Analysis and Synthesis of Robust Control Systems via Parameter_dependent Lyapunov Functions

8/6/2019 Analysis and Synthesis of Robust Control Systems via Parameter_dependent Lyapunov Functions

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Analysis and Synthesis of Robust Control Systemsvia Parameter-Dependent Lyapunov Functions

Eric Feron, Pierre Apkarian, and Pascal Gahinet

Abstruct- In this paper, the problem of robust stability of systemssubject to parametric uncertainties is considered. Sufficient conditionsfor the existence of parameter-dependent Lyapunov functions are givenin terms of a criterion which is reminiscent of, but less conservative than ,Popovs stability criterion. An e quivalent frequency-domain criterion isdemonstrated. The relative sharpness of the proposed test and existingstability criteria is then discussed. The use of parameter-dependentLyapunov functions for robust controller synthesis is then considered.It is shown that the search for robustly stabilizing controllers may belimited to co ntrollers with the sam e order as the original plant. A possiblesynthesis procedure and a numerical example are then discussed.1. PRORI F M STATFMFNT A N D TNTRODlJCTlON

In this paper, we consider the linear system

where z : R+ + R The real numbers 61, . . , ~ are uncertain butconstant in time and satisfy 16, I 5 a, = 1. . . . L with N a givenpositive number.

We address the question of robust stability: Is ( I ) stable for allpossible values of the unknown parameters 51 . . . 6 ~ ?

This question virtually a rises in any problem of autom atic control,and its solution has been the focus of intensive research over the past50 years. In particular, a great deal of effort has been spent usingLyapunov stability theory for this purpose; the notion of quadraticstability, based on the search fo r a single quadratic Lyapunov functionof the form V ( . c )= .cPn: to prove stability of ( l ) , has encounteredconsiderable academic success (we refer the reader to [ 5 ] for adetailed bibliography). However, it has long been known that suchan approach often yields overly conservative results.

As a consequence, at tent ion also has focused on using Lyapunovfunctions which exp licitly depend on the unknown parameters 6,, =1 , . . .L . However, historical context has led researchers to considerquite a rigid format both for the type of uncertain systems underconsideration and for the family of parameter-dependent Lyapunovfunction used. This is the standard Popov framework, where (1 ) iswritten in the feedback form

p , = s,c,1; i = 1. . ,L. (2)In this format, the b,s and c,s are assumed to be column and rowvectors, respectively, thus illustrating the local nature of uncertainty inmost systems of interest. The correspondence of (2 ) with the originalsystem of (1) is then given by -4, = b r c z r i= l:...L.

Manuscript received December 2, 1994; revised July 19, 1995.E. Feron is with the Department of Aeronautics and Astronautics, Massa-chusetts Institute of Technology, Cambridge, MA 02139 USA.P. Apkarian is with the DCpartement dEtudes et Recherches en Automa-tique, Centre dEtudes et Recherches de Toulouse, 31055 Toulouse Cedex,France.P. Gahinet is with the Institut National de Recherche en Informatique et enAutomatique, Domaine de Voluceau, 78153 Le C hesnay Cedex, France.Publisher Item Identifier S 0018-9286(96)04366-8.

Under these conditions, the stability of this system is then investi-gated using Lurk-Postnikov Lyapunov functions of the form

/ I, \(3)

where the A t s are real numbers. Thus, the traditional view onparameter-dependent Lyapunov functions for uncertain systems isquite restrictive and implicitly assumes identical ranks betweenthe perturbations appearing in (2) and the corresponding Lyapunovfunction (3); rank 1 perturbations on the system yield rank1 perturbations on the associated parameter-dependent Lyapunovfunct ion. Note that Haddad and Bemstein [16] have proposed anextension to Popovs stability criterion when dealing with the casewhen e , an d b; are higher rank matrices. However, the basic linkbetween the perturbation rank in the system and the correspondingLyapunov function still remains.

