Stabilization LFT Systems

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h c w d l n g e of the 3ovIConferenceon Decision and ControlBrighton, England December 1991 TI-7 - 10:40Stabilization of L37 Systems

Way M. Lu; Kemin Zhout and John C. Doyle*

AbstractThe problem of parametrizing all stabilizing controllers for general LFTsystems is studied. The LF T systems can be variously interpreted asmultidimensional systems or uncertain systems, and the controller isallowed to have the same dependence on the frequency/uncertaintystructure as plant. In the multidimensional systems case, this meansthe controller is allowed dynamic feedback, while the uncertain systemcase can be given a "gain scheduling" inter pretation. Both p and Q sta-bility are considered, altough the la tter is emphasized. In both cases,the output feedback problem is reduced by a separation argument totwo simpler problems, involving the du al problems of full information(FI) and full control (FC). In the case of Q stabi1ity;the FI and FCstabilizat ion problems can be characterized completely in terms of Lin-ear Matrix Inequalities (LMIs). I n the sta ndar d 1D system case withno uncertainty, the results in this paper reduce to the standard Youlaparametrization, although the development here appears to be muchsimpler, and does not require coprime factorizations.

1 IntroductionThe basic block diagram considered in this paper is

where G is the generalized plant with two sets of inputs: the exoge-nous inputs w and the control inputs U, nd two sets of outputs: th emeasured output s y and the regulated output s z. The control problemin this setup is to design feedback controller K such that the closedloop structure is stabilized in some sense and the signal z is specified(cf. Doyle et a l, 91; Glover and Doyle, 1989 and Zhou et al, 1990).

In this pa per we are only concerned with the basic stabilizationproblem, so we will generally ignore z and w and focus on G as anLFT on A with input u and output y:

The frequency/uncertainty stru ctur e A is in a set A E Cnxn. orconcreteness, we will assume th at

A = {diag [611q,, . JqP] : 6; E C }BA = {A E A @ (A) 5 1)

as in Doyle et al(1991). This notatio n is a direct generalization ofth e now standar d notat ion for state-sp ace realizations of transfer func-tions. One of the advantages of the use of LFTs with this notation isth at i t facilitates manipulation using state-space-like machinery. T huswe often refer to "state" and "state tra.nsformations" even when this

'Electrical Engineering 116-81,California Institute of Technology, Pasadena, C A'Department of Electrical and Computer Engineering, Louisiana Stat e Univer-

91125.sity, Baton Rouge, LA 70803.CH3076-7/91/0000-1239$01 OO0 1991 IEEE 1239

terminology does not, strictly speaking, apply, because the meaning isalways clear from context.

The aim of this paper is to apply the machinery from Doyle etal( 1989 and 1991) to p arametrize all stabilizing controllers for generalLFT systems. In this setting, both G and li are LFTs on A, which maybe interpret ed in a number of ways. One is to view the A as transformvariables in a multidimensional system. We then produce all stabilizingcontrollers for such systems. A more useful interpret ation, as in Doyleet al(1991), is to view one part of A , (eg. 61 = l / z ) as the transformvariable in an uncertain system with the remaining part of A viewedas norm-bounded perturbations. The results in this paper could thenbe given a "gain-scheduled" in terpretatio n, as the controllers dependon the same perturbations as does the plant. The terms frequency anduncertainty structures will be used interchangeably.

We will focus on Q-stabilization under th is general settin g becausethe condit ions for &-stability, stabilizability and det ectability can beelegantly characterized using LMIs. Except for these LMI results, allthe arguments and procedures can be applied to p-stabilization caseas well, although the set of controllers in the Q and p case and theirinterpretations would be different.

The construction of stabilizing controllers for th e output feedback(OF) problem is done via a sequence of special problems: full informa-tion (FI) problem, disturbance feedforward ( O F ) problem, full control(FC) problem and output estimation (OE) problem, together with aseparation argument . These special problems are also of importancein their own right. The paramete rization of all stabilizing controllerswas first introduced for th e one-dimensional case by Youla et al(197G)and extended to th e multi-dimensional case by Guiver and Bose(1985).Both used the coprime factorization technique. In this paper we con-sider the general LFT setting without the need t o introduce coprimefactorizations.

