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Dilated LMI conditions for time-varying polytopicdescriptor systems: the discrete-time caseGabriela Iuliana Bara aa LSIIT-UMR CNRS 7005 , University of Strasbourg , bd. Sébastien Brant, BP 10413, 67412Illkirch, FrancePublished online: 06 Jul 2011.

To cite this article: Gabriela Iuliana Bara (2011) Dilated LMI conditions for time-varying polytopic descriptor systems: thediscrete-time case, International Journal of Control, 84:6, 1010-1023, DOI: 10.1080/00207179.2011.586049

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International Journal of ControlVol. 84, No. 6, June 2011, 1010–1023

Dilated LMI conditions for time-varying polytopic descriptor systems: the discrete-time case

Gabriela Iuliana Bara*

LSIIT-UMR CNRS 7005, University of Strasbourg, bd. Sebastien Brant, BP 10413, 67412 Illkirch, France

(Received 31 May 2010; final version received 3 May 2011)

This article addresses the admissibility analysis and state-feedback control synthesis problems of discrete-timepolytopic descriptor systems with possibly time-varying parameters. First, we present a new necessary andsufficient strict linear matrix inequality (LMI) condition for the admissibility analysis of linear time-invariant(LTI) descriptor systems. Then, based on the concept of poly-quadratic admissibility, introduced in this article,we extend this result to the admissibility analysis of possibly time-varying parameter-dependent descriptorsystems. This extension, which applies to both uncertain and measurable parameters, uses parameter-dependentLyapunov functions and employs two slack variables. The extended conditions are also expressed as strict LMIconditions which are easily tractable numerically compared to the non-strict ones often encountered when dealingwith descriptor systems. Note that we have proposed two separate admissibility analysis conditions: one directlyexploitable for state estimation and the other for state-feedback control. The need for different admissibilityanalysis conditions for each synthesis problem is motivated by the fact that the duality between state estimationand state-feedback control, which hold in the case of measurable parameters, does not hold when dealing withuncertain ones. In our approach, we overcome the absence of such duality by considering a dilation only on thedynamical part of the descriptor system. The application of our analysis result to both robust state-feedbackcontrol and polytopic state-feedback control are also presented in this article. Our analysis results extend todescriptor systems, some existing results developed for regular systems.

Keywords: linear discrete-time descriptor systems; polytopic descriptor systems; time-varying parameters;parameter-dependent Lyapunov functions; slack variables; dilated linear matrix inequalities

1. Introduction

The class of parameter-dependent systems has received

significant attention over the past decades due to their

ability to offer a good description for many industrial

processes and real-world applications. A linear para-

meter-varying (LPV) model can be obtained from a

nonlinear system either by quasi-LPV modelling or by

means of a linearisation around a family of operating

points (see Rugh and Shamma (2000) for further

details). The time-variance of the parameters, assumed

measurable, reflects the change of operating point.

Also, the parameter-dependent/polytopic representa-

tion is often used for modelling uncertain systems. This

representation encompasses the case of uncertain

interval matrices and offers a better description for

most uncertain systems since norm-bounded parameter

uncertainties tend to overestimate the uncertainties in

the system as pointed out by de Souza, Barbosa, and

Fu (2008). In general, the analysis results developed for

parameter-dependent/polytopic systems can be applied

to LPV systems and uncertain polytopic systems

regardless of whether the parameters are measurable

(i.e. available for control purposes) or uncertain.

However, when it comes to solving a synthesisproblem, the uncertainty or measurability of theparameters plays an important role. For instance,when the parameters are uncertain a robust synthesis isperformed while a parameter-dependent controller issynthesised whenever the parameters are measurable.Note that the LPV synthesis may be more difficult thanthe robust design (regarding polytopic uncertainties)since it requires computing several gains instead of asingle one. Moreover, as shown by Blanchini andMiani (2003, 2010), the separation principle and theduality between state estimation and state-feedbackcontrol, which hold in the case of LPV synthesis, donot hold when dealing with uncertain parameters. Thismeans that the analysis results expressed in dual formcan be applied only for gain-scheduling control but notfor robust control. As a consequence, the robustsynthesis becomes more difficult than the LPV onein the sense that it requires to separately developanalysis results exploitable either for state estimationor state-feedback control without using the dualityproperty.

Descriptor or singular systems, also called general-ised state-space systems, have the ability to incorporate

*Email: Iuliana.Bara@lsiit-cnrs.unistra.fr

ISSN 0020–7179 print/ISSN 1366–5820 online

� 2011 Taylor & Francis

DOI: 10.1080/00207179.2011.586049

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algebraic constraints on physical variables in additionto differential or difference equations. Hence, para-meter-dependent descriptor systems allow a moregeneral representation with a practical appeal formany applications such as, for instance, aircraftcontrol (Masubuchi, Kato, Saeki, and Ohara 2004)and robotic systems (Mills and Goldenberg 1989). It isworth mentioning that LPV systems whose state-spacematrices are rational functions of the parameters canbe rewritten as descriptor LPV systems with affineparameter-dependent state matrices by appropriatelychoosing an augmented descriptor state vector. Thisfurther motivates our interest for the study ofparameter-dependent/polytopic descriptor systems.

Without being exhaustive, there exist, in general,two main approaches for developing analysis/synthesisresults for parameter-dependent/polytopic systemswhich are less restrictive than the classical ones basedon parameter-independent Lyapunov functions. Thefirst one employs parameter-dependent Lyapunovfunctions (PDLF) while the second one introducesadditional matrix variables usually called slack vari-ables. It has been shown by Feron, Apkarian, andGahinet (1996), de Oliveira, Bernussou, and Geromel(1999), Gahinet, Apkarian, and Chilali (1996) andDaafouz and Bernussou (2001) that the use of PDLFallows to outperform the analysis and synthesis resultsthat are based on parameter-independent Lyapunovfunctions. As a consequence, PDLF-based analysis andsynthesis methods have also been developed fordescriptor systems with either constant parameters(such as the approaches presented by Fang (2002),Gao, Chen, and Sun (2003), Sakuwa and Fujisaki(2005), de Souza et al. (2008) and Yagoubi, Bouali, andChevrel (2008)) or time-varying ones (Masubuchi,Akiyama, and Saeki 2003; Masubuchi et al. 2004;Bara 2010). The use of slack variables increases thedegree of freedom of the design variables leading toless restrictive analysis and synthesis results. Inaddition, as far as synthesis is concerned, someconstraints involving Lyapunov matrices (matrixequality constraints, for instance) are replaced byanalogous constraints involving however slack vari-ables, thus allowing to obtain less conservativesynthesis results.

