Delay-dependent stability analysis of singular linearcontinuous-time system

E.K. Boukas and Z.K. Liu

Abstract: The paper deals with the class of singular linear continuous systems. By decomposingthe singular linear system with time-delay into slow and fast subsystems, linear matrix inequality(LMI)-based delay-dependent stability and stabilisation conditions are established, and an LMI-based algorithm to design a memoryless state feedback control that stabilises the system isprovided. Two numerical examples are solved to show the usefulness and validness of thetheoretical results.

1 Introduction

Time delay arises in many industrial processes in the steelindustry, oil industry etc. It has been shown that delays inthe system are the primary cause of instability andperformance degradation.

The stability of certain and uncertain linear systems withtime-delay has retained the attention of many researchersand many results have been reported in the literature. Delay-independent and delay-dependent results have been estab-lished. The delay-independent results guarantee stability forany delay in the system, which makes them conservative,while the delay dependent results guarantee the stability ofthe considered systems only in a limited interval for thesystem delay. Among these contributions, we refer thereader to [1–3] and references therein. Singular systems canbe used to describe many engineering systems. During thepast decades, considerable attention has been devoted to theanalysis and synthesis of linear singular systems (see [3–9]and the references therein). However, to the best of ourknowledge, the stability of singular continuous-time linearsystems with time delay has received little attention so far.Our goal in this paper is to deal with the stability of singularcontinuous-time linear systems with time delay in the statevector. We will try to develop delay-dependent sufficientconditions under which the system is stable. We will alsodevelop a stabilising state feedback controller for this classof systems. The methodology we will use is mainly based onthe Lyapunov functional theory. All the sufficient conditionswe will develop will be easily solved by using the linearmatrix inequality (LMI) Matlab toolbox.

2 Problem statement

Consider a singular linear system with time delaydescribed by

E_xxt ¼ Axt þ Adxt�� þ But ð1Þ

xt ¼ �ðtÞ; 0 � t � �� ð2Þ

where xt 2 Rn and ut 2 R

m represent the state and thecontrol input of the system, respectively, �ðtÞ 2 Cð½0; 1Þ isthe initial condition, E;A;Ad 2 R

nn and B 2 Rnm are given

constant matrices, and t denotes the time-delay in the state.

Remark 2.1: The matrix E is supposed to be singular whichmakes the dynamics (1) different from the one usually usedto describe the behaviour of the time-invariant dynamicalsystems with time delay.

Definition 2.1: For any given two matrice E;A 2 Rnn;

the pencil (E, A) is called regular if there exists aconstant � 2 C such that j�E þ Aj 6¼ 0; or polynomialj�E � Aj 6¼ 0:

From [3], we know that there exist nonsingular matricesP, Q such that

QEP ¼I 0

0 N

� �; QAP ¼

A1 0

0 I

� �where N 2 R

n2n2 is a nilpotent and A1 2 Rn1n1 with

n1 þ n2 ¼ n:

Remark 2.2: For given matrice (E, A), the transformationmatrice P, Q are unique in the sense of similarity (see [3])

The goal of this paper is to (i) develop LMI-based sufficientconditions for stability of system (1) with ut 0; (ii) designa state feedback controller of the form

ut ¼ Kxt ð3Þ

with K a constant matrix that stabilises the system.In this paper, we assume that system (1) is regular, i.e. the

pair ðE;A þ Ade�s� Þ is regular. This assumption guaranteesthe existence and the uniqueness of the solution for system(1). Moreover, we assume that the system is impulse free,which ensures that the delayed system has no infinite poles.Finally, we will suppose that the state is available forfeedback.

