Automatica 44 (2008) 2902–2908

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Delay-dependent stability for discrete systems with large delay sequence basedon switching techniquesI

Xi-Ming Sun a,b,∗, Guo-Ping Liu b,c, David Rees b, Wei Wang aa Research Center of Information and Control, Dalian University of Technology, Dalian 116024, Chinab Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, United Kingdomc CSIS Lab, Institute of Automation, Chinese Academy of Sciences, China

a r t i c l e i n f o

Article history:Received 28 July 2007Received in revised form21 January 2008Accepted 2 April 2008Available online 27 September 2008

Keywords:Discrete systemsDelay-dependent criteriaPiecewise Lyapunov functionalSwitching methodLarge delay sequence

a b s t r a c t

This paper considers the exponential stability problem for a class of discrete time-varying delay systemswith large delay sequences (LDSs). A new method based on a switching technique is presented to solvethis problem. A switched delay system, in which one of the discrete subsystems may be unstable, isfirstly employed to describe such a system, and then some new concepts about LDSs are introduced.Next, using a novel piecewise Lyapunov functional, explicit delay-dependent conditions are developedto show how long and how frequent the LDSs can be and still maintain the system exponentially stable.Without LDSs, the criterion obtained in this paper includes the established one as a special case. Twonumerical examples are given to show the effectiveness of the proposed method.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

By a switched system, we mean a hybrid dynamical systemconsisting of a finite number of subsystems described bydifferential or difference equations and a switching signal thatorchestrates switching between these subsystems (Liberzon, 2003;Sun & Ge, 2005; Zhao & Hill, 2008). On the other hand, delayis a common phenomenon in practical control systems (Cooke &Grossman, 1982; Elsgolts, 1966; Gu, Kharitonov, & Chen, 2003;Nikolakopoulos, Panousopoulou, Tzes, & Lygeros, 2005). Switchedsystem with time delay is called switched delay system, wheredelay may be contained in the system state, control input orswitching signals. A typical application of this system is networkedcontrol systems (NCS) with communication delays or packetdropout (see, for example, Krtolica et al. (1994), Lin, Zhai, andAntsaklis (2003) and Xiao, Hassibi, and How (2000)). Using the

I This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Emilia Fridmanunder the direction of Editor Ian Richard Petersen. This work was supportedby the National Natural Science Foundation of China under Grants 60804011and 60534010, and by the China Postdoctoral Science Foundation under Grant20060400287.∗ Corresponding author at: Research Center of Information and Control, DalianUniversity of Technology, Dalian 116024, China. Tel.: +86 411 84707580; fax: +86411 84707581.E-mail addresses: sunxm@dlut.edu.cn (X.-M. Sun), gpliu@glam.ac.uk (G.-P. Liu).

0005-1098/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2008.04.006

augmented system method, Krtolica et al. (1994) and Xiao et al.(2000), examine an NCS with random communication delayswhich they modeled as a hybrid system, where the transition ofthe random delays satisfies a Markov chain. Stability analysis ofswitched delay systems have received much attention in recentyears (for example, see Kim, Campbell, and Liu (2006), Sun, Wang,and Xie (2006) and Sun, Zhao, and Hill (2006)).Delay-dependent stability criteria for delay systems have

received widespread attention in recent years since they canoften provide less conservative results than delay-independentcriteria, for example, refer to Fridman and Shaked (2002), Guet al. (2003), He, Wang, Lin, and Wu (2007), He, Wu, She, andLiu (2004), Jiang and Han (2005), Niculescu, Souza, Dugard, andDion (1998) and Richard (2003), for continuous time delay systemsand Fridman and Shaked (2005a,b), Gao and Chen (2007), Han(2004) and Xu, Lam, and Zou (2005) for discrete systems. Based ontraditional Lyapunov Krasovskii functional method, the search fordelay-dependent bound h is carried out so that the system stabilityor other system performance measures can be guaranteed. A basicassumption for discrete delay systems (for example, see Fridmanand Shaked (2005a,b), Gao and Chen (2007) and Xu et al. (2005)and Han (2004)) is that the delay must satisfy 0 ≤ h1 ≤ d(k) ≤ h2for ∀ k ∈ N̄ = {0, 1, 2, . . .}, because if delay bound increases toh3 > h2, and h1 ≤ d(k) ≤ h3 for ∀ k ∈ N̄ , then system maybecome unstable. However, if the large delay case, h2 < d(k) ≤ h3,occurs occasionally in some subsets of N̄ and the other subsets ofN̄ still satisfy 0 ≤ h1 ≤ d(k) ≤ h2, then the delay system may