The results developed in the literature for thecontinuous-time case generally do not apply todiscrete-time systems. This is particularly true whenthe parameters are time-varying. In the case ofdiscrete-time parameter-dependent descriptor systems,when the parameters are time-invariant, the resultsreported in the literature either make use of PDLF(Fang 2002) or slack variables (Kuo and Fang 2003).To our knowledge, however, the only work consideringtime-varying parameters and using PDLF and slack

variables was recently presented by Chadli, Darouach,and Daafouz (2008a) and Chadli, Daafouz, andDarouach (2008b). Unfortunately, their results areerroneous as shown by the counterexample presentedin Section 3. To the best of our knowledge, there are noother results dealing with the admissibility of discrete-time descriptor systems with time-varying polytopicparameters and employing PDLF and slack variables.

In this article, we address the admissibility analysisand control of discrete-time polytopic descriptorsystems with possibly time-varying parameters byusing PDLF and slack variables. First, we state a newnecessary and sufficient strict linear matrix inequality(LMI) condition for the admissibility analysis of lineartime-invariant (LTI) descriptor systems by using oneslack variable. Then, based on the concept of poly-quadratic admissibility, that we introduce in this article,we extend our new condition to the admissibilityanalysis of polytopic descriptor systems by involving,this time, two slack variables. Doing so leads to twodifferent sets of sufficient strict LMI conditions which,from a numerical point of view, are preferable to non-strict ones (Xu and Lam 2004a,b) often encounteredwhen dealing with descriptor systems. Note that thefirst set of conditions can be easily exploited for stateestimation purposes while the second one for state-feedback control. Indeed, our approach takes intoaccount the absence of duality between state estimationand state-feedback in the case of uncertain parametersby proposing separate admissibility analysis conditionsexploitable for each case. This has been achieved byconsidering a dilation only on the dynamical part of thedescriptor system. When the parameters are time-invariant, our analysis results extend to descriptorsystems the ones proposed by de Oliveira et al. (1999)and Peaucelle, Arzelier, Bachelier, and Bernussou(2000) for regular state-space systems. The applicationof our analysis result to robust state-feedback controland polytopic state-feedback control are also presentedand discussed. The numerical examples, also reportedin this article, show that, in the case of time-invariantparameters, our new admissibility analysis conditionsreduce the conservatism of the result presented by Kuoand Fang (2003). Also in the case of time-invariantparameters, we have found that there are manysituations where our method performed better thanthat of Fang (2002) by identifying a larger admissibilityregion. We have also found situations where ourmethod and the one of Fang (2002) identified thesame admissibility region as well as situations where thelatter method provided better results. However, notethat the advantage of our analysis results, over themethods presented in the literature, is that they allowtime-varying parameters while not being more restric-tive in the time-invariant case and, more importantly,

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these results are easily exploitable for synthesispurposes by taking into account the absence of dualityin the case of uncertain parameters.

This article is organised as follows. Section 2 statesthe problem under consideration and introduces theconcept of poly-quadratic admissibility. In Section 3,we present some preliminaries and discuss some recentresults. Section 4 provides our main contribution.First, we give a new necessary and sufficient conditionfor the admissibility analysis of LTI descriptor systems.Then, we extend this result to poly-quadratic admis-sibility of parameter-dependent descriptor systems andpresent its application to robust as well as polytopicstate-feedback control. Numerical examples are pre-sented in Sections 5 and 6 concludes our work.

Notations: The notations used throughout this articleare standard. The relation A4B (A5B) means thematrix A�B is positive (negative) definite. The super-script T stands for matrix transposition. The matrix Instands for the identity matrix of dimension n. (?) isused for the blocks induced by symmetry and He{A}means AþAT. Bdiag(A1, . . . ,Ai) is the block-diagonalmatrix having A1, . . . ,Ai on its main diagonal.Co(A1, . . . ,AN) denotes the convex hull generated bythe matrices A1, . . . ,AN.

2. Problem statement and definitions

Consider the class of discrete-time parameter-depen-dent singular systems defined by

Exðkþ 1Þ ¼ Að�ðkÞÞxðkÞ þ Bð�ðkÞÞuðkÞ, ð1Þ

where x(k)2 IRn is the descriptor state vector andu(k)2 IRm is the control input. The matrix E2 IRn�n isconstant and may be singular with rankE¼ r� n.

�ðkÞ ¼ �1ðkÞ . . . �NðkÞ� �T

is the vector of real andpossibly time-varying parameters. Whenever possible,the time-dependency of the parameter vector will beomitted for brevity. In the following, we assume thatthe system matrices A(�) and B(�) belong to convexpolytopic sets defined as

A ¼

�Að�ÞjAð�Þ ¼

Xi¼Ni¼1

�iAi, �i � 0

for all i 2 f1, . . . ,Ng,Xi¼Ni¼1

�i ¼ 1

�, ð2aÞ

B ¼

�Bð�ÞjBð�Þ ¼

Xi¼Ni¼1

�iBi, �i � 0

for all i 2 f1, . . . ,Ng,Xi¼Ni¼1

�i ¼ 1

�: ð2bÞ

The matrices Ai and Bi are the vertices of polytopes Aand B, respectively. We define the index set I ¼{1, . . . ,N}. Note that this class of systems is very largesince it covers many uncertain systems as well as LPVsystems.

We recall some basic definitions for unforceddiscrete-time LTI descriptor systems.

Definition 2.1 (Dai 1989; Xu and Lam 2004a):Consider the system Ex(k)¼Ax(kþ 1) with constantmatrices.

. The system is regular if det(zE�A) is notidentically zero.

. A regular system is causal ifdeg(det(zE�A))¼ rankE.

. The system is said to be stable or Schur-stableif it has Schur-stable finite modes i.e. the rootsof det(zE�A)¼ 0 lie inside the unit disccentred at the origin.

. The system is admissible if it is regular, causaland stable.

In the sequel, a descriptor system is called regularif it satisfies the regularity property as stated inDefinition 2.1. We call standard state-space system asystem without algebraic constraints i.e. E¼ In.

The goal of this article is to develop new admissi-bility analysis and control synthesis conditions forpolytopic discrete-time singular systems using PDLFs.To this end, we use quadratic PDLF of the form:

VðkÞ ¼ VðxðkÞ,�ðkÞÞ ¼ xTðkÞETPð�ðkÞÞExðkÞ,

where Pð�Þ ¼Xi¼Ni¼1

�iPi: ð3Þ

The variation of this function along the solution of theunforced system (1) is given by

DV ¼ Vðkþ 1Þ � VðkÞ

¼ xTðkÞ�ATð�ÞPþð�ÞAð�Þ � ETPð�ÞE

�xðkÞ,

where A(�)¼A(�(k)), P(�)¼P(�(k)) and Pþð�Þ ¼Pð�ðkþ 1ÞÞ ¼

Pi¼Ni¼1 �iðkþ 1ÞPi ¼

Pi¼Ni¼1 �iðkÞPi with

�i(kþ 1)¼ �i(k) for all i2I . When the parameters arenot time-varying but constant, we have Pþ(�)¼P(�).