In section 2 some notions related to &-@)stability are definedand characterized. In section 3 we will examine the &-stabilizationsof different special problems, the O F problem is solved via separationarguments. In section 4 the O,-stabilizing controllers for OF problemwill be characterized.

A longer version of this pa per is available (Lu et al, 1991).2 LFT Systems, Stabilizability and De-

tect ability2.1 Linear Systems and StabilitiesConsider G(A) and A E A from above as well as th e commutativematrix set V of A defined as

2, = {D C"'" : DA = AD,det[D] # 0) .The "state" variable transformation I I' := TI is said to be ad-missible if the transformation matrix T E V. hen the corresponding"realization" is [ ++ [-%q-Y]Note that G (A) for A E A is unchanged by the transformation, as arethe s tability properties. Because we will make frequent use of this fact,

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we will stat e it as a theorem. The proof follows immediately from thedefinitions in Doyle e t al(1991)T h e o r e m 1 The p-stability and Q-stability of LFT sys tems ar e in -variant under admissible state variable tmnsformation s.

Another important structural property of LFT systems is ex-pressed in t he following theorem.T h e o r e m 2 Let Al and A2 be two system matrices with correspond-ing uncertainty structures A: and A2 respectively. The n the system[ $ "A,' ] with any compatible dimen sioned ma trix A12 is Q (o rp)-stable with respect to structure [ : ] i f and only i f A1 andA2 are Q (or p)-stable.

Note that in this theorem, the frequency structures A: and A2may or may not depend on each other. We can easily sketch the proofof this theorem. The stat e transformation matrix T = [ y ] i sadmissible. By conducting this transformati on, the transformed sys-tem matrix T A T - ' = dA'z tends to [ 2 i z ] as tendsto 0. By continuity, the stability of the resulting system is guaranteedby the stability of A I and A z . So A is stable since the admissiblesta te transformation does not change stability. This argument holdsfor both p and Q. This theorem also implies that a cascade system isQ(or p)-stable if and only if each subsystem is Q(or p)-stable.2.2 Stabilizability an d DetectabilityThe general stabilization problem is to design a (possibly dynami-cal) output feedback controller Zi(A) = [ with frequencystructure A0 such that the feedback system with G (A) is p-stableor &-stable with respect to the induced new frequency structureA N = [ lo].ere the structure A0 depends on A , that is,it is made up of possibly multiple copies of the elements ot A .

[ 0 Az ]

Now consider the following two special struct ures,

where the frequency structures in both cases are th e same as the onefor G ( A ) .Definit ion 1 The sys tem G (A) with frequency structure A is p -stabilizable(or Q-stabilizable) if there exists a dynamical controller forthe corresponding system GsF(A):

We give following definitions:

such that the closed loop system is p-stable (or Q-stable) with respectto the induced frequency structure.L e m m a 1 The given system is p-stabilizable(or Q-stabilizable), i.e.its corresponding system GS F ca n be p-stabilized(or Q-stabilized) bysome K ( A ) = I*] with frequency structure A0 related to Ai f and only if the augmented system of GsF(A)

We can characterize this property by following fact.

r A O I B 0 1

can be p-stabilited(or &-stabilized) by static feedback F =[ 2: 2; ] with respect to frequency structure A N =

2.3 C h a r a c t e r i z a t i o n s of Q-Stabi l izabi l i ty andAssume that rank(B) = p 5 n and rank(C) = q 5 n where p is thedimension of U and q is the dimension of y .Propos i t ion 1 Consider the system G ( A ) = [*]ith fw-quency structure A and mnk(B) = p < n. A s su m e B l E RnX("-p)is such that B'Bl = 0 an d [ B B A ] is invertible. Then there existsa static feedback F such that the closed loop system matrix A + B Fis &-stable with respect to frequency structure A if and only i f thereexists a matrix P E V with P = P > 0 such that

Q-Detectability

B i A P A ' B I - B i P B l < 0 .More over, if there exists P satisfied above requirem ents, then state feed-back matrix F = -( B* P-'B)-' B' P- IA &-stabilizes the given system.

