1990.Iglesias, P.A. 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.

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Klompstra M. and T. van den Boom, A. DamenA 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.l.N. and E.M. K a s e n a l l ~ , 1. laimoukha, M.G. SafonovAll solutions to the four block general distance problem.

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In : Proc. 27 th IEEE Conf. on Decision and Control, Austin, Texas, December 7-9, 1992.New York: IEEE, 1992. P. 875-880.Maciejowski, 1.M.Multivariable feedback design.Wokingham, England: Addison Wesley, 1989.McFarlane, D.C. and K. GloverRobust controller design using normalized coprime plant descriptions.Heidelberg: Springer, 1990.Lecture notes in control and information sciences, Vol. 138.Morari, M. and E. ZaflriouRobust process control.Englewood Cliffs, N.J. : Prentice Hall, 1989.Munro, N. and C. Eng, R.S. McLeodMinimal realization of transfer function matrices using the system matrix.Proc. lEE, Vol. 118 (1971), No.9, p. 1298-1301.Safonov, M.G. and D.l.N. Limebeer, R.Y. ChiangSimplifying the H= theory via loop-shifting, matrix-pencil and descriptor concepts.Int. 1. Control, Vol. 50 (1989), No.6, p. 2467-2488.Stoorvogel, A.A. and J.H.A. LudlageThe discrete time minimum entropy H= control problem.Faculty of Mathematics and Computing Science,Eindhoven University of Technology (The Netherlands), 1993.Internal report.

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Dooren, P.M. vanThe generalizf:d 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.

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Eindhoven Universitv of TechnoloGY R8search RBoorts ISSN 0167-9708(,:den: TEUEJ)EFaculty of Electrical Engineering

i2561 Bach. A.C.P.M. ind A.A.H. DamenIDENTIFICATION fOR TH E CONTROL OF MIMO INDUSTRIAL PROCESSES.EUT Report 91--256. 1991. ISBN 90-6144-256-7

i2571 Maaat. P.J.I. de dnd H.G. ter Morsche. J.L.M. van den BroekASPATIAL RECONSTRUCTION TECHNIQUE APPLICABLE TO MICROWAVE RADIOMETRYEUT Report 92-E-257. 1992. ISBN 90-6144-257-5

il5S1 Vleeshouwers. J.N.DERIVATION OF AMODEL OF TH E EXCITER OF ABRUSHLESS SYNCHRONOUS MACHINE.EUT Report 92-F.-25S. 1992. ISBN 90-6144-258-)

i2591 Orlov. V. B.DEFECT MOTION AS TH E ORIGIN OF TH E IIF CONDUCTANCE NOISE IN SOLIDS.EUT Report 92--259. 1992. ISBN 90-6144-259-1

(2601 Rool)aciers J.E.ALGORITHMS FOR SPEECH CODING SYSTEMS BASED ON LINEAR PREDICTION.EliT Report 92-E-260. 1992. ISBN 90-6144-260-5

;2611 BOOD. T.J.J. van den and A.A.H. Oamen. Martin KlomnstraIDENTIFICATION FOR ROBUST CONTROL USING AN H-infinlty NORM.WT Report 92-E-261. 1992. ISBN 90-6144-261-3

(262; Groten. M. and W. van EttenLASER LINEWIDTH MEASUREMENT IN TH E PRESENCE OF m AND USING THE RECIRCUUTIH6 SELFHETERODYNE METHOD.EUT Report 92-E-262. 1992. ISBN 90-6144-262-1

(2631 Smolders. A.B.RIGOROUS ANALYSIS OF THICK MICROSTRIP ANTl:NNAS AND WIRE ANTENNAS EMBEDDED IN ASIJB5TRHE.EU T Renort 92-E-263. 1992 ISBN 9H144-26H

i2641 Frenks. U . and PH . CiuJtmans. M.J. van 6ilsTH E ADAPTIVE RESONANCE THEORY NETWORK: (Clustemq-I behmour In relatlOn WIth brmsteoauditory evoked potential patterns.EU T Report 92-E-264. 1992. ISBN 90-6144-264-8

i2651 Wellen. J.S. and F. Karouta, M.F.C. 5chemmann. E SBalbruaae. L.M.F. KaufmannMANUFACTURING AND CHARACTERIZATION OF GUS/ALGm MULTIPLE OUANTUMWELL RIDGE WAVEGUIDELASERS.EUT Report 92-E-265. 1992. loBN 90-6144-265-6

