Automatica 40 (2004) 1435–1439www.elsevier.com/locate/automatica

Technical Communique

Delay-dependent criteria for robust stability of time-varyingdelay systems�

Min Wua, Yong Hea ;∗, Jin-Hua Sheb, Guo-Ping Liuc;d

aSchool of Information Science and Engineering, Central South University, Changsha 410083, ChinabSchool of Bionics, Tokyo University of Technology, Tokyo, 192-0982, JapancSchool of Electronics, University of Glamorgan, Pontypridd CF37 1DL, UK

dInstitute of Automation, Chinese Academy of Sciences, Beijing 100080, PR China

Received 13 March 2003; received in revised form 6 February 2004; accepted 8 March 2004

Abstract

This paper deals with the problem of delay-dependent robust stability for systems with time-varying structured uncertainties andtime-varying delays. Some new delay-dependent stability criteria are devised by taking the relationship between the terms in the Leibniz–Newton formula into account. Since free weighting matrices are used to express this relationship and since appropriate ones are selectedby means of linear matrix inequalities, the new criteria are less conservative than existing ones. Numerical examples suggest that theproposed criteria are e;ective and are an improvement over previous ones.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Delay-dependent criteria; Robust stability; Time-varying delay; Time-varying structured uncertainties; Linear matrix inequality

1. Introduction

Stability criteria for time-delay systems have been attract-ing the attention of many researchers. They can be classi=edinto two categories: delay-dependent and delay-independentcriteria. Since delay-dependent criteria make use of infor-mation on the length of delays, they are less conservativethan delay-independent ones. For delay-dependent criteria(see, for example, Su & Huang, 1992; Li & de Souza,1997; Gu, Wang, Li, Cheng, & Qian, 1998; Cao, Sun, &Cheng, 1998; de Souza & Li, 1999; Park, 1999; Han &Gu, 2001; Kim, 2001; Moon, Park, Kwon, & Lee, 2001;Han, 2002a, b; Yue & Won, 2002; Fridman & Shaked,2002, 2003), the main approaches currently consist of fourmodel transformations of the original system (Fridman &Shaked, 2003). The =rst type is a =rst-order transformation.

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor RicuMiddleton under the direction of Editor Paul Van den Hof.

∗ Corresponding author. Tel./fax: +86-731-8836091.E-mail address: heyong08@yahoo.com.cn (Y. He).

Since additional eigenvalues are introduced into the trans-formed system, it is not equivalent to the original one(Gu & Niculescu, 2000). The second type is a neutraltransformation. The system obtained by this method is notequivalent to the original one, either (Gu & Niculescu,2001); and this method requires an additional assumptionto obtain the stability condition for the system. In addition,the inequality used to determine the stability of the sys-tem is −2aTb6 aTXa + bTX−1b; a; b∈Rn; X ¿ 0, whichis known to be conservative. Park (1999) introduced afree matrix, M , to obtain a less conservative inequality−2aTb6 (a + Mb)TX (a + Mb) + bTX−1b + 2bTMb, andMoon et al. (2001) extended it to a more general form,

−2aTb6 [ ab

]T [X

Y T−IY−IZ

] [ ab

];[ XY T

YZ

]¿ 0. The third type

of model transformation employs these inequalities andyields a transformed system that is equivalent to the originalone. However, in the derivative of the Lyapunov functional,Park (1999) and Moon et al. (2001) used the Leibniz–New-ton formula and just replaced some of the terms x(t − �)with x(t) − ∫ t

t−� x(s) ds in the derivative of the Lyapunovfunctional in order to make it easy to handle. For exam-ple, in Moon et al. (2001), x(t − �) was replaced with

0005-1098/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2004.03.004

1436 M. Wu et al. / Automatica 40 (2004) 1435–1439

x(t)− ∫ tt−� x(s) ds in the expression 2x

T(t)PA1x(t), but not

in �xT(t)Zx(t). Since both x(t − �) and x(t) − ∫ tt−� x(s) ds

a;ect the result, there must be some relationship betweenthem; and there must exist optimal weighting matricesfor those terms. However, they did not give a method fordetermining them, but just selected some =xed weightingmatrices. Fridman and Shaked (2002, 2003) combineda descriptor model transformation (Fridman, 2001) withPark and Moon’s inequalities to yield the fourth type oftransformation. This method produces less conservativecriteria. However, since the basic approach in Fridman andShaked (2002, 2003) is also based on the substitution ofx(t)−∫ t