The contribution of this note is to remark that such a viewpointis largely unjustified and that significant improvements may beobtained by considering more general parameter-dependent Lya-pun ov functions instead. Furthermore, a ll of the benefits attributed toPopovs control system analysis and synthesis may be extended to themore general class of parameter-dependent Lyapu nov functions underconsideration. In particular, recently developed convex optimizationtechniques remain applicable for the numerical implementation ofthe proposed criterion. Also, in the context of robust controllersynthesis, parameter-dependent Lyapunov functions allow one tobuild controllers with order less than or equal to the order ofthe plant. This fact, which trivially extends to the usual Popovcontroller synthesis scheme, seems to have been unnoticed before.A computationally less intensive approach to the same problem waspresented in [14]. However, it does not easily extend to controllersynthesis.

11. STABILITY TE ST VI A PARAMETER-DEPENDENTQUADRATICYAPUNOV UNCTIONS

A. The Search for Parameter-Depen dent Lyapunov FunctionsWe know from the work of Lyapunov [17], [6] that ( 1 ) IS stableif and only if for all values of 61, . . . . S L , there exists a symmetricmatrix function P(S1, . . 6 r , ) such that

. 7 P ( S 1 , . . , S L ) L > 0, z # 0 . l b z l 5 0 . i = l , . . Lan d

/

z#0.16t1 In. = l , . . . . LIn particular, if there exist symmetric matrices PO, I . . . ,PL suchthat

zT (PO+ g 6 + > 0, z # 0.16,1 5 n , i = 1,. . . L (4)an d

X f o . 1 6 t l ~ a , z = l . . . . . L ( 5 )then (1) is stable. The following lemma allows us to eliminate (4).

0018-9286/96$05.00 0 996 IEEE


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Lemma I . / : Assume that -40s stable and that ( 5 ) holds for somePO .PI.. . .PL . Then PO+ Et.=, , P, is positive-definite for allvalues of the 6, s between - 0 an d N . Hence it suffices to check

Proposition 1.1: System (1) i s stable whenever there exist sym-metric matrices PO. . S and a skew-symmetric matrix T where P, Sand T are of the form (12) such thatwhether ( 5 ) holds to prove the stability of (1 )The proof of this lemma follows similar statements made in [19]an d is therefore omitted. In the remainder of this paper, we will

+ "-" + cy2CTSC O B + - C2 ] < o[ B'P" + PCAO + T C B T C T P+ P CB - ss 0. (13)assume that A" is stable.B. Characterizing Uncertainly via QuadraticRelations and S -Procedure

Let us now introduce the variables P I . . . .p~ t R"

It is possible to rewrite ( 5 ) as

For reasons that will become clear later, we wish to characterize (6)using quadratic inequalities and equalities. We have the followinglemma.

Lemma 1.2: Let p E R" an d y E R". There exists 6 t R suchthat p = 6 q an d 101 5 CL if and only ifp r s p 5 n2q"sq,vs= s > 0 (8)

The proof of this lemma is similar to one found in [lo] and willtherefore not be repeated here.

Applying this lemma to (6 ) an d ( 7 ) ,we conclude that ( 1 ) is stableif there exist symmetric matrices PO. I. . .PL such that

pTTq = O.VT = -T'

whenever

.r # 0p j s z p c5 nLrTSp.r , vsz= si > Op i T,,,:= 0. V T z = -T:. 1 = l:.. . L .

The replacement of (6) by (8) is a technique which appears, forexample, in the works of Yakubovich [22], [12] The S-procedure isa method to check (9) (possibly conservatively): assume there existS, = ST > 0 and T, -T:,z = 1, . . L, such that

L-5 ,:stpt- az.c's,*.) + 2 - y p T T , * . < 0

2 = 1 i = l( N . p I : " ? p L ) # (O,O....O) (10)

Then, (9) is satisfied and (1) is stable. Proving th is fact is trivial an dmay b e found in [5], for example. For simplicity of notation, we nowintroduce the following matrices:B = [A , " ' A L ] . cT= [I " ' I ] (11)S = diagS,. T = diagT,, P = diagP,.

an dL L L

L = l ? = I i= 1 (12)We have the following proposition that states a sufficient conditionfor stability.

P r o o j It suffices to check that (13) i s the same as (10) . 0Note that the problem of finding PO, ,S: n d T atisfying (13) is

a finite-dimensional, convex feasibility problem similar to the onesdescribed in [SI. I t can be solved using, for instance, the LM I ControlToolbox [15]. Th e reader may remark that there is a certain amoun tof redundancy when using both scalings S an d T ; emoving the Tscalings from the above criterion may increase the conservatism ofthe overall criterion in some cases. On the other hand, the fewernumber of involved variables definitely accelerates computations.