X.-M. Sun et al. / Automatica 44 (2008) 2902–2908 2903

still be stable. Existing methods in the above-mentioned paperswill fail once this large delay case occurs since it can only dealwith the global case, that is d(k) ≤ h2 for k ∈ N̄ . In fact, thislarge delay phenomenon is often encountered in practical systems.For example, it often appears in the study of NCS for the reasonof network-induced delay and packet dropout phenomenon (Yue,Han, & Lam, 2005). Thus, it is important to check the stabilityproblem of systems with large delays. Unfortunately, to the bestof our knowledge, this problem has been unsolved to date sincetraditional methods used in the study of delay systems fail for suchsystems.In this paper, we use a switched delay system consisting of

two subsystems to describe the system with large delays. Onestable subsystem is used to describe the small delay case; theother subsystem, which may be unstable, is used to describethe large delay case. Next, to enable the stability analysis of theconsidered system, some new concepts, such as the length rateand frequency of the large delay sequence (LDS) are introduced.Then, based on two lemmas and a novel piecewise Lyapunovfunctional, the exponential stability conditions for the systemwithLDS are developed. Without considering LDS, the result in thispaper contains the existing result Gao and Chen (2007) as a specialcase. Two numerical examples are given to show the effectivenessof the proposed method.This paper is organized as follows. Section 2 introduces model

description and preliminaries; Section 3 gives two lemmas and themain result in this paper; in Section 4, two numerical examples aregiven to show the effectiveness of the proposed method; Section 5draws the conclusions.

Notation. P > 0 is used to denote a positive definite matrix P .λmin(P) denotes the minimum eigenvalues of P . diag{. . .} standsfor a block-diagonal matrix. Matrices, if their dimensions arenot explicitly stated, are assumed to be compatible for algebraicoperations.

2. Preliminaries

Consider the following system model

x(k+ 1) = Ax(k)+ Bx(k− d(k)),x(k) = φ(k), k = −h3,−h3 + 1, . . . , 0, (1)

where x(k) is the state vector; A and B are constant matrices; φ(k)denotes given initial condition, h3 > 0; d(k) denotes the time-varying delay satisfying

0 < h1 ≤ d(k) ≤ h3. (2)

The following assumptions are adopted:

Assumption 1. System (1) is stable if delay d(k) satisfies h1 ≤d(k) ≤ h2 for ∀k ∈ N̄ based on existing methods (for example,see Gao and Chen (2007)), where h2 < h3.

Assumption 2. The stability of system (1) is not guaranteed basedon existing methods or system (1) itself is unstable if delay d(k)satisfies h2 < d(k) ≤ h3 for ∀k ∈ N̄.

Definition 1. Given time sequence ℘ = {i, i+ 1, i+ 2, . . . , i+ j},i, j ∈ N̄ , ℘ is called a large delay sequence (LDS) if for ∀k ∈ ℘, itholds that h2 < d(k) ≤ h3. Time sequence ℘ is called small delaysequence (SDS), if it holds that h1 ≤ d(k) ≤ h2 for ∀k ∈ ℘. And thenumber of elements in ℘, j+ 1, is called the length of sequence ℘.

Suppose the LDS occurs occasionally, then system (1) can bedescribed by the following switched delay system{x(k+ 1) = Ax(k)+ Bx(k− dσ(k)(k))x(k) = φ(k), k = −h3,−h3 + 1, . . . , 0

(3)

where σ(k) : N̄ → {1, 2} is a piecewise constant functionand called switching signal; σ(k) = 1 implies that system (3) isrunning in small delay sequences (SDSs), and σ(k) = 2 denotesthat system (3) is running in large delay sequences (LDSs); h1 ≤d1(k) ≤ h2 and h2 < d2(k) ≤ h3.We use time sequence k1 < k2 < · · · to denote time switching

sequence of switching signal σ(k), where k1 > 0. Without lossof generality, we assume that the time sequence ℘1 = {k2j, k2j +1, . . . , k2j+1 − 1} (∀j ∈ N̄) denotes SDSs and ℘2 = {k2j+1, k2j+1 +1, . . . , k2j+2− 1} denotes LDSs for system (3), where k0 = 0. Then,system (3) is equivalent tox(k+ 1) =

{Ax(k)+ Bx(k− d1(k)) k ∈ ℘1Ax(k)+ Bx(k− d2(k)) k ∈ ℘2,

x(k) = φ(k), k = −h3,−h3 + 1, . . . , 0.(4)