3 Stability

This Section addresses the stability problem of system (1)with uðtÞ 0: For the given system parameters E, A, Ad;

q IEE, 2003

IEE Proceedings online no. 20030635

doi: 10.1049/ip-cta:20030635

This work is supported by the Natural Sciences and Engineering ResearchCouncil of Canada under grants OGP0036444.The authors are with the Mechanical Engineering Department, EcolePolytechnique de Montreal, P.O. Box 6079, station ‘Centre-ville’,Montreal, Quebec, Canada, H3C 3A7

Paper received 17th January 2003

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003 325

using Jordan canonical form decomposition, we concludethat there exist nonsingular matrices Q, P such that

QEP ¼I 0

0 N

� �

where N is a nilpotent.Let us define a transformation of the state vector as

follows:

xt ¼ Px1;t

x2;t

� �ð4Þ

where x1;t 2 Rn1 ; x2;t 2 R

n2 : Then we obtain the followingequivalent system:

I 00 N

� �_xx1;t

_xx2;t

� �¼ QAP

x1;t

x2;t

� �þ QAdP

x1;t��

x2;t��

� �ð5Þ

The components x1;t; x2;t are called the slow and fast statesof the system (1).

Throughout this paper, we will assume that the slowand fast states satisfy the following assumption:

Assumption 3.1: There exists a symmetric matrix M suchthat the fast and slow states of the system satisfy thefollowing:

kx2;tk2 � x>

1;tMx1;t ð6Þ

Remark 3.1: When M ¼ h is a constant scalar, then (6)becomes:

kx2;tk2 � hkx1;tk

2 ð7Þ

Hence, (6) gives a relation between the slow states and thefast states. As M can be chosen arbitrarily, this does notmean a strong restriction.

Before providing the main results of this Section, let usgive a lemma on the S-procedure for quadratic forms(see [2, 9]).

Lemma 3.1: Let T0; T1 be symmetric matrices, then

j>T0j > 0; 8j 6¼ 0; s:t: j>T1j � 0

holds if and only if there exist w � 0 such that

T0 � wT1 > 0

The following theorem states the result on stability forthe class of systems considered in this paper.

Theorem 3.1: If there exist symmetric and positive-definite matrice X1 2 R

nn; X2 2 Rn2n2 ; R1;R2 2 R

n1n1

and a positive scalar w such that

Q¼

J1 E>X1AdPI1 E>X1AdPI>0 E>X1AdPI>0

I>1 P>A>d X1E �X2 0 0

I0P>A>d X1E 0 � 1

� R1 0

I0P>A>d X1E 0 0 � 1

� R2

0BBBBBB@

1CCCCCCA

< 0

ð8Þ

where

J1 ¼ E>X1A þ A>X1E þ E>X1AdPI>0 I0P�1

þ ½P�1>I>0 I0P>A>d X1E þ �A>Q>I>0 R1I0QA

þ �A>d Q>I>0 R2I0QAd þ ½P�1>

0 0

0 X2

!P�1

þ w½P�1> I>0 MI0 � I1I>1

h iP�1;

I0 ¼ I 0 �

;

I1 ¼0

I

!

holds for some symmetric and positive semidefinitematrix M � 0; then system (1) with uðtÞ 0 is asympto-tically stable.

Proof: See Appendix Section 8.

Remark 3.2: Note that Y is increasing with respect to �in the positive-definite sense, i.e. if inequality Y < 0holds for �1; then it will remain valid for any time delay� less than �1; i.e. 0 < � < �1: Thus, the upper bound ofthe time delay for which the system (1) remains stablecan be obtained by one-dimensional search or by solvinga generalized eigenvalue problem.

Example 3.1: To show the usefulness of the results of thistheorem, let us consider a singular system described by(1) with uðtÞ 0 with the following parameters:

E ¼

1 0 0 0

0 1 0 0

0 0 0 1

0 0 0 0

0BBBB@

1CCCCA; A ¼

�3 0 0 0:2

0 �4 0:1 0

0 0 �0:1 0

0:1 0:1 �0:2 �0:2

0BBBB@

1CCCCA

Ad ¼

�0:5 0 0 0

0 �1 0 0

0 0:1 �0:2 0

0 0 0 0

0BBBB@

1CCCCA

Noting that the elements in the last row of E and Ad are zero,we have

0:1 0:1 �

x1;t ¼ 0:2 0:2 �

x2;t

from which it follows that

1 1

1 1

� �kx1;tk

2 ¼ 41 1

1 1

� �kx2;tk

2

This means that assumption 3.1 holds with h ¼ 1=4:Moreover, matrice E þ A and E � A are of full rank, so(E, A) is regular.