111. A N EQUIVALENT F REQUENCY-DOMAIN CRJTERIONCOMPARISON WITH OTHER STABILJTY CRJTERlA

Inequality (13) can also be equivalently expressed in the frequencydomain using the results of Willems [20, Proposition 31. In thissection, we assume that the pair ( - 4 0 . B ) s controllable with B givenby (1 1) . The next corollary follows directly from the application of[20,Proposition 31 to (13). Let H be the transfer function given by

C o r o l l a q 2.1: System (1) is stable if there exist symmetric ma-trices P an d S , a skew-symmetric matrix T , and a real number tsuch that

H ( s ) = C ( s 1 - l o ) - l B .

S. an d T satisfy (12):S > 0, E > 0.V L E R. r \ ' H H ( i d ) * S H ( i w )- S +

x ( ( 2 . P - t T ) H ( i u )- H ( i w ) " ( ~ ~ l ' +T ) )+ 15 0. (14)The famil iar form of (14) suggests that i t may b e com pared withother stability criteria.

A. Compa rison with Popo v's Stability CriterionIn this section, we compare the newly developed criterion with

Popov's stability criterion, and we show that it always performsbetter. We consider (2),where the uncertainties 6, are assumed tosatisfy -n 5 , 5 a , = 1 ,. . ,L. Note this setting slightly differsf rom the usual one, where the 6,'s are assumed to lie between zeroand some upper bound k , fo r i = 1 , . . . , L [5], [16], [ 2 2 ] . t isreadily shown via loop-transformations [7] that both viewpoints areequivalent. Popov's stability criterion is based on the search for Lurequadratic Lyapunov functions of the for m of ( 3 ) ,where the symmetr icmatrix PO and the scalars XI, . . . XL are to be determined. Thus,Popov's stability criterion and the new stability criterion only differby the kind of parameter-dependent Lyapunov function sought.

Following th e sa me lines as in S ection 11-B, the condition ford V ( r r ) / d t to be negat ive-defini te can b e wri t ten asS' ( i l " + B,AC,)T(Po + C,T:lAC,)n: < 0 vz # 0VA = diag(6,) with 15,1 5 y, i = 1 , . . L (15)

where h = diag(Al:. . . ,XL),B, = [ b l , . . . : b ~ ] nd CT =[c:?. . , .E]. A resul t similar to Lemma 1.1 holds whenever A Ois a stable matrix, that is, (15) ensures that V(z) is positive-definitefor any possible perturbation A . Defining p = AC,z, (15) can beexpressed as

( 4 " Z + B,p)T(Pos + C%"lp)< 0.


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Using a result similar to Lem ma 1.2, an equivalent stability conditionis

IV . PARAMETER-DEPENDENTY A P U N O VUNCTIONSA ND ROBUST ONTROL YSTEM YNTHESIS

In the second part of this note, we show some of the moreinteresting features of parameter-dependent Lyapunov functions forcontroller synthesis. It is important to remember that controller

( i l o z + Bpp)T(Pos c,Tnpj < 0whenever

synthesis for plant subject to real parametric uncertainty 15 no ta convex problem. However, the concept of parameter-dependentT R p 5 a' c C i RC,r. V R diagonal and positive-definite.Thus, using the S-procedure, (2) is stable if there exist symmetricmatrices P,:I.nd R such that R > 0. R, an d A diagonal Lyapunov functions can again be exploited to lead to a (possiblyconservative) constructive technique offering some potential advan-tages. In particular, we point out that one can limit the search for

robustly stabilizing controllers to controllers with the same order asthe original plant, a fact seemingly left unnoticed in the literature.A. Robust

-4TPo + I'oAo + ~V'C'C,TRC, PoB, + &CC,TA[ B; Po+ hC,AoWe now co mpa re Popov's stability criterion with our new criterion.