Based on the framework developed by de Oliveiraet al. (1999) and Daafouz and Bernussou (2001) foruncertain polytopic discrete-time systems, we introducethe concept of poly-quadratic admissibility which canbe seen as a generalisation to singular systems of thepoly-quadratic stability concept for standard state-space systems.

Definition 2.2: The unforced polytopic system (1) ispoly-quadratically admissible if it is regular andcausal over the polytope A and there exists a

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parameter-dependent positive-definite quadraticLyapunov function (3) whose difference is negative-definite and decrescent.1

Remark 1: When the parameters �i are uncertain, thepoly-quadratic admissibility implies the robust admis-sibility of the system.

When the parameters are uncertain and based onthe previous definition, we introduce the notion ofrobust poly-quadratic causal stabilisability by state-feedback control.

Definition 2.3: The polytopic system (1) is calledrobustly, poly-quadratically and causally stabilisable bystate-feedback if there exists a control law u(k)¼Kx(k)such that the closed-loop system is poly-quadraticallyadmissible.

When the parameters are not uncertain and aresupposed measurable, we define the notion of poly-quadratic causal stabilisability by polytopic state-feedback control.

Definition 2.4: The polytopic system (1) is calledpoly-quadratically and causally stabilisable by polytopicstate-feedback if there exists a parameter-dependentdescriptor state-feedback control law

uðkÞ ¼ Kð�ðkÞÞxðkÞ, where Kð�Þ ¼Xi¼Ni¼1

�iKi ð4Þ

such that the closed-loop system is poly-quadraticallyadmissible.

3. State transformation and comments on previous

results

3.1 State transformation

As the matrix E is of rank r, there exist invertiblematrices S and T such that

SET ¼Ir 0

0 0

� , where T ¼

T11 T12

0 T22

� : ð5Þ

These matrices are not unique and, for instance, theQR factorisation of E can be employed in order toobtain the decomposition (5). Note that this decom-position is a particular case of the singular valuedecomposition (SVD) presented by Dai (1989) andhence leads to a particular equivalent SVD coordinatesystem. Based on the coordinate transformationxðkÞ ¼ T �xðkÞ, the matrices of the equivalent systemare SA(�)T and SB(�) where

SAð�ÞT ¼A11ð�Þ A12ð�Þ

A21ð�Þ A22ð�Þ

� ¼Xi¼Ni¼1

�iSAiT ð6Þ

was partitioned following the partitioning in (5). Notethat these equivalent system matrices belong to poly-topes SAT and SB, respectively. Using the notations

�xTðkÞ ¼ �xT1 ðkÞ �xT2 ðkÞ� �T

and

�Pð�Þ ¼ S�TPð�ÞS�1 ¼�P11ð�Þ �P12ð�Þ

�P21ð�Þ �P22ð�Þ

" #

it follows that the PDLF (3) can be rewritten as

VðkÞ ¼ �xTðkÞðSET ÞT �Pð�ÞðSET Þ �xðkÞ ¼ �xT1 ðkÞ�P11ð�Þ �x1ðkÞ:

Therefore, the singular system is poly-quadraticallyadmissible (as defined in Definition 2.2) if and only ifthe matrix A22(�) is invertible for any value of � and

�P11ð�Þ4 0,

~ATð�Þ �P11þð�Þ~Að�Þ � �P11ð�Þ5 0,

where ~Að�Þ ¼ A11ð�Þ � A12ð�ÞA22ð�Þ�1A21ð�Þ,

i.e. ~A(�) is poly-quadratically stable (Daafouz andBernussou (2001) for more details on poly-quadraticstability). Hence, based on the transformation intro-duced above, the poly-quadratic admissibility isequivalent to the poly-quadratic stability of a parameter-dependent system without algebraic constraints which,as pointed out by de Souza et al. (2008) (see Remark 2),does not carry a polytopic structure. Consequently, theavailable poly-quadratic stability results developed forpolytopic state-space systems cannot be exploited forpoly-quadratic admissibility. Moreover, as shown anddiscussed next, the extension of poly-quadraticstability results developed for polytopic state-spacesystems to singular systems is a difficult problem.

3.2 Comments on previous results

Recently, it has been claimed by Chadli et al. (2008a,b)that system (1) is stable whenever there exist non-singular symmetric matrices Pi such that

ETPiE � 0 for all i 2 I , ð7aÞ

ATi PjAi � ETPiE5 0 for all ði, j Þ 2 I � I : ð7bÞ

Unfortunately, this result is incorrect. In order to showthis, consider the following system matrices:

E ¼ BdiagðI2, 0Þ, A1 ¼

0:08 �0:06 0:5

0 �0:2 0

0:2 0 2:2

264

375

and A2 ¼

0:02 0:06 0:5

0 �0:2 0

0:2 0 �0:2

264

375:

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The LMI conditions (7) were found feasible. However,

the system does not have stable finite modes for all

possible values of �. Indeed, for a constant �, the

characteristic polynomial is

detðzE�Að�ÞÞ ¼ ðzþ 0:2Þ

z� 0:08�1� 0:02�2

�0:1

0:2�2� 2:2�1

�, where �1þ �2 ¼ 1:

If, for instance, �1¼ 0.1 the system has an unstable

finite mode at �2.474. There are many values of � for

which the system has an unstable finite mode. This

counterexample shows that conditions (7) and, there-

fore, their dilated LMI extensions also proposed by

Chadli et al. (2008a,b) are incorrect. This discrepancy

can be explained as follows. In the case of standard

state-space systems i.e. E¼ In, the stability of the

polytopic system is equivalent to conditions (7b) with

Pi4 0 for all i2I (Daafouz and Bernussou 2001). This

can be proved by applying Schur complement since Pi

is positive-definite. In the case of descriptor systems,

Schur complement cannot be applied anymore since

the term ETPiE is only positive semi-definite and not

strictly positive-definite. In fact, conditions (7) are

necessary in order to guarantee the admissibility of

(1) with time-varying parameters but they are not

sufficient.