(2661 Clultmans. L.J.M.USING GENETIC ALGORITHMS FOR SCHEVULING DATA FLOW GRAPHSEU T Renort 92-E-266. 1992. ISBN 90-6144-266-4

(2671 Jomai L. and !P.H. van DlliAMETHOD FOR GENERAL SIMULTANEOUS FULL DECOMPOSITION (iF SEQUENTIAL MACHINESAI90rltnms and ImplementatIon.EU T Report 92-E-267. 1992. ISBN 90-6144-267-2

12681 Boon. H. van den and W. van Etten. W.H.C. de Krom. P. van Benneko. F. HUI)siens.~ N ~ ~ ~ : ~ ~ ~ . A;K ~ : D L ~ : ~ e ~ H A S E DIVERSITY T R A N g M ~ S S I O N SYSTEM.EU T Heport 9H-268. 1m. 15BN '10-6144-268-0

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O:indhoven University (,f Toechnoioov Research Reoorts ISSN 0167-9708Coden: TEUEDEFacul tv of E l e ~ t r l c a 1 Enaine.eYina

i2691 Putten. I.H.i. vaTl aerMUhTIDI5CiPLiNAiR SPECIFlCEREN EN ONTWERPEN VAN MICROELEKTRONICA IN PRODUKTEN iJn Dutchl.EIIT Report 93-E-269. 1993. ISBN 90-6144-26H

mOi Blois. RHj.PROGRIL. A dnauaoe for the deflnltion of orotoeol gfdlllllars.Wi Repnrt \3--270. 1993. ISBN 91i-6144-27i>-1

ii711 Sloks KH JCODE GENERATiON fGH THE AiiRiBiiiE EVALUATOR OF THE PROTOCOL ENGINE GRAMMAR PROCESSOR UNIT.Wi Renort 9;--271. 1993 ISBN 90-6144-271-[112721 Yan. Keplno ,nd U . van VeldhuIZen

fLUE GA S CLflNING BY PULSE CORON! STREAMER.WT Repon 9H-272 1993. ISbN 90-6144-272-912731 5 m o i d e r ~ . A ,.

FINITE STACKED MICR05TRIP ARRAYS WITH THICK SUBSTRATES.EliT Report 93-E-173. 1993. ISBN 90-(,144-173-7

1274; Boller!. M.U. and M.i. van HoutenON INSljLAR POWER SY5TEMS: Drawlng up an Inventory of phenomena and research possibilities.EUT Report 93--274. 1993. ISBN 90-6144-274-5

iii)) Demen. U j . mELECTROMAGNETIC COMPATiBILITY Part 5. installation and mltlgatlon gUidelines. sect JOn 3.cabling and wiringElIT Report 93-E-275. 1993. iSBN 90-6144-275-3

12761 Bollen. MH.JLITERATURE SEARCH FOR RELIABILITY DATA OF COMPONENTS IN ELECTRIC DISTRIBUTION NETWORKS.fUT ReDort 93--276. 1993. ISBN 90-6144-276-1

i2m Weiland. SieDA EHilvIORAL APPROACH TO BALANCED REPRESENTATIONS OF DYNAMICAL SYSTEMS.EUT Reoort 93-E-277. 1993 ISBN 90-6144-277-)12781 1;(Irshkov Yu A. ,nd V. i. VladimlrovLINE REVERSAL GAS fLOW iEMPERATURE mSUREMENTS: Evaluations of the optical arrangements torthe Instrument.

WT Reoort 9H-278 1993. ISBN 90-6144-278-81279) CrevqbtoTl. LL.M. and U Rutcrers. E,M. van VeldhUIZen

IN-SITU INVESiIGATION OF PULSED CORONA DISCHARGE.EliT Reoan 93-[-271 1993. ISBN 90-6144-279-&

12801 LI. H.Q and RP , SmeetsGAP-LENGTii DEPENDENT PHENOMENA OF HIGH-FREQUENCY VACUUM ARCS