Let M ¼ 0:25I and using these data, by one-dimensionalsearch technique we obtain an upper bound, 0.6633, forthe time-delay for the LMI (8) to remain feasible. With� ¼ 1:150, solving (8) gives the following solution:

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003326

X1 ¼

37:0212 �0:6967 �2:9142 3:3157

�0:6967 0:5907 1:3751 �10:0422

�2:9142 1:3751 8:2161 11:9149

3:3157 �10:0422 11:9149 424:1673

0BBBB@

1CCCCA

X2 ¼1:2338 0:6719

0:6719 2:5290

!; R1 ¼

17:3934 �0:1641

�0:1641 0:1120

!

R2 ¼51:1653 �0:3012

�0:3012 0:4479

!

Therefore, according to theorem 3.1 the system under studyis stable for all � 2 ½0; 1:15:

4 Delay-dependent stabilisation

This Section deals with the design of a controller of the form(3) that stabilises system (1). Substituting (3) into (1) yieldsthe following closed-loop dynamical system:

E_xxt ¼ �AAxt þ Adxt�� ð9Þ

where �AA ¼ A þ BK:Employing Theorem 3.1 we get the following result.

Proposition 4.1: Under Assumption 3.1, for a givencontroller of the form (3), if there exist symmetric andpositive-definite matrices X1 2 R

nn; X2 2 Rn2n2 ; R1; R2

2 Rn1n1 such that

�QQ < 0 ð10Þ

holds, where �QQ is obtained from Q by replacing A with �AA;then the closed-loop system is stable under control (3).

Proof: The proof of this proposition follows the samelines as the one of theorem 3.1, and so the details areomitted. A

As K is a design matrix, �QQ is nonlinear in the designparameters X1 and K, thus in this case (10) cannot besolved directly by using LMI toolbox. To cast theproblem of designing a stabilising controller (3) into theLMI formulation, let us define U1 ¼ R�1

1 ;U2 ¼ R�12 :

Then, pre- and post-multiplying both sides of (10) bydiag{I, U1; U2} yields

�JJ1 E>X1AdPI1 E>X1AdPI>0 U1 E>X1AdPI>0 U2

I>1 P>A>d X1E �X2 0 0

U1P>A>d X1E 0 �1

�U1 0

U1P>A>d X1E 0 0 �1

�U2

0BBBBB@

1CCCCCA

< 0 ð11Þ

holds, where

�JJ1 ¼ E>X1�AA þ �AA

>X1E þ E>X1AdPI>0 I0P�1

þ ½P�1>I>0 I0P>A>d X1E þ � �AA

>Q>I>0 U�1

1 I0Q �AA

þ �A>d Q>I>0 U�1

2 I0QAd þ ½P�1>0 0

0 X2

!P�1

þ w½P�1> I>0 MI0 � I1I>1

h iP�1

Let

J ¼ E>X1�AA þ �AA

>X1E þ E>X1AdP

I 0

0 0

!P�1

þ ½P>�1 I 0

0 0

!P>A>

d X1E þ ½P�1>0 0

0 X2

!P�1

þ w½P�1> I>0 MI0 � I1I>1

h iP�1

then

J1 ¼ J þ �AA>Q>I>0U1

�

� ��1

I0Q �AA þ �A>d Q>I>0

U2

�

� ��1

I0QAd

ð12Þ

Therefore, using Schur complement, (11) is equivalent to

�YY1 ¼

J E>X1AdPI1 D

I>1 P>A>d X1E �X2 0

D> 0 �1�U

0BB@

1CCA < 0 ð13Þ

where

D¼ E>X1AdPI>0 U2 E>X1AdPI>0 U1 A>d Q>I>0 �AA>Q>I>0

� �U¼diagfU2;U1;U2;U1g

From this discussion, we obtain the following theorem.