Remark 2: Note that, in general, the generalisation

of standard state-space systems theory to singular

systems is a difficult problem. For instance, the

stability analysis is more complicated in the case of

descriptor systems than it is when dealing with

standard state-space ones. It requires to consider

simultaneously the stability as well as the regularity

and either the absence of impulses (continuous-time

systems) or the causality (discrete-time systems). In

addition, as the Lyapunov matrix appearing in the

study of singular systems is indefinite, the analysis

results developed for standard state-space systems

exploiting the symmetry and the positive-definiteness

of the Lyapunov matrix cannot be easily generalised

to descriptor systems. Therefore, as also pointed out

in Xu, Yang, Niu, and Lam (2001), Xu and Lam

(2004a) and Rejichi, Bachelier, Chaabane, and Mehdi

(2008), developing necessary and sufficient conditions

for the stability analysis and stabilisation of uncertain

discrete-time singular systems is more difficult than

when dealing with state-space systems. This is

particularly true in the case of descriptor systems

with time-varying parametric uncertainties for which

necessary and sufficient stability analysis and stabili-

sation conditions still have not been proposed in the

literature.

In the following, we develop strict LMI conditionsfor poly-quadratic admissibility analysis (Definition2.2) as well as causal stabilisability by state-feedback(Definitions 2.3 and 2.4).

4. Main results

In this section, we provide the main contribution ofour article. First, we present a necessary and sufficientcondition for the admissibility analysis of discrete-timeLTI singular systems. This condition is expressed as astrict LMI feasibility problem and hence easilytractable by convex optimisation techniques (Boyd,El Ghaoui, Feron, and Balakrishnan 1994). Then, thisresult is extended to poly-quadratic admissibilityanalysis of polytopic singular systems. The applicationof this extension to the synthesis of state-feedbackcontrollers guaranteeing the poly-quadratic admissi-bility of the closed-loop system is discussed in eithercase where the parameters are uncertain or availablefor control purposes.

4.1 LTI systems

In the following, we provide a new necessary andsufficient LMI condition for the admissibility analysisof discrete-time LTI descriptor systems. Assuming thatthere is no parameter-dependency in the system, theunforced descriptor system reduces to

Exðkþ 1Þ ¼ AxðkÞ þ BuðkÞ: ð8Þ

Theorem 4.1 (Admissibility): The pair (E,A) is admis-sible if and only if there exist a symmetric matrixQ2 IRr�r and general matrices G2 IRr�r, X12 IR

r�(n�r)

and X22 IR(n� r)� (n� r) such that

He0 0

0 X2

� SAT

� �þ

Q 0

0 0

� ð?Þ

G X1

� �SAT Gþ GT �Q

264

3754 0

ð9Þ

where S and T are given by (5).

Proof: Based on the change of state variablesintroduced in the previous section, the equivalentstate matrix is

SAT ¼A11 A12

A21 A22

� :

The pair (E,A) is admissible if and only if A22 isinvertible and ðA11 � A12A

�122 A21Þ is Schur stable (Lewis

1986; Dai 1989). The invertibility of A22 guarantees theregularity of the system and its causality while the

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second condition guarantees the stability of the finite

modes.

Sufficiency: The inequality (9) can be rewritten as

He

Q=2 0 0

X2A21 X2A22 0

GA11þX3A21 GA12þX3A22 G�Q=2

2664

3775

8>><>>:

9>>=>>;40:

ð10Þ

We deduce from (10), based on Schur complement,

that He{X2A22}5 0 which implies, by Lemma A.2,

that A22 is invertible. Moreover, by left- and right-

multiplying inequality (10) by

Bdiag Ir,0 Ir

In�r 0

� �

and its transpose, respectively, and by applying

Lemma A.1, we obtain

Q ð?Þ

G ~A Gþ GT �Q

" #4 0,

where ~A ¼ A11 � A12A�122 A21: ð11Þ

Based on the stability results presented in de

Oliveira et al. (1999), this last inequality implies the

stability of ~A.

Necessity: The matrix ~A ¼ A11 � A12A�122 A21 is stable

if and only if there exist a symmetric matrix Q and a

general matrix G of appropriate dimensions such that

(11) holds (de Oliveira et al. 1999). Therefore, there

exists a sufficiently small positive scalar �4 0 such that

Q ð?Þ

G ~A Gþ GT �Q

" #4�

2B diagðAT

21A21, 0Þ: ð12Þ

Let X2 ¼ �AT22. Then, as A22 is invertible, we

have X2A22 þ AT22X

T2 ¼ 2�AT

22A22 positive-definite.

Moreover, we have

�

2AT

21A21 ¼ AT21X

T2

�X2A22 þ AT

22XT2

��1X2A21:

Hence, the inequality (12) is equivalent, by Schur

complement, to

Q ~ATGT AT21X

T2

G ~A Gþ GT �Q 0

X2A21 0 X2A22 þ AT22X

T2

2664

37754 0:

Let X1 ¼ �GA12A�122 then 0¼GA12þX1A22 and

G ~A¼GA11þX1A21. Now, we replace the blocks

(2,1), (3,2), (1,2) and (2,3) of the last inequality by

these relations. Left- and right-multiplications by

Bdiag Ir,0 Ir

In�r 0

� �

and its transpose, respectively, lead to the

inequality (9). œ

Remark 3: The dual form of (9) can be obtained by

replacing SAT with (SAT)T.

A necessary and sufficient condition for the

admissibility analysis of LTI descriptor systems has

been proposed by Lee, Chen, and Fang (2004) using a

slack variables approach. Their condition involves two

slack variables which are coupled with the system state

matrix. Therefore, because of these two couplings, the

use of this condition for control design causes a

numerical solvability issue. For instance, the synthesis

of state-feedback controllers becomes a nonlinear

design problem. Compared to the result proposed by

Lee et al. (2004), our condition of Theorem 4.1

involves only one coupling between one slack variable

and the system state matrix. This advantage allows to

obtain a necessary and sufficient LMI condition for the

state-feedback control design as stated in the following

theorem.

Theorem 4.2: There exists a descriptor state-feedback

control law u(t)¼Kx(t) guaranteeing the admissibility of

the closed-loop system if and only if there exist a

symmetric matrix Q2 IRr�r and general matrices

G2 IRr�r, X12 IR(n� r)�r, X22 IR

(n� r)� (n� r) and

R2 IRm�n such that the following LMI condition holds:

He

(SAT

0 0

0 X2

" #

þSBR0 0

0 In�r

" #)þ

Q 0

0 0

" #8>>>>><>>>>>:

9>>>>>=>>>>>;

ð?Þ

SATG

X1

" #þSBR

Ir

0

" # !T

GþGT�Q

26666666666664

3777777777777540:

ð13Þ

Proof: By applying the dual form of Theorem 4.1 to

the closed-loop system whose dynamical equation is

Ex(kþ 1)¼ (AþBK )x(k) and by making the change of

variable R ¼ KT G 0X1 X2

h i, we obtain the condition (13)

since

0 0

0 X2

� ¼

G 0

X1 X2

� 0 0

0 In�r

�

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and

G

X1

� ¼

G 0

X1 X2

� Ir

0

� :

œ

Remark 4: When the matrix X2 provided by theLMI condition (13) is nonsingular, the descriptor

state-feedback gain is computed as

K ¼ R G 0X1 X2

h i�1T�1. When X2 is singular, we can

obtain an invertible solution as follows. The matricesQ, G, X1, X2 and R, solutions of (13), are used in order

to obtain a scalar �2 (0, 1) such that (13) holds whenX2 is replaced by X2þ �In�r. Note that such a � alwaysexists and the feedback gain is given by

K ¼ RG 0

X1 X2 þ �In�r

� �1T�1:

4.2 Polytopic systems

In order to distinguish the case of time-varying

parameters from the case of time-invariant ones, letus define the following indicator function q:

q ¼1 for time-invariant parameters,

0 for time-varying parameters.