It is usually difficult to exactly test when and where switchingoccurs. But it may be easy to check the number and the length ofLDS in some time intervals, for example, for the case of a periodicalswitching signal. In this paper, it is necessary to introduce thefollowing time sequence {p1, p2, . . .} belonging to a sub-sequenceof switching sequence {k1, k2, . . .}, where

k0 = p0 < p1 < p2 < p3 < · · · , (5)

which satisfies the conditions:⋃+∞

i=0 [pi, pi+1) = [k0,+∞),[pi, pi+1) ⊇ [kl, kj), i ≤ l < j and i, l, j ∈ N̄ , and,

pi+1 − pi ≤ ηi ≤ η < +∞, (6)

for some positive constant numbers η and ηi, i ∈ N̄.Inspired by average dwell time used in switched systems

(Hespanha & Morse, 1999; Liberzon, 2003), we will introduce theconcept of the frequency of LDS to denote the appearance numberof LDS for each unit time interval.

Definition 2. For any integer T2 > T1 ≥ 0, let Nf (T1, T2) denotethe number of LDSs in sequence ℘ = {T1, T1 + 1, . . . , T2 − 1}.Ff (T1, T2) =

Nf (T1,T2)T2−T1

is referred to as frequency of LDS in sequence℘ = {T1, T1 + 1, . . . , T2 − 1}.

If Nσ (T1, T2) is used to denote the number of switchings of σ(k)in sequence ℘ = {T1, T1 + 1, . . . , T2 − 1}, then it is obvious thatin time interval (k0, k), the number of switchings is not more thantwo multiples of the number of LDS, that is,

Nσ (k0, k) ≤ 2Nf (k0, k). (7)

Similar to the concept of the unavailability rate of the controller inZhai and Lin (2004), we introduce the concept of length rate of LDS.

Definition 3. For time sequence ℘ = {T1, T1 + 1, . . . , T2 − 1},denote the total time length of LDSs during ℘ by T+(T1, T2) anddenote the total time length of SDSs by T−(T1, T2). We call

T+(T1,T2)T−(T1,T2)

length rate of LDS in time sequence ℘.

Problem formulation. Under the restriction of the length rateof LDS and the frequency of LDS in each sequence ℘ = {pi, pi +1, . . . , pi+1 − 1},∀i ∈ N̄ , search for the condition to make system(1) exponentially stable.

2904 X.-M. Sun et al. / Automatica 44 (2008) 2902–2908

3. Stability analysis

In this section, two lemmaswill be first developed. Consider thefollowing delay system

x(k+ 1) = Ax(k)+ Bx(k− d(k)),x(k) = φ(k), k = −h3,−h3 + 1, . . . , 0, (8)

where A, B are constantmatrices, d(k) satisfies 0 ≤ h1 ≤ d(k) ≤ h2while it may hold that h1 ≤ d(k + 1) ≤ h3. Choose the Lyapunovfunctional candidate of the following form

V1(k) =7∑i=1

V1i, (9)

where

V11(k) = xT(k)P1x(k),

V12(k) =k−1∑

i=k−d(k)

xT(i)eα1(i−k+1)Q1x(i)

V13(k) =k−1∑i=k−h2

xT(i)eα1(i−k+1)R1x(i),

V14(k) =−h1∑

j=−h3+1

k−1∑i=k+j

xT(i)eα1(i−k+1)Q1x(i),

V15(k) =−1∑j=−h2

k−1∑i=k+j

ηT(i)eα1(i−k+1)(Z1 + Z2)η(i),

V16(k) =k−h2−1∑i=k−h3

xT(i)eα1(i−k+1)R2x(i),

V17(k) =−h2−1∑j=−h3

k−1∑i=k+j

ηT(i)eα1(i−k+1)Z3η(i),

η(k) = x(k + 1) − x(k), and P1 > 0,Q1 > 0, Ri > 0, Zj > 0 (i =1, 2, j = 1, 2, 3), are matrices to be determined.

Lemma 1. For given constants α1 > 0, h3 > h2 > h1 ≥ 0, if thereexist matrices P1 > 0,Q1 > 0 Ri > 0, (i = 1, 2), Zj > 0 (j =1, 2, 3), and any matrices M,N, S and L such that[Ξ1 + Ξ2 + Ξ

T2 + Ξ3 Ξ4

∗ Ξ5

]< 0, (10)

then along the trajectory of the system (8), we have

V1(k+ 1) ≤ e−α1V1(k), (11)

where

Ξ1 =

Ξ11 ATP1B 0 0∗ Ξ22 0 0∗ ∗ e−α1h2R2 − e−α1h2R1 0∗ ∗ ∗ −e−α1h3R2

,Ξ2 =

[M + N −M + S −S − N + L −L

],

Ξ3 =[A− I B 0 0

]T(h2(Z1 + Z2)+ h23Z3)

×[A− I B 0 0

].