Theorem 4.1: If there exist matrices Z1 > 0; Z2 > 0; W,and scalar > 0 such that the following LMIs hold:

Z1E> ¼ EZ1; ð14Þ

JJ EAdPI1Z2 D2 Z1½P�1>I>0 ½M1=2> Z1½P

�1>I1

I>1 P>A>d E>X�1

2 �Z2 0 0 0

D>2 0 �1

�U 0 0

M1=2I0P�1Z1 0 0 �I 0

I>1 P�1Z1 0 0 0 �Z2

0BBBBBBBBBB@

1CCCCCCCCCCA

< 0 ð15Þ

where

JJ ¼E½AZ1 þBWþ ½AZ1 þBW>E>þEAdPI 0

0 0

!P�1Z1

þZ1½P>�1 I 0

0 0

!P>A>

d E>

�Z1½P�1>I1I>1 � I1I>1 P�1Z1 þI;

ð16Þ

D2 ¼

EAdPI>0 U2 EAdPI>0 U1 Z1A>d Q>I>0 ½AZ1 þ BW>Q>I>0

� �ð17Þ

then controller (3) with K ¼ WZ�11 stabilises system (1).

Proof: To verify the stability of the system if suffices tocheck whether (13) is satisfied. Suppose

X1E ¼ E>X1 ð18Þ

Then, �YY1 becomes

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003 327

J2 X1EAdPI1 D1

I>1 P>A>d E>X1 �X2 0

D>1 0 �1

�U

0B@

1CA < 0

ð19Þ

where

J2 ¼ X1E �AA þ �AA>E>X1 þ X1EAdPI 0

0 0

!P�1

þ ½P>�1 I 0

0 0

!P>A>

d E>X1 þ ½P�1>0 0

0 X2

!P�1

þ w½P�1>½I>0 MI0 � I1I>1 P�1

D1 ¼ X1EAdPI>0 U2 X1EAdPI>0 U1 A>d Q>I>0 �AA>Q>I>0

� �Pre- and post-multiplying (19) by diagfX�1

1 ;X�12 ; I; I; I; Ig

yields

X�11 J2X�1

1 EAdPI1X�12 X�1

1 D1

I>1 P>A>d E>X�1

2 �X�12 0

D>1 X�1

1 0 �1�U

0BB@

1CCA < 0

ð20Þ

Note that

X�11 J2X�1

1 ¼E �AAX�11 þX�1

1�AA>E>þEAdP

I 0

0 0

!P�1X�1

1

þX�11 ½P>�1 I 0

0 0

!P>A>

d E>þX�11 ½P�1>I1X2I>1 P�1X�1

1

þwX�11 ½P�1>½I>0 MI0 � I1I>1 P�1X�1

1

Letting Z1 ¼ X�11 ; Z2 ¼ X�1

2 ;W ¼ KX�11 ; ¼ 1=w and using

Schur complement, (20) is equivalent to

J3 EAdPI1Z2 D2 Z1½P�1>I>0 ½M1=2> Z1½P

�1>I1

I>1 P>A>d E>X�1

2 �Z2 0 0 0

D>2 0 �1

�U 0 0

M1=2I0P�1Z1 0 0 �I 0

I>1 P�1Z1 0 0 0 �Z2

0BBBBBBB@

1CCCCCCCA

<0

ð21Þ

where

J3 ¼E½AZ1 þBWþ ½AZ1 þBW>E>þEAdPI 0

0 0

!P�1Z1

þZ1½P>�1 I 0

0 0

!P>A>

d E>þZ1½P�1>I1X2I>1 P�1Z1

�1=Z1½P�1>I1I>1 P�1Z1

Nothing that

J3 �E½AZ1 þBWþ ½AZ1 þBW>E>þEAdPI 0

0 0

!P�1Z1

þZ1½P>�1 I 0

0 0

!P>A>

d E>

�Z1½P�1>I1I>1 � I1I>1 P�1Z1 þI ¼ JJ

Therefore, (21) follows from (15). Moreover, (18) isequivalent to (14). From this derivation, we conclude that,

if there exist matrices Z1 > 0; Z2 > 0; W and a scalar > 0satisfying (14) and (15), then X1 ¼ Z�1

1 ;X2 ¼ Z�12 ;

K ¼ WZ�11 and the scalar w ¼ 1= satisfy (13). This

completes the proof of theorem 4.1. A

Theorem 4.1 provides a method for designing a controllerunder the assumption that the delay in the system is known.In fact, in case of unknown constant delay, a controller thatstabilises the system and maximises the time delay can besolved using LMI technique.