�

The following theorem states sufficient LMI con-ditions for poly-quadratic admissibility of singular

discrete-time polytopic systems involving two slackvariables.

Theorem 4.3 (Poly-quadratic admissibility): Thesystem (1) is poly-quadratically admissible if one of the

following conditions is satisfied:

(i) There exist symmetric positive-definite matrices

Q1,Q2, . . . ,QN2 IRr�r and general matrices

F,G2 IRr�r, X1,X32 IR(n� r)�r and X22

IR(n� r)� (n� r) such that

HeF X3

0 X2

" #SAiT

( )þ

Qi 0

0 0

" #ð?Þ

G X1

� �SAiTþ FT 0

� � GþGT��qQi

þð1� qÞQj

�( )

2666664

3777775

40 8ði, j Þ 2 I �I ð14Þ

where S and T are given by (5).(ii) There exist symmetric positive-definite matrices

Q1,Q2, . . . ,QN2 IRr�r and general matrices

F,G2 IRr�r, X1,X32 IRr�(n�r) and X22

IR(n� r)� (n� r) such that

He SAiTF 0

X3 X2

" #( )

þqQi þ ð1� qÞQj 0

0 0

" #8>>>>><>>>>>:

9>>>>>=>>>>>;

SAiTG

X1

" #þ

FT

0

" #

ð?Þ GþGT�Qi

2666666664

3777777775

40 8ði, j Þ 2 I � I : ð15Þ

Proof

Condition (i): As the matrices Qi are positive-definte,

any polytopic combination of these matrices is also

positive-definite which shows the positive-definiteness

of Q(�). Multiplying inequalities (14) by �i�j and

summing for i from 1 to N and for j from 1 to N we

obtain

HeF X3

0 X2

� SAð�ÞT

� �

þQð�Þ 0

0 0

� 8>>><>>>:

9>>>=>>>;

ð?Þ

G X1

� �SAð�ÞTþ FT 0

� � GþGT��qQð�Þ

þ ð1� qÞQþð�Þ�

( )

26666666664

37777777775

40, ð16Þ

where Qð�Þ ¼Pi¼N

i¼1 �iPi and Qþð�Þ ¼Pj¼N

j¼1 �jPj.

Inequality (16) can be rewritten as

Using similar arguments to the ones presented in

the proof of Theorem 4.1, this last inequality implies

the invertibility of matrix A22(�) and

He�F ~Að�Þ

þQð�Þ ð?Þ

G ~Að�Þ þ FTGþ GT �

�qQð�Þ

þ ð1� qÞQþð�Þ�

( )2664

37754 0:

Left- and right-multiplying this inequality by

Ir � ~ATð�Þ� �

and its transpose, respectively, implies

that Q(�)� ~AT(�)((qQ(�)þ (1� q)Qþ(�)) ~A(�)4 0

He

FA11ð�Þ þ X3A21ð�Þ þQð�Þ=2 FA12ð�Þ þ X3A22ð�Þ 0

X2A21ð�Þ X2A22ð�Þ 0

GA11ð�Þ þ X1A21ð�Þ þ FT GA12ð�Þ þ X1A22ð�Þ G��qQð�Þ þ ð1� qÞQþð�Þ

�=2

264

375

8><>:

9>=>;4 0:

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which guarantees the stability of ~A(�) whether the

parameters are constant or time-varying.

Condition (ii): Following similar arguments to the

ones of the proof of condition (i), inequalities (15)

imply the invertibility of A22(�) and

He�

~Að�ÞF þ qQð�Þ

þ ð1� qÞQþð�Þ

( )~Að�ÞGþ FT

ð?Þ Gþ GT �Qð�Þ

2664

37754 0:

Left- and right-multiplying this inequality by

Ir � ~ATð�Þ� �

and its transpose, respectively, we

deduce the stability of ~A(�) by applying Schur

complement. œ

Remark 5: Note that in the case of standard state-

space systems i.e. E¼ In, when the parameters are time-

invariant, Theorem 4.3 covers the results presented by

de Oliveira et al. (1999), by fixing F¼ 0, and also those

presented by Peaucelle et al. (2000).

Remark 6: Theorem 4.3 presents two alternative sets

of LMI conditions for poly-quadratic admissibility

analysis which are not dual to each other. In the case of

robust design, unlike the case of LPV synthesis, as

mentioned in the introduction, it has been shown by

Blanchini and Miani (2003, 2010) that the duality

between state estimation and state-feedback control

does not hold. Our Theorem 4.3 tackles this fact by

proposing separate sets of admissibility analysis con-

ditions exploitable for each case: the first one can be

exploited for state estimation purposes while the

second one for state-feedback control. Note that this

has been achieved by considering a dilation only on the

dynamical part of the system.

The conditions of Theorem 4.3 involve the trans-

formation matrices S and T. As these matrices are not

unique, we show in the following that the feasibility of

(14) or (15) is independent of the particular realisation

of these matrices.

Theorem 4.4: If conditions (14) or (15) of Theorem 4.3

are feasible for a pair (S,T ) satisfying (5) then they are

also feasible for any pair (S,T) satisfying (5).