Ξ4 =[√c1M

√c2S

√c1N

√c3L],

Ξ5 = diag{ −Z1 −Z1 −Z2 −Z3 },

Ξ11 = ATP1A− e−α1P1 + (1+ h12 + h23)Q1 + R1 ,

Ξ22 = BTP1B− e−α1h2Q1,

c1 =eα1(h2+1) − eα1

eα1 − 1, c2 =

eα1(h2+1) − e(h1+1)α1

eα1 − 1,

c3 =eα1(h3+1) − e(h2+1)α1

eα1 − 1,

h12 = h2 − h1, h23 = h3 − h2.

Proof. See the Appendix. �

Consider the following system,

x(k+ 1) = Ax(k)+ Bx(k− d(k)),x(k) = φ(k), k = −h3,−h3 + 1, . . . , 0, (12)

where A, B are constant matrices, d(k) satisfies h2 ≤ d(k) ≤ h3,while d(k + 1) may vary from h1 to h3. Choose the Lyapunovfunctional candidate of the following form

V2(k) =7∑i=1

V2i, (13)

where

V21(k) = xT(k)P2x(k),

V22(k) =k−1∑

i=k−d(k)

xT(i)eα2(k−1−i)Q2x(i)

V23(k) =k−1∑i=k−h2

xT(i)eα2(k−1−i)R3x(i),

V24(k) =−h1∑

j=−h3+1

k−1∑i=k+j

xT(i)eα2(k−1−i)Q2x(i),

V25(k) =−1∑j=−h2

k−1∑i=k+j

ηT(i)eα2(k−1−i)(Z4 + Z5)η(i),

V26(k) =k−h2−1∑i=k−h3

xT(i)eα2(k−1−i)R4x(i),

V27(k) =−h2−1∑j=−h3

k−1∑i=k+j

ηT(i)eα2(k−1−i)Z6η(i),

and P2 > 0,Q2 > 0, Ri > 0 (i = 3, 4), Zj > 0 (j = 4, 5, 6) arematrices to be determined.

Lemma 2. For given constants α2 > 0, h3 > h2 > h1 ≥ 0, ifthere exist matrices P2 > 0,Q2 > 0, Ri > 0 (i = 3, 4) andZj > 0 (j = 4, 5, 6), and any matrices E, F and G such that[Π1 +Π2 +Π

T2 +Π3 Π4

∗ Π5

]< 0, (14)

then, along the trajectory of the system (12), we have

V2(k+ 1) ≤ eα2V2(k), (15)

where

Π1 =

Π11 ATP2B 0 0∗ Π22 0 0∗ ∗ eα2h2R4 − eα2h2R3 0∗ ∗ ∗ −eα2h3R4

,Π11 = ATP2A− eα2P2 + (1+ h13)Q2 + R3Π22 = BTP2B− eα2h2Q2,Π2 =

[E −F + G −E + F −G

],

Π3 =[A− I B 0 0

]T(h2(Z4 + Z5)+ h23Z6)

×[A− I B 0 0

].

X.-M. Sun et al. / Automatica 44 (2008) 2902–2908 2905

Π4 =[√c4E

√c5F

√c5G],

Π5 = diag{ −(Z4 + Z5) −Z6 −Z6 },

c4 =1− e−h2α2

eα2 − 1, c5 =

e−h2α2 − e−α2h3

eα2 − 1, h13 = h3 − h1.

Proof. See the Appendix. �

Remark 1. The terms Vi3(k) and Vi6(k) and Vi7(k) (i = 1, 2) in (9)and (13) are called large delay sum terms, which are necessary inthat they are used to deal with the large delay bound h3 in system(8) and (12).

Theorem 1. For given constants α1 > 0, α2 > 0, h3 > h2 >h1 ≥ 0, if there exist matrices Pi > 0,Qi, (i = 1, 2), Rj > 0, (j =1, 2, 3, 4), Zl > 0, (l = 1, 2, 3, 4, 5, 6), and any matrices Mi,Ni, Si,Li, Ei, Fi and Gi (i = 1, 2) such that LMIs (10) and (14) hold, thensystem (1) with LDS (S1) is exponentially stable and the state decayestimation is given as

‖x(k)‖ ≤

√V1(k0)eceαη

ρe−0.5(α

∗−α)(k−k0), (16)

where LDS (S1) satisfies the following two conditions

Cond1.T+(pi,pi+1)T−(pi,pi+1)