The proof of this prop osition needs the following lemma:L e m m a 2 Consider the pair (A ,B ) E RnX" x RnXP and runk(B) =p < n. L et B l E RnX("-P) and BO RPxn are such that B f B = 0 and[ BO B l ] is unitary. T hen

inf a ( A + B F ) = a ( B 1 A )F'ERPX"and the infimum is attained by F = - (B, 'B)- 'B, 'A.Proof. Since U := [ B O B l ] is unitary, then

inf 8 ( A+ B F ) = inf a ( U * ( A+ B F ) )F E R P X n FERPXn

Moreover the infimum is attained by B,'A + B , + B F= 0 or F =UB,' B ) 1 B,' A .

Proof. of Propos i t ion 1 There exists a static feedback F suchthat the closed loop system matrix A + B F is Q-stable with respectto frequency structure A if and only if

1 > inf 8 ( D ( A+ B F ) D - ' ) = inf a ( D A D - ' +D B F D - ' )F,D F.D

where infimum is obtain ed over possible F E RPx" and D E V.check that V i V l = I and V i ( D B )= 0.Now take V i = ( B f ( D * D ) - ' B l ) - ? B f D - ' then it is easy to

Now by lemma 2, we have1 > inf U ( D ( A+ B F ) D - ' ) = inf a ( V T D A D - ' )F,D

or there exists a D E V such that( V i D A D - ' ) ( V i D A D - ' ) * < Z

Take P = ( D *D ) - ' then P E V and P = P' > 0 , hence we have(B;PB~)-?B;APA*B~(B;PB~)-?z < o

or B i A P A ' B l - B i P B l < 0 UUsing the above result we can easily get

T h e o r e m 3 G ( A ) = [%]s Q-stabilizable if and only if thereexist s a static feedback matrix F such that the closed loop system matrixA + B F is Q-stable with respect to the same frequency structure.We can also define detectability in the obvious (dual) way.

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Proof. If B is square and of full rank, then th e result is trivial. NOWwe only consider th e case where r a n k ( B ) = p < n .

Assume that the system can be &-stabilized by a dynamical con-troller I


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resulting interconnected system Gb,, then its realization is

with respect to the frequency structure 11. The resultingtransfer matrix is exactly GFI as claimed. Since A -BIG2 is &-stable,so the theorem follows.

0[

Remark 1 Note that, given a feedback matrix K D F or the DF struc-ture, then the corresponding FI controller can be obtained from (i)a s K D F Cz I 1 ; and now we return it to D F problem by (ii),then ~ ~ ( P D F ,D F [ c2 I 1 ) = K D F . w e also have the fact thatf i (P DF ,K F I ) [ c2 I ] = I ~ F I .Remark 2 This theorem shows that if A - B1C2 is &-stable, thenproblems F I and DF ape equivalent in the above sense.3. 3 FC and OE ProblemsF C ( 0 E ) problem is dual to th e FI( DF) case and has t he dual solvabilitycondition t o FI(F D) problem. The solutions to this kind of controlproblems can be obtained by first transposing GFC(GOE nd solvingthe corresponding FI(DF) problem then transposing it back.Consider the following F C and O E diagrams

we have the dual results to the ones in the last subsection:Theorem 5 ( i ) Given an O E feedback matrix K O E which &-stabilizes 0Estructure, then the corres nding &-stabilizing

FC controller can be chosen as [ F~oE ,hich makes~ ~ ( G o E ,~ o E ) ~ ~ ( G F c ,T ICOE).

(ii) Suppose that A - BzCl is &-stable. For any &-stabilizing FCcontroller K F C , a &-stabilizing O E controller can be taken as~ ~ ( P o E ,F C )with

P oE(A) = [ ~ ]Momover, ~ ( ( G F C ,~ F c ) ~ ~ ( G o E , ~ ~ ( P o E .~ F c ) ) .