t−� x(s) ds for x(t−�), it does not entirely overcomethe conservatism of the methods given by Park (1999) andMoon et al. (2001).This paper presents new criteria based on a new method

with some interesting features. First, it deals with the systemmodel directly and does not employ any system transforma-tion, thus avoiding the conservatism that results from such atransformation. Second, it does not use the above inequalityor the improved inequality to estimate the upper bound of−2aTb. This also reduces the conservatism in the derivationof the stability condition. Third, some free weighting matri-ces are employed to express the inKuence of the terms in theLeibniz–Newton formula, in contrast to existing methods,which preselect =xed ones. The matrices are determined bysolving linear matrix inequalities (LMIs). This is the mainadvantage of our method, and is the essential di;erence be-tween existing methods and ours. Compared with Moonet al. (2001), and Fridman and Shaked (2002), our new crite-ria overcome some of the main sources of conservatism, andcontain the criteria in Moon et al. (2001) as a special case.Furthermore, the new criteria also contain the well-knowndelay-independent stability condition in Gu, Kharitonov,and Chen (2003) and Hale and Verduyn Lunel (1993).For two examples studied numerically, the new criteria areshown to be e;ective, o;ering signi=cant improvements overpreviously published criteria.

2. Preliminaries

Consider a nominal system �0 with a time-varying delaygiven by

�0 :

{x(t) = Ax(t) + Bx(t − d(t)); t ¿ 0;

x(t) = �(t); t ∈ [− �; 0];(1)

where x(t)∈Rn is the state vector. The time delay, d(t), isa time-varying continuous function that satis=es

06d(t)6 �; d(t)6 �¡ 1; (2)

where � and � are constants and the initial condition, �(t),is a continuous vector-valued initial function of t ∈ [− �; 0].

When the system contains time-varying structured uncer-tainties, it can be described by

�1 :

x(t) = (A+OA(t))x(t)

+(B+OB(t))x(t − d(t)); t ¿ 0;

x(t) = �(t); t ∈ [− �; 0]:

(3)

The uncertainties are assumed to be of the form

[OA(t) OB(t)] = DF(t)[Ea Eb]; (4)

where D, Ea and Eb are constant matrices with appro-priate dimensions, and F(t) is an unknown, real, andpossibly time-varying matrix with Lebesgue-measurableelements satisfying

FT(t)F(t)6 I; ∀t: (5)

The following lemma is employed to handle the time-varyingstructured uncertainties in the system.

Lemma 1 (Xie, 1996). Given matrices Q = QT; H; E andR= RT¿ 0 of appropriate dimensions,

Q + HFE + ETFTHT¡ 0

for all F satisfying FTF6R, if and only if there existssome �¿ 0 such that

Q + �HHT + �−1ETRE¡ 0:

The Lyapunov functional candidates for �0 and �1 arechosen to have the same form and are given by

V (xt) := xT(t)Px(t) +∫ t

t−d(t)xT(s)Qx(s) ds

+∫ 0

−�

∫ t

t+ xT(s)Zx(s) ds d ; (6)

where P = PT¿ 0; Q = QT¿ 0 and Z = ZT¿ 0 are to bedetermined.

3. Main results

First, the nominal system, �0, is discussed. The Leibniz–Newton formula is employed to obtain a delay-dependentcondition, and the relationship between the terms in the for-mula is taken into account. Speci=cally, the terms on the leftside of the equation

2[xT(t)Y + xT(t − d(t))T ]

×[x(t)−

∫ t

t−d(t)x(s) ds − x(t − d(t))

]= 0 (7)

are added to the derivative of the Lyapunov functional,V (xt). In this equation, the free weighting matrices Y and Tindicate the relationship between the terms in the Leibniz–Newton formula. As is shown in the following theorem,they can easily be determined by solving the correspondinglinear matrix inequalities.

M. Wu et al. / Automatica 40 (2004) 1435–1439 1437

Theorem 2. Given scalars �¿ 0 and �¡ 1, the nominalsystem �0 is asymptotically stable if there exist symmet-ric positive-de<nite matrices P= PT¿ 0, Q=QT¿ 0 andZ = ZT¿ 0, a symmetric semi-positive-de<nite matrixX =

[X11X T12

X12X22

]¿ 0, and any appropriately dimensioned

matrices Y and T such that the following LMIs are true.