≤α1−α

∗

α2+α∗, α∗ ∈ (0, α1),∀i ∈ N̄,

Cond2. Ff (pi, pi+1) ≤ α

ln(µ2µ1), α ∈ (0, α∗);

µ ≥ 1 satisfies

Pl ≤ µPh, Ql ≤ µQh, ∀l, h ∈ {1, 2},Ri ≤ µRj, ∀{i, j} or {j, i} ∈ {{1, 3}, {2, 4}},Zm ≤ µZn,∀{m, n} or {n,m} ∈ {{1, 4}, {2, 5}, {3, 6}}; (17)

and

µ1 = e(α1+α2)h3 , (18)

c = (α2 + α∗)α1 − α

∗

α1 + α2η, ρ = min{λmin(P1), λmin(P2)}.

Proof. Construct piecewise Lyapunov functional candidate asfollows

V (k) = Vσ (k) ={V1(k) k ∈ ℘1,V2(k) k ∈ ℘2

(19)

where V1(k) and V2(k) are defined in (9) and (13), respectively.From (17) and (18), it is easy to see that

V1(k) ≤ µV2(k), V2(k) ≤ µµ1V1(k). (20)

Considering the piecewise Lyapunov functional candidate (19), wehave from Lemmas 1 and 2

V (k) ≤{e−α1(k−k2j)V1(k2j) k ∈ ℘1,eα2(k−k2j+1)V2(k2j+1) k ∈ ℘2.

(21)

Without loss of generality, we assume that k ∈ {k2l+1, k2l+1 +1, . . . , k2l+2 − 1} and k ∈ {pm, pm + 1, . . . , pm+1 − 1}, wherel ≥ 1,m ≥ 1. Based on (20) and (21), along the trajectory of system(3), the piecewise Lyapunov functional candidate (19) satisfies

V (k) ≤ V2(k2l+1)eα2(k−k2l+1)

≤ µµ1V1(k2l+1)eα2(k−k2l+1)

≤ µµ1V1(k2l)e−α1(k2l+1−k2l)eα2(k−k2l+1)

= µµ1V1(k2l)e−α1T−(k2l,k)+α2T+(k2l,k)

≤ · · ·

≤ µNσ (k0,k)µNf (k0,k)1 V1(k0)e−α1T

−(k0,k)+α2T+(k0,k). (22)

Next, we will show

−α1T−(k0, t)+ α2T+(k0, t) ≤ −α∗(k− k0)+ c, (23)

and

µNσ (k0,t)µNf (k0,k)1 ≤ eα(k−k0)+αη (24)

where c is defined in Theorem 1.It holds that

−α1T−(k0, k)+ α2T+(k0, k)= −α∗(k− k0)− (α1 − α∗)T−(k0, k)+ (α∗ + α2)T+(k0, k)= −α∗(k− k0)− (α1 − α∗)[T−(p0, pm)+ T−(pm, k)]+ (α∗ + α2)[T+(p0, pm)+ T+(pm, k)]≤ −α∗(k− k0)+ [−(α1 − α∗)T−(p0, pm)

+ (α∗ + α2)T+(p0, pm)] + (α∗ + α2)T+(pm, k). (25)

From Cond1, it holds that

−(α1 − α∗)T−(p0, pm)+ (α∗ + α2)T+(p0, pm)

=

m−1∑q=0

[−(α1 − α∗)T−(pq, pq+1)+ (α∗ + α2)T+(pq, pq+1)]

≤ 0, (26)

and

T+(pm, pm+1) ≤α1 − α

∗

α2 + α1ηm ≤

α1 − α∗

α2 + α1η. (27)

Applying (26) and (27) to (25) leads to (23). Next, we will showinequality (24). From Cond2 and Definition 2, it holds that

Nf (pi, pi+1) ln(µ2µ1) ≤ α(pi+1 − pi), ∀i ∈ N̄,

and thus

Nf (k0, k) ln(µ2µ1)

= ln(µ2µ1)

[m−1∑i=0

Nf (pi, pi+1)+ Nf (pm, k)

]

≤

m−1∑i=0

α(pi+1 − pi)+ ln(µ2µ1)Nf (pm, k)

≤ α(pm − k0)+ αηm ≤ α(k− k0)+ αη. (28)

Based on (7), it holds that

µNσ (k0,k)µNf (k0,k)1 ≤ µ2Nf (k0,k)µ

Nf (k0,k)1 = eNf (k0,k) ln(µ

2µ1). (29)

Applying (28) to (29) leads to (24). Then substituting (23) and (24)into (22) leads to