Using the correspondence between (1) and (2) through the relation-4, = b z c L ,we have the following proposition.Proposition 2.1: If Popov's criterion (16) is satisfied, then thecriterion given in Proposition 1.1 is satisfied.B. Comparison with the Real -p Upper Bound

(16)system Synthesis Setup

We consider the uncertain linear system

(18)!I= C7JxIn this section, we compare (14) with the one obtained by Fan etal . in [ lo ]. We first rewrite ( 1 ) in the feedback form

tl-C = Aox +Bp,d t ~ ( 0 ) 20(1 = CTL

Z = lp = Aq. A = diag6,I .

[B an d C are given in (1 l ) . ] Let H ( s ) = C(sI- A O ) ~ * B .e havethe fol lowing theorem [ lo] .Theorem 2.1: System (1) is stable if for any frequency U), thereexist a positive-definite Hermitian matrix S(w) and an Hermitianmatrix G(w) that satisfy

a"H(iU))*S(w)H(iw) - S ( w )+ ( G ( w ) H ( i w j- H ( Z U ) * G ( U ) ) O (17)

where z: R+ + R" is the state vector, U : R+ + Rnu s thecontrol input, and .y: R+ + Rny is the measurement vector. Theuncertainties 6; re constant real parameters and satisfy IS, 5 CL for agiven positive real number a , = 1;. . L. We address the problemof robust stabilization: does there exist a linear output feedbackcontroller

d- Z K = A K X I C + B I C ~at (19)U = CKZ JC D I C ~

where XI


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Our choice for parameter-dependent quadratic Lyapunov functionsfollows from the previous developments and is of the form

When partitioned conformally with the dimensions of the plant andthe controller, we assume the matrices PO,. . .PL an be written as

The reader will notice that the matrices PI... .PL ave somestructure. They only "read' the plant 's states and not the controller 'sstate. An alternate expression for V(z,l. A) isV(Z,l. A) = &Po + CTAPC)z, , (24)

with P = diagf=,P,. Following the sam e S-procedure argument asbefore, the positivity o f V and the negativity o f its time-derivativeare implied by the matrix inequality

AzPo +PoA,i + a2CTSC &CTP + Poi3 - CTT[ PCA,-I + B?'PO+TC B T C T P+ PCB - sPo > O.S > 0. (25)Constraint (25) now allows us to prove the following propositionregarding the order of robustly stabilizing controllers.Proposition 3.1: Assume there exists a controller I< that robustly

stabilizes (1 8) as well as a parameter-dependent Lyapunov functionof the form (24) that proves it . Then such a controller can always betaken of order less than or equal to the order of the plant (18).

The proof of this proposition relies on the following intermediateresult whose proof may be found in [9] or [13], for example.Lemma 3.1; Given a symmetric matrix 9 nd two matrices L- an dV with suitable dimen sions, there exists a matrix I< such that

(26)P + C J T I i V+ vTI< TC T < 0if and only if

,v:Q.Gr < 0. N,TQ.tC < 0 ( 2 7 )where Jlir..ljv are any bases of the null spaces of 1- and I - ,respectively.

Indeed, it is possible to rewrite ( 2 5 ) in the form (26) by definingAiPo + PoAo + 02Cf C A,TCTP+ o f3 - CTT

XP = [ PCAo + BTPo +TC B T C T P+ PCU - s (28)an dT/ = [C, 01, U = [U:? B Z C T P ] . (29)

Eliminating the controller variables, and after algebraic manipula-tions, the second inequality in (27) can then be written

1/AT X + X A o + a 2 CTS C X B + A,ICTP- CT TBTX + PCAo + T C B T C T P+ P C B - Sx p ;] o

where !Vu denotes any basis of the nullspace of C,, an d PO asbeen partitioned according to the dimension of the system and thecontroller, respectively, that is

where the * ' s are irrelevant quan tities. Likewise, the first inequalityin (27) can be rewritten as

Y.4TCTP + B - Y C r T aYCT0 -S-l I

T C Y B T C T P+ PCB - Sx [ ;] 0

where -ITudenotes any basis of the nullspace of [BT B:CTP], an dPC1 as been partitioned according to the dimension of the systemand the controller as

Finally, it is easy to show through algebraic manipulations that thepositivity requirement on PO mplies the condition

Conversely, if (30)-(32) are satisfied for some matrices X an d Y , tmay be shown, using a reasoning similar to [13], that a kt h ( k 5 n)order controller I< can be built , where n + k is the rank of the matrixappearing in (32). Such a controller may be built as follows:

Compute full-column-rank matrices M,Ar E R" ' uch thatM N T = I - Y X .