Remark 3 Given controllers I ~ F C nd K O E or FC a nd O E s t ruc -tures, res pectively, we have the following facts:

Remark 4 W e can see that if A - B2Cl is Q-stable, then FC an d OEproblems are equivalent in the above sense.3.4Consider th e general OF problem. Wit hou t loss of generality, we fur-ther assume the realization of G (A) has D22 = 0.Let z denote the "state" of system G ( A ) . Since ( A , & ) is &-stabilizable, there is a constant m atrix F such tha t A+ B2 F is &-stable.Note that [ F 0 ] is actually a special FI stabilizing controller. Nowlet v = U - Fx, hen t he system can be broken into two subsystemsG I and Gtmnas shown below

OF Problem and Separation Property

where

is &-stable, and

From theorem 2, we know that K &-stabilizes G if and only if K&-stabilizes Gtmp. Now Gt,, has an OE structure, so let L be suchthat A+ LC2 is Q-stable then is a &-stabilizing controller for thecorresponding FC problem. Since A + B2 F is &-stable by construction,by Theorem 5 we have a controller given by K ( A ) = 3 4 J ,where

[:I [r A + B z F I O - B2 1 1

This proves the following theorem, which is stated without theassumption that D Z Z= 0.Theorem 0 Consid er the geneml OF problem. Le t F and L be suchthat A + LC 2 and A + B2F are &-stable, then the controller

A + B2 F + LC2 + L D zzF I - LF( A ) =with the frequency structure A &-stabilizes the given system.

We can also get the same result by the dual procedure to aboveconstruction.

Now we examine the separa tion property. We will denote the st at evariable of G ( A ) as x and state variable of resulting controller I i ( A )as E. We will take the state variable of the closed loop system asr 7z = , and the corresponding realization is

Now we conduct the admissible state transformation Z H T% =1 , then T = [ JI 1. After the transformation, the real-[ Iization isi.e. th e system is decoupled into two separated subsystems whichare &-stable with respect to the same frequency structure A by as-sumptio n. Hence the closed-loop system after admissible st at e vari-able transforma tion is also &-stable with respect to t he new frequencystructure A N = [ 1 by theorem 2, so is the original closed-loopsystem as desired.

The separation property is clear now, since the synthesis of O Fproblem can be reduced to F I and F C problems, i.e. the latt er twoproblems can be designed independentlv.

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4 The Characterization of All Q-StabilizingControllers For Output Feedback Problem s

4. 1 Problem Statement and MotivationsConsider again the standard system block diagram

with

Suppose ( A ,Bz ) is Q-stabilizable and (Cz,A ) is &-detectable, inGiven a plant G ( A ) , parametrize all controllers Zi(A) thatQ-stabilize G.

More specially, the characterization problem is to find a suitablematrix J ( A ) such that any admissible controller I i ( A ) for G ( A ) canbe expressed as an LFT with respect to a Q-stable parameter matrixQ ( A ) with coefficient matr ix J ( A ) , i.e. K = & ( J , Q ) .

Note that for a Q-stable parameter matrix Q ( A ) , it will be as-sumed that its realizations are & -stabilizable and Q -detectable.

In this section, we only consider the characterization of all ad-missible controlle rs for O F problems. For this purpose it is unneces-sary t o parametrize all admissible controllers for the special problems.Instead, we only parameterize a set of controller equivalence classeswhich generate the same control action. We say two controllers, Iian d K, are of equivalent control actions i f their corresponding closedloop tmnsfer matrices are identical, i.e. Fl(G,Ii ) = Fr(G, Ii) , writtenas A E A. The controller equivalence is an equivalence relation. Wewill see that, for different special problems, we have different refinedversions of this relation. We will also see that the characterizationsof equivalent classes of admissible controllers for special problems aregood enough to construct the characterization of all admissible con-trollers for O F problem.4. 2 Admissible Controllers for FI and FC ProblemsWe first examine the FZ stru ctur e as given in the last section. Wesay two controllers I ~ F Ind ~ F Ire equivalent if they produce thesame closed loop transfer function from w to U . Obviously, it is alsoguaranteed that ~ ~ ( G F I ,F I )= F ~ ( G F I , F I ) .