V (k) ≤ V1(k0)eceαηe−(α∗−α)(k−k0). (30)

Noting that V (k) ≥ ρ‖x(k)‖2, we have from (30) that

ρ‖x(k)‖2 ≤ V1(k0)eceαηe−(α∗−α)(k−k0), (31)

where ρ = min{λmin(P1), λmin(P2)}. And thus, (16) follows from(31). The proof is completed. �

For the case without LDS, that is h1 ≤ d(k) ≤ h2 for ∀k ∈ N̄ , orh2 = h3, then it holds that Nσ (k0, k) = Nf (k0, k) = 0, T+(k0, k) =0, T−(k0, k) = k− k0. Thus, from (22), it is obtained that

V1(k) ≤ V1(k0)e−α1(k−k0), (32)

then we can get

‖x(k)‖ ≤

√V1(k0)λmin(P1)

e−0.5α1(k−k0). (33)

2906 X.-M. Sun et al. / Automatica 44 (2008) 2902–2908

Thus, the following corollary is obtained.

Corollary 1. If the delay satisfies h1 ≤ d1(k) ≤ h2 for ∀k ∈ N̄ , thenunder the LMI (10), system (1) is exponentially stable.

Remark 2. If α1 → 0, then c1 → h2, c2 → h12 and c3 → h23,where c1, c2 and c3 are defined in Lemma 1. Thus, if we let α1 → 0,R2 → εI , (ε is a sufficiently small positive constant) and h23 = 0and L = 0, then LMI (10) is reduced to matrix inequality (2) in Gaoand Chen (2007). Thus, Corollary 1 contains Theorem 1 in Gaoand Chen (2007) as a special case. It is obvious Corollary 1 cangive conditions of exponential stability with decay degree 0.5α1while Theorem 1 in Gao and Chen (2007) only give conditions ofasymptotical stability.

Remark 3. In Theorem 1, Cond1 is used to restrict the length ofthe LDS in each interval [pi, pi+1); Cond2 is used to restrict thefrequency of LDS. Theorem 1 shows that by making a certainrestriction to these two conditions, exponential stability for system(3) with LDS can be guaranteed.

Remark 4. From Theorem 1, it can be seen that the maximumallowable delay bound (MADB) guaranteeing exponential stabilityof system (3) can be given as h3 = h2 + h23. Also, from (27), it canbe seen that T+(pi, pi+1), which denotes the permitted length ofLDS during sequence ℘ = {pi, pi + 1, . . . , pi+1 − 1}, can be givenas α1−α

∗

α2+α1ηi. Existing literature (Fridman & Shaked, 2005a,b; Gao &

Chen, 2007; Gao, Lam, Wang, & Wang, 2004; Han, 2004; Xu et al.,2005)mainly concentrate on the size of h2, but the important partsh23 and

α1−α∗

α2+α1ηi caused by LDS can not be dealt with.

Remark 5. In Theorem 1, to get MADB, we need to solve matrixinequalities (10), (14) and (17). Usually, we select a positiveµ suchthat (17) becomes an LMI. First, give a larger µ so that LMIs (10),(14) may have a feasible solution, and then repeat this process tolook for a smaller µ so as to relax Cond2.

Remark 6. To check exponential stability of system (3) with LDS,the following steps are needed:(I) Give α1, α2, h1, h2, h23, µ, and checkwhether LMIs (10), (14)

and (17) have feasible solutions.(II) If (I) holds, choose α∗ ∈ (0, α1) to calculate the bound of

length rate, α1−α∗

α2+α∗.

(III) Choose α ∈ (0, α∗), and then calculate the bound offrequency, α

ln(µ2µ1).

(IV) Under the Cond1 and Cond2, system (3) is exponentiallystable. The allowed T+(pk, pk+1) in each sequence ℘ = {pi, pi +1, . . . , pi+1 − 1} is given as (27), and Nf (pi, pi+1) is given as ηi ×Ff (pi, pi+1) according to Definition 2.

4. Examples

Example 1. In this section, we use two examples from Gao andChen (2007) to show the benefits of our result. Consider system(1) with

A =[0.8 00.05 0.9

], B =

[−0.1 0−0.2 −0.1

].