Compute PO s the unique solution to the linear equationX I I Y

( L v T 0 ) = F0 ( o M T )Compute any I< such that (25) is satisfied. Such a solution al-ways exists and is obtained using standard convex optimizationcodes.

C. One Robust Controller Synthesis Algorithmand Numerical ExampleBased on the previous developments, a simple scheme to effec-

tively compute robustly stabilizing controllers K is now discussed.The proposed synthesis procedure is reminiscent of p synthesisalgorithms as it alternates analysis phases and synthesis phases.Such schemes may converge to a local minimizer only, or evenno minimizer at all, but have proven efficient in a number ofinteresting applications [SI, [21], [18], [3], [l ] . Unlike p-synthesistechniques, parameter-dependent Lyapunov functions, rather thandynamic (frequency-dependent) scalings, allow us to bypass thecritical phase of curve fitting. Moreover, the technique proposed inthis paper always yields controllers having at most the order of theplant.We now propose a controller synthesis algorithm to maximize thestability margin of a given system:

Step 0)

Step 1)Step 2)Step 3)

Compute a stabilizing controller IC for the nominal plantusing any available method (LQG, for example). If nostabilizing controller can be found, then stop.I< given, maximize N over PO,, S , an d T subject to(25).P.s, given, maximize CY over X an d Y subject to(30)-(32). Reconstruct a corresponding controller I


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H, synthesisComplex p synthesis

Real U synthesis

1045

11. CY2.54 0.39372.53 0.39531.1 0.9009

Note that Steps 1 and 2 are generalized eigenvalue minimizationproblems which may be solved efficiently using recently developedpositive-sem idefinite programming codes. Step I may be very de-manding since it involves a number of scalar variables which isroughly 2n2 + 3%. This number may be prohibi t ive for largesystems with available solvers. Step 2 bears the same computationalcomplexity as LMI-based H, -synthesis problems.

We now illustrate the proposed synthesis procedure on a simplenumerical example. The system under consideration is a simpleflexible mechanical system, and it is presented in [231. The uncertainplant G ( s ) s given by the transfer function

s2 - j L u Y + d 2 j l + 6,)SL + 2CLd.S +d (+ 6,)G(s ) = ,

with natural frequency U = 1 and damping C = 0 .2 , an d 61.62 arereal uncertain parameters.

The standard setting for this problem is easily derived as

y = (0 l ) x + Uwhere

Note that the uncertainty appears not only in the dynamics matrix,but also in the control input matrix; the foregoing results thusneed to be adapted in a straightforward, yet tedious man ner. Itis important to recall that small-gain-based techniques as well asquadratic approaches [4], [24] are generally very conservative forproblems of this type. Thus, this problem is a valuable test for ourtechnique. We used the algorithm described in the previous sectionto find the controller

4.Gi4.5 + 8 . 0 3 3 ~ 5 s+ . 67e6s 2 + 2 . 2OGc. 5~- 7 . 0 8 3 ~ 7i l y a p =

The evolut ion of y = l / n , the inverse of the guaranteed stabilitymargin, is depicted in Fig. 1 and its minimal value was obtained as1/nOpt 1.08.The accuracy of this bound an d the calculat ions waschecked a posteriori using real-p analysis on the closed-loop systemand yielded the same guaranteed stability margin. The algorithm onthis problem behaves very well; convergence occurred after eightiterations.Real i.1synthesis was used in [23] and led to 1/a about 1.1 after fiveiterations and a controller having 14 states before further reduction.The results based on the various techniques are recapped in Table I.Summing up, our approach performs very well, bears comparisonwith the real ,LL synthesis, and directly produces a simpler robustcontroller with no more apparent conservatism.