Since we have full information for feedback, our controller willhave th e following general form

this section we discuss the following problem:

IiFr(A) = [ Zii (A) I iz (A) ]with I i l (A) &-stabilizing[w]nd arbitrary Q-stable Ziz(A).Proposit ion 2 Le t F be a constant matrix such that A + B 2 F is Q-stable. The n all admissible controllers, in the sense of generating allQ-stabilizing control, for F I can be parametrized as

I < F Z(A) [ F Q ( A ) ]with an y Q-stable Q ( A ) .Proof. It is easy to see that the controller given in the above for-mula Q-stabilizes the system G F I (A) . Hence we only need to showthat the given set of controllers parameterize all Q-stabilizing con-trol action, U, .e., there is a choice of Q-stable Q ( A ) such that thetransfer functions from w to U for any stabilizing controller Iipr(A) =[ I i l (A) Ziz(A) ] and for I i F z (A)= [ F Q (A) ] are the same. Toshow that, make a change of control variable as v = U - Fs , wherez denotes the sta te of the system G F / ( A ) . Consider the system withthe controller Z ~ F ~ A ) .et Q ( A ) be the transfer matrix from w to

U ; it is Q-st able by the Q-stability of the closed loop system. Then0= F X U = FZ+ Qw , so I < F ~ ( A ) [ F Q ( A ) 1 .

Remark 5 In the above proof, we have used a standard technique ofchanging variables. A simi lar change of variables appeared in Doyle etal(1989).Now we consider the dual F C problem whose diagram is given

in the last section. We say controllers K F C and ~ F Cre equivalentin the sense that the same injection inputs YFCS p d u c e th e s am eoutputs 2s. This also guarantees the identity of their resulting closedloop transfer matrices from w to z . We also haveProposit ion 3 Le t L be a constant matrix such that A + LC2 is &-stable. The n the set of equivalent classes of all admissible controlle rsfo r FC in the above sense can be parameterized as

with any Q-stable Q ( A )4. 3The D F block diagram is given as in the last section. We will furtherassume that A - BlCz is Q-stable in this subsection. This will simplifythe solution, and can be relaxed. We say that the controllers K DF (A)and IibF(A) are equivalent for the D F problem i f the two transfermatrices from w to U in the above diagram are the same . Of coursethe r esulting two closed loop transfer matrices from w to z are identical.Remark 6 Since the equivalence between FZ and D F problems asdiscussed in the last section, it is easy to show that if I i D F 2 IibFin the D F structure, then I


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Jtmp can be o btained by using composition formula of LF T as

A + B z F + LCz + L D z z F-(cz + D z z F )J ( A )=

Hence F ~ ( J D F ,o) = Ft( tmprK D F )= K D F . This shows thatany admissible controller can be expressed in t he form of ~ ~ ( J D F ,o )for some &-stable Qo . 0

Since for any admissible X D F , we can get two parametrizationsby the above two formulae. We relate them asProposition 5 Le t K D F= Ft( D F ,Qo) for some &-stable Q o be ob-tained from the c onstructive proof of Proposition 4 and let I i b F =F ~ ( J D F , Q )KD F or some Q-s table Q be obtained from its corre-sponding F I structure. Then Q o = Q . Moreover I ~ D F K b F .

Now we turn to the O E problem, the system structure is given inthe last section. We will assume that A - B2C1 is &-stable. We havethe following resultProposition 6 All admissible controllers fo rO E problem can be char-acterized as Ft( OE , Q o ) with any Q-stable Qo, where J O E i s defined

- L B z + LDzz0

with L such that A + LC2 is &-stable.4. 4Consider th e O F problem given in section 4.1, we haveTheorem 7 Le t F an d L be such that A + LC2 and A + BZF areQ-stable, then all controllers &-stabilize G(A ) can be Characterized asF,( J ,Q ) with

All Admissible Controllers for Problem OF

and with any &-stable Q ( A ) uch that the structum is well-posed.Proof.Let x denote the stat e of system G. Since (A , B z ) is &-stabilizable,there is a constant matrix F such tha t A + B z F is &-stable. Note that[ F 0 ] is actually a special FI &-stabilizing controller. Now letv = U - F x as in the proof of Theorem 6, we have K ( A ) &-stabilizesG ( A ) f and only if it &-stabilizes

We will assume again D zz = 0 for simplicity.