The problem of maximising � can be formulated as

maxZ1>0; Z2>0; W>0; >0

�

s:t: ð14Þ; ð15Þð22Þ

Let v ¼ 1=�; then (22) can be cast into a standardgeneralised eigenvalue problem. For this purpose, let usintroduce two auxiliary matrices G1 > 0; G2 > 0 and write(15) as

JJ EAdPI1Z2 D2 Z1½P�1>I>0 ½M1=2> Z1P�1I1

I>1 P>A>d E>X�1

2 �Z2 0 0 0

D>2 0 �G 0 0

M1=2I0P�1Z1 0 0 �I 0

I>1 P�1Z1 0 0 0 �Z2

0BBBBBBBBB@

1CCCCCCCCCA

< 0 ð23Þ

G � vU ð24Þ

where G ¼ diagfG2;G1;G2;G1g: Hence, the optimisationproblem (22) is equivalent to the following generalisedeigenvalue problem:

minZ1>0; Z2>0; W>0; >0; G1>0; G2>0; U1>0; U2>0

v

s:t: ð14Þ; ð23Þ; ð24Þð25Þ

which can be solved by using LMI toolbox.

5 Numerical example

To show the usefulness of the results of the precedingSection, let us consider a singular system described by thedynamics of the form (1), with the following data:

E ¼

1 0 0 0

0 1 0 0

0 0 1 1

0 0 0 0

0BBBBB@

1CCCCCA; A ¼

1:2 0:1 0 0

0 1:1 0:1 0:1

0:1 0 �0:1 0:1

0 0 0:1 0:4

0BBBBB@

1CCCCCA

Ad ¼

0:1 0 0:1 0

0 0:1 �0:1 0

�0:1 0 0 0:1

0 0 �0:1 �0:1

0BBBBB@

1CCCCCA; � ¼ 6:41;

B ¼

0:1 0 0 0

0 0:1 0 0

0 0 0:2 0

0 0 0 0:2

0BBBBB@

1CCCCCA

With these sets of data, solving feasible problem (14) and(15) yields the following solution:

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003328

¼ 573:07458; U1 ¼1192:3728 �0:0349452

�0:0349452 1348:2848

!

U2 ¼1204:1353 �6:0125363

�6:0125363 1361:3943

!;

W ¼

�2913:1 �210:5 42:4 1:5

8 �2674:5 �476:6 �4:7

715:5 0 336:4 �522

0 10:0 730:8 �730:8

0BBBB@

1CCCCA

Z2 ¼559:074 0:1177

0:1177 558:907

!;

Z1 ¼

212:65 �0:0336 �6:0476 0

�0:0337 212:6074 2:4072 0

�6:0476 2:4072 680:7541 �231:6453

0 0 �231:6453 231:645

0BBBB@

1CCCCA

K ¼ WZ�11 ¼

�13:701 �0:991 �0:0814 �0:0750

0:0073 �12:568 �1:0041 �1:0241

3:3541 0:005 �0:3681 �2:6216

0 0:0469 0 �0:0002

0BBBB@

1CCCCA

Hence, according to theorem 4.1, controller (3) with gainK given in the preceding text stabilises system (1).

Remark 5.1: Based on the results of the theorems givenrespectively in Section 3 and Section 4, we were able tocheck whether a system is stable and to design a controllerin the state feedback form that stabilises the class of systemwe are considering in this paper. Our results are notconservative because they are delay-dependent and theassumptions we have used are standard ones.