Proof: Since the pairs (S,T ) and (S,T) satisfy (5),

then

SET ¼ SET ¼Ir 0

0 0

" #which implies that

Ir 0

0 0

" #¼ SS

�1Ir 0

0 0

" #T�1T:

ð17Þ

By denoting

SS�1¼S11 S12

S21 S22

" #and T

�1T ¼T 11 T 12

0 T 22

" #

sinceT andT are upper block-triangular, it follows from

equality (17) that S11T 11¼ Ir, S11T 12¼ 0, S21T 11¼ 0

and S21T 12¼ 0. As the invertibility of T 11 is

guaranteed by the invertibility of T�1T, we obtain that

S11 ¼ T�111 , S21 ¼ 0 and T 12 ¼ 0: ð18Þ

From these equations, we deduce that:

SAiT ¼ SS�1

SAiTT�1T

¼T�111 S12

0 S22

" #SAiT

T 11 0

0 T 22

" #: ð19Þ

Assume that conditions (14) are feasible for the pair

(S,T ). Denote Q1,Q2, . . . ,QN, F,G, X1, X3 and X2 the

solutions of these LMI conditions. Then, by pre- and

post-multiplying inequalities (14) by

B diagT�T11 0

0 T�T22

" #, T �T11

!

and its transpose, respectively, and using relations

(18)–(19), we obtain

HeF X3

0 X2

" #SAiT

( )þ

Qi 0

0 0

" #ð?Þ

G X1

� �SAiTþ F

T 0� � GþG

T� qQi

þð1� qÞQj

( )2666664

3777775

40 8ði, j Þ 2 I �I ð20Þ

where F ¼ T�T11 F T �111 , G ¼ T

�T11 GT

�111 , Qi ¼

T�T11 QiT

�111 , X1 ¼ T

�T11 ðGS12 þ X1S22Þ, X3 ¼ T

�T11 �

ðFS12 þ X3S22Þ and X2 ¼ T�T22 X2S22. Hence, it follows

that (14) are also feasible for the pair (S,T).We can prove in a similar way that the feasibility of

(15) for a pair (S,T ) satisfying (5) implies its feasibility

for any pair (S,T) satisfying (5). œ

Based on Theorem 4.3 and assuming that

the parameters are uncertain, we present a robust

poly-quadratic causal stabilisability condition for

system (1).

Corollary 4.5 (Robust state-feedback control): The

system is robustly, poly-quadratically and causally

stabilisable by state-feedback if there exist symmetric

positive-definite matrices Qi2 IRr�r for i2 {1, . . . ,N}, a

scalar � and general matrices G2 IRr�r, X12 IR(n� r)�r,

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X22 IR(n� r)� (n� r) and R2 IRm�n such that the follow-

ing conditions hold:

He

( SAiT

G 0

X1 X2

" #

þSBiR

!�Ir 0

0 In�r

" #)

þqQiþð1� qÞQj 0

0 0

" #

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

ð?Þ

SAiT

G

X1

" #þSBiR

Ir

0

" #�T

þ� G 0� �

8>><>>:

9>>=>>; GþGT�Qi

266666666666666666664

377777777777777777775

40

8ði, j Þ 2 I �I : ð21Þ

Remark 7: The inequality conditions (21) ofCorollary 4.5 are not linear in the unknown variablesbut they become linear whenever the scalar � is fixeda priori. Since � is a scalar, a solution to these inequalityconditions can be obtained by iteratively increasing thevalue of � between predefined minimum and maximumvalues while solving for the other unknown variables.Note that �¼ 0 corresponds to the choice F¼ 0 andX3¼ 0 in Theorem 4.3 while �¼ 1 corresponds to F¼Gand X3¼X1 in Theorem 4.3. Introducing the scalar �may be seen as a trade-off between these two cases.

When the solution X2 obtained by solving condi-tions (21) is nonsingular, the descriptor state-feedbackgain is given by

K ¼ RG 0

X1 X2

� �1T�1:

When X2 is singular, the solutions Qi, G, X1, X2 and Rof (21) are used in order to obtain a scalar �2 (0, 1)such that (21) holds when X2 is replaced by X2þ �In�r.Note that such a � always exists. Therefore, thefeedback gain is given by

K ¼ RG 0

X1 X2 þ �In�r

� �1T�1:

When the parameters are not uncertain and areassumed measurable, we can exploit the parametersmeasurability and design a polytopic state-feedbackcontroller such that the closed-loop system is poly-quadratically admissible. This, first, requires thetransformation of the polytopic system (1) into apolytopic system with a constant control matrix byintroducing an augmented descriptor state vector (seeMasubuchi et al. (2003, 2004) for more details). Then,as a second step, we apply the following synthesisconditions.

Corollary 4.6 (Polytopic state-feedback control): The

system is poly-quadratically and causally stabilisable by

polytopic state-feedback if there exist symmetric posi-tive-definite matrices Qi2 IR

r�r for i2I , a scalar � andgeneral matrices G2 IRr�r, X12 IR

(n� r)�r, X22

IR(n� r)� (n� r) and Ri2 IRm�n for i2I such that the

following conditions hold:

He

�SAiT

G 0

X1 X2

�

þSBRi

��Ir 0

0 In�r

� �

þqQiþð1� qÞQj 0

0 0

�

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

ð?Þ

SAiT

G

X1

� þSBRi

Ir

0

� �T

þ� G 0� �

8><>:

9>=>; GþGT�Qi

266666666666666664

377777777777777775

40

8ði, j Þ 2 I �I : ð22Þ

When the solution X2 obtained by solving the LMIconditions (22) is nonsingular, the descriptor state-

feedback gains are given by

Ki ¼ Ri

G 0

X1 X2

� �1T�1:

When X2 is singular, the solutions Qi, G, X1, X2 and Ri

of (22) are used in order to obtain a scalar �2 (0, 1) suchthat (22) holds when X2 is replaced by X2þ �In�r.Therefore, the feedback gains are given by

Ki ¼ Ri

G 0

X1 X2 þ �In�r

� �1T�1:

Remark 8: Some results dealing with the robust

analysis of uncertain polytopic discrete-time descriptorsystems have been reported by Fang (2002) and Kuo

and Fang (2003) These results are applicable only forsystems with time-invariant uncertainties. To our

knowledge, there are no results reported in the

literature in the case of time-varying polytopicuncertainties (except the ones proposed by Chadli

et al. (2008a,b), which have been shown to beerroneous in Section 3). Fang (2002) employed

PDLF in order to obtain less conservative robustadmissibility analysis conditions which are formulated

as a non-strict LMI feasibility problem. Note that therobust control design problem has not been addressed

and the proposed analysis conditions are very difficultto apply for control design. Indeed, due to numerous

couplings between the Lyapunov matrices and thevertices of the system state matrix polytope, these

analysis conditions lead, when employed for control

purposes, to trilinear inequalities which are very

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difficult to linearise. Kuo and Fang (2003) presented

an admissibility analysis condition, also proposed byLee et al. (2004) for LTI systems, that involves two

slack variables. Unfortunately, this condition is basedon a common Lyapunov function for the wholeuncertainty domain and therefore, it is conservative.

Moreover, only the robust static output feedbackcontrol has been discussed. Our admissibility analysisresults admit time-varying parameters and use PDLF

as well as two slack variables. These results have beendeveloped with the objective in mind of easilydesigning robust state-feedback controllers. In order

to deal with the absence of duality between state-feedback control and state estimation problems in thecase of uncertain parameters, our approach employs a

dilation only on the dynamical part of the system. As aconsequence, the results proposed in our article can beexploited for robust design as well as LPV synthesis.