From a computational point of view, our technique may be verydemanding for problems of much larger size that requires numerousanalysis/synthesis iterations. This may be prohibitive for designs thatneed repeated trials and errors to determine the problem parameters,the performance filters, etc. The development of different synthesisschemes to sub stantially improve the convergence speed is currentlyunder investigation. Note that as with many local optimizationalgorithms, convergence properties are very data-dependent, and thepresented results should thus be taken with caution.

evolution of gammaz:I \-\\\\

AnalysidSynthes is iterationsO ,

Fig 1 Evolution of 1 /a versus iterationsTABLE I

SIZE OF PARAMETEROX OBTAINED WITH VARIOUS TECHNIQUES

, Proposed approach I 1.08 I 0.9259v. CONCLUS ION AND EXTENSIONS

In this paper, we have considered the problem of determining thestability of linear systems subjec t to real parametric uncertainties. Wehave shown how the combined use of parameter-dependent Lyapunovfunctions and the S-procedure yields a stability criterion that canbe expressed in the form of convex inequalities. These inequalitiescan b e solved using recent ly developed opt imizat ion codes and h avea frequency-domain counterpart leading to immediate comparisonswith existing criteria. Numerical experiments show that the newcriterion performs well compared to existing criteria, both from theaccuracy and computational viewpoints. Extension of the developedcriterion to robust controller synthesis problems show s that the searchfor robustly stabilizing controllers may be limited to controllerswith the same order as the original plant. Moreover, the proposedapproach bypasses scaling interpolation steps encountered in ,L/-synthesis algorithms. It may be easily extended to handle additionalproperties or characteristics, such as H, performance indexes andslowly time-varying systems; we invite the interested reader to consultreferences [ 111 and [2] for further results.

REFERENCES[11 R. J. Adam5 and S. S. Banda, Robust flight control design usingdynamic inversion and structured singular value synthesis, IEEE Trans.Circuits Syst., vol. I , no. 2, pp. 80-92, June 1993.[ 21 P. Apkdrian, E. Feron, and P. Gahinet, Parameter-dependent Lyapunovfunctions for robust control of systems with real parametric uncertainty,in Proc. European Contr. Cor$, Roma, Italy, Sept. 1995,pp . 2275-2280.[3] G. J. Balas, J. Reiner, and W. Garrard, Design of flight control systemfo r a highly maneuverable aircraft using p synthesis, in Proc. A IMGuid. Contr. Cony!, Monterey, CA, 1993.[4] B. R. Barmish , Stabilization o f uncertain systems vi a linear control,IEEE Truns. Automat. Contr.,vol. AC-21, n o . 8, pp . 848-850, Oct. 1982.


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151 S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear matrixinequalities in system and control theory, SIAM Studies Appl. Math.,SIAM, vol. 15, 1994.[6] F. M. Callier and C. A. Desoer, Multivariable Feedback Systems. NewYork: Springer-Verlag, 1982.[7] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-OutputProperties. New York: Academic, 1975.[8] J. C. Doyle, K. Lenz, and A. Packard, Design examples using p

synthesis: Space shuttle lateral axis FCS during reentry, in Proc. IEEECon$ Decision Contr., Athens, Greece, 1986, pp. 2218-2223.191 J. C. Doyle, A. Packard, and K. Zhou, Review of LFTs, LMIs andp, in Proc. ZEEE Con$ Decision Contr., Brighton, England, 1991, pp.122171232.

[ I O1 M. K. H. Fan, A. L . Tits, and J. C. Doyle, Robustness in the presence ofmixed parametric uncertainty and unmodeled dynamics, ZEEE Trans.Automat. Contr., vol. 3 6 , no. 1, pp. 25-38, Jan. 1991.[ I I ] E. Feron, P. Apkarian, and P. Gahinet, S-procedure for the analysis ofcontrol systems with parametric uncertainties via parameter-dependentLyapunov functions, in Proc. European Contr. Conj?,Roma, Italy, Sept.[12] A. L. Fradkov and V. A. Yakubovich, The S-procedure and dualityrelations in nonconvex problems of quadratic programming, VestnikLeningrad Univ. Math., vol. 6 , no. 1, pp. 101-109, 1979; in Russian ,