6 Conclusion

This paper considers the stability and stabilisation problemsof singular linear systems with time delay. An LMI-basedsufficient stability condition has been developed and an LMIalgorithm to design a stabilising controller has beenprovided. The numerical example shows that the proposedcontroller design method works very well. The resultsestablished here can be extended to the class of uncertainlinear singular systems.

7 References

1 Boukas, E.K., and Liu, Z.K.: ‘Deterministic and stochastic time-delaysystem’ (Birkhauser Boston, 2002)

2 Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V.: ‘Linear matrixinequalities in system and control’ (SIAM, Philadelphia, 1994)

3 Dai, L.Y.: ‘Singular control systems’. Lecture Notes in Control andInformation Sciences 118 (Springer-Verlag, New York, 1988)

4 Kolmanovskii, V.B., and Richard, J.P.: ‘Stability of some linear systemswith delays’, IEEE Trans. Autom. Control, 1999, 44, (5), pp. 984–988

5 Li, X., and De Souza, C.E.: ‘Delay-dependent robust stability andstabilization of uncertain linear delay systems: A linear matrix inequalityapproach’, IEEE Trans. Autom. Control, 1997, 42, (8), pp. 1144–1148

6 Lin, J.L., and Chen, S.J.: ‘Robustness analysis of uncertain linearsingular systems with output feedback control’, IEEE Trans. Autom.Control, 1999, 44, (10), pp. 1924–1928

7 Liu, X.: ‘Input-output decoupling of linear time-varying singularsystems’, IEEE Trans. Autom. Control, 1999, 44, (5), pp. 1016–1021

8 Shi, P., Boukas, E.K., and Agarwal, R.K.: ‘Control of Markovian jumpdiscrete-time systems with norm bounded uncertainty and unknowndelay’, IEEE Trans. Autom. Control, 1999, 44, (11), pp. 2139–2144

9 Uhlig, F.: ‘A recurring theorem about pairs of quadratic forms andextensions: A Survey’, Linear Algebr. Appl., 1979, 25, pp. 219–237

8 Appendix: Proof of theorem 3.1

Note that

x1;t � x1;t�� ¼Zt

t��_xx1;sds ð26Þ

Substituting (5) into (26) yields

x1;t�� ¼ x1;t �Zt

t��½I0QAxs þ I0QAdxs�� ds ð27Þ

In turn, substituting (27) into (5) produces an equivalentsystem of (5) as follows:

E_xxt ¼Axt þAdPx1;t��

x2;t��

!

¼Axt þAdPx1;t �

Rtt�� ½I0QAxs þ I0QAdxs�� ds

x2;t��

!

¼Axt þAdPI>0 x1;t þAdPI1x2;t��

�Zt

t��AdPI>0 I0Q½Axs þAdxs�� ds ð28Þ

Let us consider the following Lyapunov functional:

VðxtÞ ¼ V1ðxtÞ þ V2ðxtÞ ð29Þ

where

V1ðxtÞ ¼ x>t E>X1Ext; xt ¼ fxs; t � � � s � tg;

V2ðxtÞ ¼Zt

t��x>2;sX2x2;sds þ

Z0

��

Zt

tþsx> X3xdds

þZ0

��

Zt

t��þsx> X4xdds ð30Þ

with X3 ¼ A>Q>I>0 R1I0QA and X4 ¼ A>d Q>I>0 R2I0QAd:

Direct manipulation gives the following expressions for_VV1ðxtÞ and _VV2ðxtÞ :

_VV1ðxtÞ ¼ _xx>t E>X1Ext þ x>t E>X1E_xxt

¼ x>t ½E>X1A þ A>X1Ext þ 2x>t E>X1AdPI>0 x1;t

þ 2x>t E>X1AdPI1x2;t�� þ hðtÞ

¼ x>t E>X1A þ A>X1E þ E>X1AdPI 0

0 0

!P�1

"