5. Numerical examples

5.1 Numerical examples for time-invariantparameters

We consider here that the uncertain parameters aretime-invariant and we provide several numerical

examples in order to evaluate the performance of ouradmissibility analysis results in comparison with theones proposed by Fang (2002) and Kuo and Fang

(2003). These numerical examples were randomlygenerated for different values of N (number ofvertices), n (dimension of the state vector) and r

(number of dynamical state variables). In the follow-ing, we investigate the admissibility of the convex hull

Co(�(A1, . . . ,AN)) for different values of � with thepurpose of evaluating the admissibility domain guar-anteed by the methods in Fang (2002) and Kuo and

Fang (2003) and our Theorem 4.3(i)–(ii).First, consider a polytopic uncertain descriptor

system with N¼ 2 vertex matrices

A1 ¼

0:2594 0:0018 0:0590 �0:1570 0:0102

0:2595 �0:0292 0:1574 0:1396 0:1095

0:0121 �0:1374 0:0562 �0:0293 0:0628

0:0926 �0:1718 �0:1722 0:1146 0:0722

0:0027 �0:1101 �0:0296 0:0729 0:0542

26666664

37777775

and

A2 ¼

�0:3612 �0:2867 0:3450 0:3837 0:5281

0:3114 �0:4251 �0:1279 �0:2867 �0:1527

0:4565 �0:1678 �0:1794 0:2067 �0:0681

�0:3353 0:4063 0:0141 0:6519 0:1313

�0:0218 0:7138 �0:1451 0:0992 0:4533

26666664

37777775

and E¼Bdiag(I3, 0). Based on the root-locus of the

finite modes of the descriptor system, we identified the

largest possible admissibility domain which is given by

�RL¼ 1.999. The method in Fang (2002), exploiting

PDLFs, succeeded for the maximum bound �¼ 1.703

while the method in Kuo and Fang (2003), involving

only slack variables, succeeded for all �� 1.068. Our

Theorem 4.3(i) and (ii) guarantees the admissibility of

the system for the maximum bound �¼ 1.827 and

�¼ 1.801, respectively.Second, consider a polytopic uncertain descriptor

system defined by the following N¼ 3 vertex

matrices Ai:

A1 ¼

0:0248 0:2201 �0:1472 �0:2277 0:0387

0:2275 �0:0767 0:0767 �0:0626 �0:0440

0:0442 0:0466 0:0938 �0:1470 0:0949

0:0223 0:1898 0:0151 �0:1052 �0:2549

�0:0125 �0:2178 0:0815 �0:2472 0:1321

266666664

377777775,

A2 ¼

0:3567 0:3709 0:7896 �0:3370 1:1710

0:1948 0:3811 0:1587 0:0232 0:5549

�0:5563 �0:1224 0:4842 �0:1492 0:1469

0:3336 0:8812 �0:5565 0:6562 0:0446

1:1669 �0:3536 0:0632 0:2750 0:9339

266666664

377777775

and

A3 ¼

0:1949 0:1114 0:0837 �0:3888 �0:0733

�0:0248 �0:4704 0:3497 0:0521 0:1290

�0:0252 �0:1563 �0:2602 0:2649 0:3457

0:0482 �0:0678 0:0674 �0:1122 �0:2558

0:0017 0:2108 �0:0577 0:0213 0:1153

266666664

377777775:

The singular matrix E is B diag(I4, 0). Based on the

root-locus of the finite modes of the descriptor system,

we were able to find the largest value �RL¼ 1.399 for

which the system is admissible. The method proposed

by Kuo and Fang (2003) provided the maximum bound

�¼ 0.923 while the application of the result of Fang

(2003) using PDLFs gave �¼ 1.293. The implementa-

tion of LMI conditions of Theorem 4.3(i)–(ii) provided

�¼ 1.360 and �¼ 1.367, respectively.Third, consider the descriptor system defined by

N¼ 4 vertices

A1 ¼

0:3628 �0:0255 �0:0081

�0:0599 0:1126 0:1863

0:3583 0:0902 0:1055

264

375,

A2 ¼

�0:2347 �0:1528 �0:1482

�0:4586 0:1375 0:1526

0:0819 0:2396 0:0240

264

375,

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A3 ¼

�0:0462 0:0966 0:2321

0:1239 0:1246 0:1067

�0:0407 0:1644 0:0826

264

375,

A4 ¼

�0:0408 �0:0675 0:0981

�0:0137 0:0476 0:0718

�0:0730 0:0314 0:0262

264

375

and E¼Bdiag(I1, 0). Kuo and Fang (2003) and Fang

(2002) provided the maximum bound �¼ 0.935 and

�¼ 1.107, respectively. Theorem 4.3(i) and (ii) suc-

ceeded for all �� 1.184. Note that for this example, our

analysis results identified the largest possible admissi-

bility domain which is given by the root-locus of the

finite modes of the descriptor system.All these results are summarised in Table 1. In

addition, in this table, we also indicate the percentage

of the largest possible admissibility radius �RL,obtained by using the root-locus of the finite modes

of the system, identified by each method. From several

tens of tests we have conducted on randomly generated

systems, we can say that, in the case of time-invariant

parameters, our admissibility analysis conditions are

less restrictive than the ones proposed by Kuo and

Fang (2003). For the numerical examples presented

above, our admissibility analysis results performed

better than the one proposed by Fang (2002). We can

see that our approach improved the relative admissi-

bility radius by about 5–6.5% compared to the one of

Fang (2002). However, there are also situations, as the

one presented in the next example (Section 5.2), where

our conditions and the result of Fang (2002) provided

the same admissibility domain as well as situations

where Fang (2002) provided a larger admissibility

radius than our approach. In this latter case the

improvement is quite comparable with the improve-

ment provided by our approach for the above

numerical examples. Therefore, we can say that our

admissibility analysis results and the ones of Fang(2002) are alternative results. Note also that anotheradvantage of the admissibility analysis conditionsproposed in our paper, compared to the ones ofFang (2002), is that they can be easily exploited forsynthesis purposes. Indeed, employing the admissibilityconditions of Fang (2002) for state-feedback controldesign or state estimation leads to trilinear matrixinequalities which are very difficult to linearise in orderto solve the synthesis problem. This is not the casewhen using our analysis conditions which can be easilyexploited for state estimation or state-feedback controlas shown next.