"=

"11 "12 �ATZ

"T12 "22 �BTZ

�ZA �ZB −�Z

¡ 0; (8)

# =

X11 X12 Y

X T12 X22 T

Y T TT Z

¿ 0; (9)

where

"11 = PA+ ATP + Y + Y T + Q + �X11;

"12 = PB − Y + TT + �X12;

"22 =−T − TT − (1− �)Q + �X22:

Proof. Using the Leibniz–Newton formula, we can write

x(t − d(t)) = x(t)−∫ t

t−d(t)x(s) ds: (10)

Then, for any appropriately dimensioned matrices Y and T ,we have Eq. (7). On the other hand, for any semi-positive

de=nite matrix X =[X11X T12

X12X22

]¿ 0, the following holds.

�$T(t)X$(t)−∫ t

t−d(t)$T(t)X$(t) ds¿ 0; (11)

where $(t) = [xT(t) xT(t − d(t))]T. Then, for X = X T¿ 0,and any matrices Y and T , using Eqs. (7) and (11) andcalculating the derivative of V (xt) in (6) for �0 yields

V (xt) = xT(t)[PA+ ATP]x(t)

+ 2xT(t)PBx(t − d(t)) + xT(t)Qx(t)

− (1− d(t))xT(t − d(t))Qx(t − d(t))

+ �xT(t)Zx(t)−∫ t

t−�xT(s)Zx(s) ds

6 xT(t)[PA+ ATP]x(t)

+ 2xT(t)PBx(t − d(t)) + xT(t)Qx(t)

− (1− �)xT(t − d(t))Qx(t − d(t))

+ �xT(t)Zx(t)−∫ t

t−d(t)xT(s)Zx(s) ds

+2[xT(t)Y + xT(t − d(t))T ]

×[x(t)−

∫ t

t−d(t)x(s) ds − x(t − d(t))

]

+ �$T(t)X$(t)−∫ t

t−d(t)$T(t)X$(t) ds

:= $T(t)%$(t)−∫ t

t−d(t)&T(t; s)#&(t; s) ds; (12)

where

&(t; s) := [xT(t) xT(t − d(t)) xT(s)]T;

% :=

["11 + �ATZA "12 + �ATZB

"T12 + �BTZA "22 + �BTZB

];

"11; "12 and "22 are de=ned in (8) and # is de=ned in(9). If %¡ 0 and #¿ 0, then V (xt)¡ 0 for any $(t) �= 0.Applying the Schur complement (Boyd, El Ghaoui, Feron,Balakrishnan, 1994) shows that Eq. (8) implies %¡ 0. So�0 is asymptotically stable if LMIs (8) and (9) are true. Thiscompletes the proof.

Remark 1. For a time-invariant delay system, according tothe procedure of the proof of Theorem 2, it is clear thatsetting X12 = 0, X22 = 0 and T = 0 in Theorem 2 yieldsprecisely Theorem 1 in Moon et al. (2001). So, Theorem 1in this paper is an extension of Theorem 1 in Moon et al.(2001). Instead of choosing X12, X22 and T to be =xed matri-ces, Theorem 2 selects them by solving LMIs. So, it alwayschooses suitable ones, thus overcoming the conservatism ofTheorem 1 in Moon et al. (2001).

Remark 2. If the matrices Y , T and X in Eq. (9) are set tozero, and Z='I (' is a suTciently small positive scalar), thenTheorem 2 is identical to the well-known delay-independentstability criterion in and Gu et al. (2003) (Proposition 5.14on p. 169) and Hale and Verduyn Lunel (1993) (Eq. (2.4)on p. 134), which is stated as follows:

Corollary 3. When � = 0, system �0 is asymptoticallystable if there exist real symmetric matrices P¿ 0 and Qsuch that[PA+ ATP + Q PB

BTP −Q

]¡ 0

is satis<ed.

So, any system that exhibits delay-independent stability,as determined by Corollary 3, is, for all practical purposes,asymptotically stable for any delay satisfying 06d(t)6 �,where � is a positive real number.Now, extending Theorem 2 to systems with time-varying

structured uncertainties yields the following theorem.

1438 M. Wu et al. / Automatica 40 (2004) 1435–1439

Theorem 4. Given scalars �¿ 0 and �¡ 1 and assuming(2), the uncertain system �1 is robustly stable if there ex-ist symmetric positive-de<nite matrices P = PT¿ 0, Q =QT¿ 0 and Z=ZT¿ 0, a symmetric semi-positive-de<nitematrix X =

[X11X T12

X12X22

]¿ 0, any matrices Y and T such that

LMI (9) and the following LMI are true:

( =

"11 + ETa Ea "12 + ETa Eb �ATZ PD

"T12 + ETb Ea "22 + ETb Eb �BTZ 0

�ZA �ZB −�Z �ZD

DTP 0 �DTZ −I

¡ 0; (13)

where "11; "12 and "22 are de<ned in (8).