Gtmp(A)= [ - F 0cz Dz iNow Gtmp(A)s of the OE structure, let L be such that A + LC2

is &-stable then by proposition 6 all controllers &-stabilizing Gtmp( )are given by

K ( A )= F e( J( A ) ,Q ( A ) )

- 1A + Bz:+ Lc2 I - L I 2 11 4 2 1 1 0 1

This concludes our proof. 0

This theorem shows that any admissible controller K ( A ) can becharacterized as an LFT of a Q-stable parameter matrix & ( A ) , .e.,K ( A )= Ft(J(A) ,Q(A)) . here is an alternative direct proof that thisparametrization produces all stabilizing controllers. To see this, recallfrom th e inversion formulas for LFT s in Doyle et al( 1991) that we cansolve the equation K ( A )= F t ( J ( A ) ,Q ( A ) )or Q ( A ) o give

Q = F " ( J - ~ ,f ) = 3 [ ( j , )where a little algebra shows that

J-' =[y]F I 0Note that Q is stable if and only if K stabilizes &. But j22 = Gzz,so Q is stab le if and only if Zf stabilizes G, as desired.

This last manipulation illustrates the power and simplicity of theLFT machinery. In contrast, the standard proof that the formulaK(A) = Fp(J(A),Q ( A ) ) ields all stabilizing controllers requires in-volved arguments using coprime factorizations. I t is also possible to ob-tain similar results using the cha in-scatter ing version of LFTs ( Kimura,private communication).5 Concluding RemarksWe have considered the problems of stabilization and t he paramet riza-tion of all stabilizing controllers for LFT systems. Although the focuswas on &-stability, most of the results are identical in the p-stabilitycase via simple change of notation. Th e exception is that the FI andOE solutions in the p case are not simply characterized by LMIs, andstabiliza tion by dynamic feedback is not equivalent t o stabilizat ion byconstant gain. The separation theory holds in the p case exactly as nth e & case, although the notation is a bit more cumbersome.

The separation proof in this paper holds in greater generality thanfor just the & and p stability problems. All that is required for theseparation proof is tha t t he notion of stability satisfy two requirements:1) stability invariance under a sufficiently rich set of similarity trans-formations, as in Theorem 1, and 2) a certain structural property asgiven in Theorem 2. It would clearly be possible to develop a moreabstract axiomatic stabilization theory using these 2 properties.61.

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ReferencesDoyle,J.C. (1984), Lecture Notes in Advances in MultivariableControl, ONR/Honeywell Workshop, Minneapolis, MN.Doyle ,J.C., K.Glover, P.Khargonekar an d B.Francis (1989), State-Space Solutions to Standard 71 2 and 71, Control Problems, IEEETrans., Vol.AC-34, pp.831 - 847.Doyle,J.C., A.Packard a nd K. Zhou(1991), Review of LFTs, LMIsand p , 1991 I E E E C D C , England.Glover,K. and J.C.Doyle (1989), A Sta te Space Approach t o U,Optimal Control, in Three Decades of Mathematical System The-ory (H.Nijmeijer, J.M.Schumacher Eds.), Springer-Verlag, Berlin.Guiver,J.P. and N.K.Bose (1985), Causal and Weakly Causal 2-D Filters with Applications in Stabilization, in MultidimensionalSys tems Theory(N.K.Bose ed.), D.Reide1 Publishing Co., Holland,1985.Lu,W.M., K.Zhou and J.C.Doyle(lggl) , On stabilization and Sta-bilizing Controller Characterization of LFT Systems, Report.Youla,D.C., H.A.Jabr and J.J.Bongiorno (1976), Modern Wiener-Hopf Design of Optimal Controllers -- The Multivariable Case,IEEE Trans., Vol.AC-21, pp.319 N 338.Zhou,K., J.C.Doyle, K.Glover and B.Bodenheimer (1990) , Mixed'HZand 1-1, Control, Proc. of 1990 ACC.

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