According to Remark 6, the following steps are needed:(I) Given h1 = 2, h2 = 10, α1 = 0.06, α2 = 0.2, h3 =

30, µ = 1.05, it is found that LMIs (10), (14) and (17) have feasiblesolutions;(II) Given α∗ = 0.052 < α1, it holds that

T+(pi,pi+1)T−(pi,pi+1)

≤ 0.0317according to Cond1;

Table 1MADB for different h1 under different methods (Example 1)

h1 = 2 h1 = 4 h1 = 6

MADB (Gao et al., 2004) 7 8 9MADB (Gao & Chen, 2007) 13 13 14

Table 2

The bounds of T+(pi,pi+1)T−(pi,pi+1)

, Ff (pi, pi+1), and MADB under different d1(k) (Example 1)

d1(k) [2, 10] [4, 10] [6, 10]

T+(pi,pi+1)T−(pi,pi+1)

0.0317 0.0317 0.0317Ff (pi, pi+1) 0.0063 0.0038 0.0027MADB 30 50 70

Table 3

The bounds of T+(pi,pi+1)T−(pi,pi+1)

, Ff (pi, pi+1) and MADB for d1(k) ∈ [1, 2] under differentd2(k) (Example 2)

d2(k) (2, 9] (2, 20] (2, 30]

T+(pi,pi+1)T−(pi,pi+1)

0.0490 0.0408 0.0306Ff (pi, pi+1) 0.0105 0.0040 0.0020MADB 9 20 30

(III) Givenα = 0.05 < α∗, it holds that Ff (pi, pi+1) ≤ α

ln(µ2µ1)=

0.0063 according to Cond2;(IV) Thus, if T

+(pi,pi+1)T−(pi,pi+1)

≤ 0.0317 and Ff (pi, pi+1) ≤ 0.0063during each sequence℘ = {pi, pi+1, . . . , pi+1−1}, the consideredsystem is exponentially stable under the delay 2 ≤ d(k) ≤ 30.Suppose pi+1 − pi = η for ∀i ∈ N̄ and choose η =

200 s (second). It can be seen that T+(pi, pi+1) ≤α1−α

∗

α1+α2× 200 =

6.1538 s. Also, from Cond2 it is obtained that Nf (pi, pi+1) ≤ η ×

Ff (pi, pi+1) ≤ 1.2662. It means that the LDS, in which d2(k)satisfies 10 < d2(k) ≤ 30, can be permitted to occur once in each200 s and the permitted length of LDS can reach to 6.For the case of h2 = 10 and different h1, the bounds of

T+(pi,pi+1)T−(pi,pi+1)

and Ff (pi, pi+1), andMADB canbe obtained by a similar process andcan be seen in Table 2. Without considering the case of LDS, MADBobtained by using different methods for different h1 can be seenin Table 1. Comparing Tables 1 and 2, it can be seen that for LDSoccurring in a certain frequency, themethodproposed in this paperprovides much larger MADB while existing methods do not. It isalso noticed that with MADB increasing, the bounds of Ff (pk, pk+1)decrease.

Example 2. Consider an example of an inverted pendulum systemwith delayed control input, described by the following equation(Gao & Chen, 2007)

x(k+ 1) =[1.0078 0.03010.5202 1.0078

]x(k)+

[−0.0001−0.0053

]u(k),

where u(k) is the control input and is taken the same as in Gao andChen (2007), that is u(k) =

[102.91 80.7916

]x(k−d(k)). For the

case of 1 ≤ d1(k) ≤ 2, the bounds ofT+(pk,pk+1)T−(pk,pk+1)

and Ff (pk, pk+1),and MADB can be seen in Table 3 under different d2(k). It is worthnoting that MADB obtained in Gao and Chen (2007) and Gao et al.(2004) are 2 and 3, respectively, while it can be even as high as30 under a certain restriction in the length rate and the frequencyof LDS by using the method proposed in this paper. It is clearlyshown that the method in this paper can deal with the case of LDSeffectively while existing literature can not.

X.-M. Sun et al. / Automatica 44 (2008) 2902–2908 2907

5. Conclusions

This paper has considered exponential stability problem fora class of time-varying delay systems with LDS. Such a systemwith LDS has been firstly modeled as a switched delay system, inwhich a discrete subsystem may be unstable due to the effect of alarge delay. By constructing a novel piecewise Lyapunov functionalcontaining large delay sum terms, exponential stability with adecay degree for the considered system are developed under therestriction of frequency and length rate of LDS. We also haveshown in the theory that the obtained result contains existing oneswithout considering LDS as a special case. Two examples have beengiven to show the effectiveness of the proposed method.