Proof. Replacing A and B in (8) with A+DF(t)Ea and B+DF(t)Eb, respectively, we =nd that (8) for �1 is equivalentto the following condition:

"+

PD

0

�ZD

F(t)[ Ea Eb 0 ]

+

ETa

ETb

0

FT(t)[DTP 0 �DTZ ]¡ 0:

By Lemma 1, a suTcient condition guaranteeing (8) for�1 is that there exists a positive number �¿ 0 such that

�"+ �2

PD

0

�ZD

[DTP 0 �DTZ ]

+

ETa

ETb

0

[ Ea Eb 0 ]¡ 0: (14)

Replacing �P; �Q; �Z; �X; �Y and �T with P; Q; Z; X; Yand T , respectively, and applying the Schur complementshows that (14) is equivalent to (13). This completes theproof.

Remark 3. Moon et al. (2001) presented an algorithm forconstructing a controller with a suboptimal upper bound onthe delay based on themethod of convex optimization, whichstabilizes the system for all admissible uncertainties. Theresults in this paper combined with that algorithm provide amethod of solving the synthesis problem for control systemswith a time-varying delay.

Table 1Allowable time delay (Example 5)

� 0 0.5 0.9

Li and de Souza (1997) 0.2013 — —Kim (2001) 0.2412 ¡ 0:2 ¡ 0:1Yue and Won (2002) 0.2412 0.2195 0.1561Moon et al. (2001) 0.7059 — —Fridman and Shaked (2002) 1.1490 0.9247 0.6710Theorem 4 1.1490 0.9247 0.6954

4. Numerical examples

In this section, some examples are used to demonstratethat the method presented in this paper is e;ective and is animprovement over existing methods.

Example 5. Consider the uncertain system �1 with thefollowing parameters:

A=

[ −2 0

0 −1

]; B=

[ −1 0

−1 −1

];

Ea = diag{1:6; 0:05}; Eb = diag{0:1; 0:3}; D = I:

This example was given in Kim (2001) and Yue and Won(2002). The upper bounds on the time delay for di;erent �obtained from Theorem 4 are shown in Table 1. For com-parison, the table also lists the upper bounds obtained fromthe criteria in Li and de Souza (1997), Kim (2001), Yue andWon (2002), Moon et al. (2001) and Fridman and Shaked(2002). Note that the results for Fridman and Shaked (2002)were obtained by combining Lemma 1 in their paper withLemma 1 in this paper. It is clear that Theorem 4 gives muchbetter results than those obtained by Li and de Souza (1997),Kim (2001), Yue and Won (2002) or Moon et al. (2001),and the same or better results than Fridman and Shaked’s(2002).

Example 6. Consider the robust stability of the uncertainsystem �1 with the following parameters:

A=

[ −0:5 −21 −1

]; B=

[ −0:5 −10 0:6

];

Ea = Eb = diag{0:2; 0:2}; D = I:

The upper bounds on the time delay obtained from Theorem4 are listed in Table 2. The results for Fridman and Shaked(2002) in the table were also obtained by combiningLemma 1 in their paper with Lemma 1 in this paper. It is clearthat our results are signi=cantly better than those in Fridmanand Shaked (2002). In particular, when �=0:9, the methodin Fridman and Shaked (2002) fails; but the upper bound0.2420 is obtained by using Theorem 4 in this paper.

M. Wu et al. / Automatica 40 (2004) 1435–1439 1439

Table 2Allowable time delay (Example 6)

� 0 0.5 0.9

Fridman and Shaked (2002) 0.6812 0.1820 —Theorem 4 0.8435 0.2433 0.2420

5. Conclusion

This paper presents a new method of determiningdelay-dependent stability criteria that takes the relationshipbetween x(t − d(t)) and x(t)− ∫ t

t−d(t) x(s) ds into account.Some free weighting matrices that express the inKuenceof these two terms are determined based on linear matrixinequalities, which makes it easy to choose suitable ones.It was shown that the criteria in Moon et al. (2001) are aspecial case of this new method, and that the new method isless conservative than existing ones. Finally, some numeri-cal examples suggest that the method presented here is verye;ective and is a signi=cant improvement over existingones.

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