5.2 Numerical example for time-varying parameters

Consider a parameter-dependent descriptor systemwhose state matrices are

E ¼ BdiagðI3, 0Þ,

Að�Þ ¼

0:5 �0:1� 0:7 1

0 �0:2 0 �2

0 �0:5� 0:5� 0 0

0:3 0 0 1

2666664

3777775 and

Bð�Þ ¼

1

�0:2�

0:3

0:5þ �

2666664

3777775:

We assume that the parameter is bounded j�j � �. Thismeans that the parameter-dependent descriptor systemallows a polytopic representation as in (1) withA1¼A(��), A2¼A(�), B1¼B(��) and B2¼B(�).

When the parameter is time-invariant, based on theroot-locus of the finite modes of the descriptor system,

Table 1. Largest admissibility radius evaluated based on the root locus, Fang (2002) and Kuo and Fang (2003) andTheorem 4.3.

System 1 System 2 System 3

N 2 3 4

Maximum � by root-locus ¼ �RL 1.999 1.399 1.184� maximum based onKuo and Fang (2003)

Absolute value 1.068 0.923 0.935Per cent of �RL 53.4% 65.9% 78.9%

� maximum based onFang (2002)

Absolute value 1.703 1.293 1.107Per cent of �RL 85.1% 92.4% 93.5%

� maximum based onTheorem 4.3(i)

Absolute value 1.827 1.360 1.184Per cent of �RL 91.4% 97.2% 100%

� maximum based onTheorem 4.3(ii)

Absolute value 1.801 1.367 1.184Per cent of �RL 90.1% 97.7% 100%

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the largest admissibility region corresponds to

�¼ 3.3604. The method proposed by Kuo and Fang

(2003) provided the maximum bound �¼ 3.2519 while

the application of the result of Fang (2002) and our

Theorem 4.3(i)–(ii) provided the maximum bound

�¼ 3.3601. For this system, our method and the one

in Fang (2002) identified the same admissibility region.

However, unlike the result of Fang (2002) whose

application to state-feedback control leads to trilinear

matrix inequalities, our analysis result is easily exploi-

table for control design purposes as shown in the

following. When the parameters are time-varying, the

root-locus of finite modes cannot be used anymore in

order to evaluate the maximum value of � guaranteeingthe stability. Our conditions of Theorem 4.3(i)–(ii)

guarantee a maximum bound �¼ 3.307.Assuming the parameters are time-varying and

uncertain, a robust state-feedback controller can be

designed using Corollary 4.5. The maximum value of �for which the system is robustly, poly-quadratically

and causally stabilisable by state-feedback based on

Corollary 4.5 is 3.5707 and it has been obtained for

�¼ 0.1.When the parameters are not uncertain but

measurable, a parameter-dependent state-feedback

controller may be obtained by applying Corollary 4.6.

In order to do so, we transform the system into a

descriptor system with a constant control matrix.

By introducing the additional variables 1¼ u and

2¼ �u¼ �1, we obtain an augmented descriptor state

vector ~x ¼ xT 1 2� �T

. Therefore, the parameter-

dependent system is equivalent to the augmented

descriptor system defined by the following matrices:

~E ¼E 0

0 0

� , ~Að�Þ ¼

Að�Þ 0 B01

0 1 0

0 �� 1

264

375 and

~B ¼

B00

�1

0

264

375,

where B00 and B01 are such that B(�)¼B00þ �B01.~A(�) evolves into a polytope whose vertices are ~A(��)and ~A(�). The maximum value of � for which the

conditions of Corollary 4.6 are feasible is 3.5772 and it

has been obtained for the same �¼ 0.1. For this

maximum value of �, the vertices of the polytopic

controller gain are

K1 ¼ �0:2873 1:6666 �0:0027 �0:5896�124:6454 �0:7939

�,

K2 ¼ �0:3001 2:0224 �0:0333 �0:7485�118:4508 �0:7496

�,

and the polytopic parameters are �1 ¼���2� and

�2 ¼�þ�2� . This provides the parameter-dependent gain

Kð�Þ ¼ ½�0:2937�0:0018� 1:8445þ0:0497�

�0:0180�0:0043� �0:6690�0:0222�

121:5481�0:8658� �0:7717þ0:0062� �

for the state-feedback control of the augmenteddescriptor system. Since the control law uðkÞ ¼Kð�Þ ~xðkÞ uses the augmented descriptor vector, wecan eliminate the additional descriptor variables 1 and2 in order to obtain an explicit formulation of thecontrol law as a function of the descriptor statevector x. This provides

uðkÞ ¼1

�0:0062�2 þ 1:6376�� 120:5481

� ½�0:2937� 0:0018� 1:8445þ 0:0497�

�0:0180� 0:0043� �0:6690� 0:0222� �xðkÞ:

6. Conclusion

In this article, the problem of admissibility analysis andstate-feedback control for polytopic discrete-timedescriptor systems has been addressed. Using a slackvariables approach, we have proposed a new necessaryand sufficient condition for the admissibility analysis ofdiscrete-time LTI descriptor systems. This condition isexpressed as a strict LMI feasibility problem and leadsto a linear necessary and sufficient condition for thestate-feedback control design. This new condition hasbeen extended to poly-quadratic admissibility, whichinvolves PDLFs, of polytopic systems with possiblytime-varying parameters. Considering a dilation onlyon the dynamical part of the descriptor system allowedus to tackle the absence of duality between stateestimation and state-feedback control in the case ofuncertain parameters and to propose separate admis-sibility analysis conditions exploitable for each case.The application of these analysis results to the robuststate-feedback control and polytopic state-feedbackcontrol has also been presented in this article. The mainadvantage of the approach proposed in this article is thedevelopment of analysis results which are not morerestrictive than the results already proposed in theliterature for the time-invariant case as supported byour numerical examples and, especially, are easilyexploitable for robust design as well as LPV synthesis.

Acknowledgements

The author thanks the Associate Editor, Professor DavidWilson, who handled this article and the anonymous

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reviewers for their valuable and helpful comments andsuggestions for improving the quality of this article.

Note

1. A function V(x) is decrescent if there exists a positiveconstant � such that V(x)5��xTx.

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Appendix A: Complement on matrix manipulation

In the following lemmas, we state some results on matrixmanipulation.

Lemma A.1: Let M be a general matrix partitioned asA BC D

� �. If M is negative-definite (not necessarily sym-

metric) i.e. He{M}5 0 then D is invertible and theSchur complement of M relative to D is also negative-definite (not necessarily symmetric) i.e. He{A�BD�1C}5 0.

Proof: If He{M}5 0 then, by Schur complement,He{D}5 0 which implies by Lemma A.2 that D is invertible.The proof of the positive-definiteness of the Schur comple-ment of M relative to D can be found in Xu and Yang(2000). œ

Lemma A.2: Let X and Y be real square matrices. IfHe{XY}5 0 then X and Y are invertible.

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