þ ½P�1>I 0

0 0

!P>A>

d X1E

#xt

þ 2x>t E>X1AdPI1x2;t�� þ hðtÞ

ð32Þ

where

hðtÞ ¼ �2Zt

t��x>t E>X1AdPI>0 ½I0AQAxs þ I0QAdxs�� ds

and

_VV2ðxtÞ ¼ x>2;tX2x2;t � x>2;t��X2x2;t�� þ �x>t X3xt

þ �x>t X4xt �Zt

t��x>s X3xsds �

Z0

��x>tþs��X4xtþs��ds ð33Þ

By using the following inequality:

�2x>y � x>Xx þ y>X�1y;8X > 0; x; y 2 Rr ð34Þ

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003 329

we obtain

hðtÞ � �x>t E>X1AdPI>0 R�11 I0P>A>

d X1Ext

þZt

t��x>s A>Q>I>0 R1I0QAxsds

þ �x>t E>X1AdPI>0 R�12 I0P>A>

d X1Ext

þZt

t��x>s��A>

d Q>I>0 R2I0QAdxs��ds

¼ �x>t E>X1AdPI>0 R�11 I0P>A>

d X1Ext þZt

t��xsX3xsds

þ �x>t E>X1AdPI>0 R�12 I0P>A>

d X1Ext

þZt

t��x>s��X4xs��ds

ð35Þ

Combining (32)–(35) yields

_VVðxtÞ � x>t

hE>X1AþA>X1Eþ �E>X1AdPI>0 ½R�1

1 þR�12

I0P>A>d X1Eþ �X3 þ �X4 þE>X1AdP

I 0

0 0

!P�1

þ½P�1>I 0

0 0

!P>A>

d X1Eixt þ2x>t E>X1AdPI1x2;t��

þ x>2;tX2x2;t � x>2;t��X2x2;t��

¼ x>t

hE>X1AþA>X1E

þ �E>X1AdPI>0 R�11 þR�1

2

ihI0P>A>

d X1E

þ �A>d Q>I>0 R2I0QAd þ �A>Q>I>0 R1I0QA

þE>X1AdPI 0

0 0

!P�1 þ½P>�1 I 0

0 0

!P>A>

d X1E

þ½P�1>0 0

0 X2

!P�1

ixt

þ2x>t E>X1AdPI1x2;t�� �x>2;t��X2x2;t��

¼ �>t X�t

ð36Þ

where

�>t ¼ ðx>t x>2;t�� Þ and X ¼J0 E>X1AdPI1

I>1 P>A>d X1E �X2

� �with

J0 ¼ E>X1A þ A>X1E þ �E>X1AdPI>0 ½R�11 þ R�1

2

I0P>A>d X1E þ �A>

d Q>I>0 R2I0QAd

þ �A>Q>I>0 R1I0QA þ E>X1AdPI 0

0 0

!P�1

þ ½P>�1 I 0

0 0

!P>A>

d X1E þ ½P�1>0 0

0 X2

!P�1

Noting that (6) is equivalent to

x>t ½P�1>½I>0 MI0 � I1I>1 P�1xt � 0 ð37Þ

that is

�>t½P�1>½I>0 MI0 � I1I>1 P�1 0

0 0

� ��t � 0: ð38Þ

By virtue of lemma 3.1, we conclude that

�>t X� t < 0 ð39Þ

i.e.

�>t ½�X�t > 0

holds for all �t 6¼ 0 satisfying (38), if and only if

�X � w½P�1>½I>0 MI0 � I1I>1 P�1 0

0 0

� �> 0 ð40Þ

is feasible for some w � 0; i.e.

~JJ0 E>X1AdPI1

I>1 P>A>d X1E �X2

� �< 0 ð41Þ

where ~JJ0 ¼ J0 þ w½P�1>½I>0 MI0 � I1I>1 P�1: Using Schurcomplement, (41) holds if and only if (8) is feasible.

From the given derivation, it follows that LMI (8) implies(39) and (38). Equation (39) implies that _VVðxtÞ < 0; yieldingthe fact that the system (1) is stable. Equation (38) impliesthat (6) holds, i.e. system (1) with ut 0 is stable and (6) issatisfied. This completes the proof of Theorem 3.1.

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003330