1973.[13] P. Gahinet and P. Apkarian, A linear matrix inequality approach to H,control, lnt. J. Robust No d. Contr ., vol. 4, no. 4, pp. 421448, 1994.[14] P. Gahinet, P. Apka rian, and M. Chilali, Parameter-dependent Lya-punov functions for real parametric uncertainty, IEEE Trans. Automat.Contr., vol. 41, no. 3, pp. 436442, 1996.[15] P. Gahinet, A. N emirovskii, A. J. Laub, and M. Chilali. LM I ControlToolbox. The Mathworks, 1994, 1995.[I61 W. Haddad and D. Bernstein, Explicit construction of quadratic Lya-punov functions for the small gain, positivity, circle, and Popov theo-rems and their application to robust stability. Part I: Continuous-timetheory, Int. J. Robust Nonlinear Contr., vol. 3 , pp. 313-339, 1993.[171 A. M. Lyapunov, Problkme gtnCral de la stabilitt du mouvement,Annals qfMathematics Studies, vol. 17. Princeton, NJ : Princeton Univ.Press, 1947.1181 J. C. Morris, P. Apkarian, and J. C Doyle, Synthesizing robust modeshapes with p and implicit model following, in Proc. IEEE Con6 Contr.Appl., Dayton, Ohio, Sept. 1992.[I91 K. S. Narendra and J. H. Taylor, Frequency domain criteria for absolutestability, Electrical Science.[20] J. C. Willems, Least squares stationary optimal control and the alge-

braic Riccati equation, IEEE Trans. Automat. Contr ., vol. AC-16, no.6, pp. 621-634, Dec. 1971.1211 K. Wise and K. Poolla, Missile autopilot design using H, optimalcontrol with p-synthesis, in Proc. Amer. Contr. Con$, May 1990, pp.2362-2367.[22] V. A. Yakubovich, The S-procedure in nonlinear control theory,Vestnik Leningrad Univ. Math., vol. 4, pp. 73-93, 1977; in Russian,1971.[23] P. M. Young, Robustness with parametric and dynamic uncertainty.Ph.D. dissertation, California Inst. Technol., 1993.[24] K. Zhou and P. P . Khargonekar, On the stabilization of uncertainlinear systems via bound invariant Lyapunov functions, SIAM J . Contr.Optimiz., vol. 26, pp. 1265-1273, 1988.

1995, pp. 141-146.

New York: Academic, 1973.

A New Approach to EigenstructureAssignment by Output FeedbackA. T. Alexandridis and P. N. araskevopoulos

Absb-act- The eigenstructure assignment problem for a linear timeinvariant multi-input-multi-ou tput system using output feedback is con-sidered. A new approach is developed which identifies the eigenspaces forthe desired set of all the closed-loop eigenvalues. For the assignment ofthis desired set, necessary and sufficient conditions are established. Theseconditions contain two coupled Sylvester matrix equations, one of whichis proven to be a reduced-order square Sylvester matrix equation. Thisresults in an efficient analytical procedure, numerically superior to knowntechniques, for the determination of the output feedback gain-matrix.

I. INTRODUCTIONConsider the completely controllable and observable linear dynam-

ical system

where .r E R. U E R, y E R, an d -4;B , an d C ar econstant matrices of appropriate dimensions with rank B = m an drank C = r . To (1) apply the proportional output feedback lawU = A y . ( 2 )

Then, the closed-loop system has the following form:j: = (A + BliC)z , y = Cx. ( 3 )

Le t .I = (-11.12} be an arbitrarily selected set, where A I ={ X I . A z :... A,} an d 122 = {Xr+l,Xr+2;..,An}. It is assumedthat -1 1 an d -1 2 are self-conjugate sets and A contains distinct values.Then, the problem considered in this paper is to find a real matrix Irsuch that the eigenvalues of (3) are those of the set A.

The present problem has attracted great attention the last twodecades, wherein several interesting results have been derived[1]-[15]. In most cases, a main restriction was imposed on the orderof the system with respect to the number of the inputs and outputs,namely m + 2 n+1.Other results prove that there exists a solutionto the output feedback pole-assignment problem, as stated above,under the significantly less restrictive condition m r 2 [16]-[19].These last results require an additional constraint that is eitherrank E = n , where E s an ri x mr matrix with each row formed fromthe rows of the matrix C-4kB fo r k = O . l , . . . , n- 1) [16], [17],or full rank of the Plucker matrix [19]. In [20]similar conditions forlocal complete assignabi li ty ar e der ived by per turbing th e feedbackgain-matrix I

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Analysis and Synthesis of Robust Control Systems via Parameter_dependent Lyapunov Functions