Appendix

Proof of lemma 1. Define 1V1(k) = V1(k + 1) − e−α1V1(k), andζ =

[xT(k) xT(k− d(k)) xT(k− h2) xT(k− h3)

]T. Along thesolution of system (8), it holds that

1V1(k)≤ [Ax(k)+ Bx(t − d(k))]TP1[Ax(k)+ Bx(t − d(k))]− xT(k)e−α1P1x(k)+ xT(k)Q1x(k)− xT(k− d(k))

× e−α1h2Q1x(k− d(k))+k−h1∑

i=k+1−h3

xT(i)eα1(i−k)Q1x(i)

+ xT(k)R1x(k)− xT(k− h2)e−α1h2R1x(k)

+ (h3 − h1)xT(k)Q1x(k)−k−h1∑

i=k−h3+1

xT(i)eα1(i−k)Q1x(i)

+ h2ηT(k)(Z1 + Z2)η(k)−k−1∑

i=k−d(k)

ηT(i)eα1(i−k)Z1η(i)

−

k−d(k)−1∑i=k−h2

ηT(i)eα1(i−k)Z1η(i)−k−1∑i=k−h2

ηT(i)eα1(i−k)Z2η(i)

+ xT(k− h2)e−α1h2R2x(k− h2)− xT(k− h3)e−α1h3R2x(k− h3)

+ (h3 − h2)ηT(k)Z3η(k)−k−h2−1∑i=k−h3

ηT(i)eα1(i−k)Z3η(i)

+ 2ζ T(k)M

[x(k)− x(k− d(k))−

k−1∑l=k−d(k)

η(l)

]

+ 2ζ T(k)S

[x(k− d(k))− x(k− h2)−

k−d(k)−1∑l=k−h2

η(l)

]

+ 2ζ T(k)N

[x(k)− x(k− h2)−

k−1∑l=k−h2

η(l)

]

+ 2ζ T(k)L

[x(k− h2)− x(k− h3)−

k−h2−1∑l=k−h3

η(l)

]≤ ζ T(k)[Ξ1 + Ξ2 + Ξ T2 + Ξ3 + c1MZ

−11 M

T

+ c2SZ−11 ST+ c1NZ−12 N

T+ c3LZ−13 L

T]ζ (k). (34)

Here the following inequality is used,

2ζ T(k)Mk−1∑

l=k−d(k)

η(l) ≤k−1∑

l=k−d(k)

ηT(l)eα1(l−k)Z1η(l)

+

k−1∑l=k−d(k)

ζ T(k)e−α1(l−k)MZ−11 MTζ (k)

≤

k−1∑l=k−d(k)

ηT(l)eα1(l−k)Z1η(l)+ c1ζ T(k)MZ−11 MTζ (k).

By using Schur Complement, we know Ξ1 + Ξ2 + ΞT2 + Ξ3 +

c1MZ−11 MT+ c2SZ−11 S

T+ c1NZ−12 N

T+ c3LZ−13 L

T < 0 from matrixinequality (10). Thus, 1V1(k) ≤ 0 from inequality (34) and hence(11) holds. The proof is completed. �

Proof of lemma 2. Define1V2(k) = V2(k+ 1)− eα2V2(k). Alongthe solution of system (12), it holds that

1V2(k)≤ [Ax(k)+ Bx(t − d(k))]TP2[Ax(k)+ Bx(t − d(k))]T

− xT(k)eα2P2x(k)+ xT(k)Q2x(k)− xT(k− d(k))

× eα2h2Q2x(k− d(k))+k−h2∑

i=k+1−h3

xT(i)eα2(k−i)Q2x(i)

+ xT(k)R3x(k)− xT(k− h2)eα2h2R3x(k− h2)

+ (h3 − h1)xT(k)Q2x(k)−k−h2∑

i=k−h3+1

xT(i)eα2(k−i)Q2x(i)

+ h2ηT(k)(Z4 + Z5)η(k)−k−1∑i=k−h2

ηT(i)eα2(k−i)(Z4 + Z5)η(i)

+ xT(k− h2)eα2h2R4x(k− h2)− xT(k− h3)eα2h3R4x(k− h3)

+ (h3 − h2)ηT(k)Z6η(k)−k−h2−1∑i=k−d(k)

ηT(i)eα2(k−i)Z6η(i)

−

k−d(k)−1∑i=k−h3

ηT(i)eα2(k−i)Z6η(i)+ 2ζ T(k)E

×

[x(k)− x(k− h2)−

k−1∑l=k−h2

η(l)

]

+ 2ζ T(k)F

[x(k− h2)− x(k− d(k))−

k−h2−1∑l=k−d(k)

η(l)

]

+ 2ζ T(k)G

[x(k− d(k))− x(k− h3)−

k−d(k)−1∑l=k−h3

η(l)

]≤ ζ T(k)[Π1 +Π2 +ΠT2 +Π3 + c4E(Z4 + Z5)

−1ET

+ c5FZ−16 FT+ c5GZ−16 G

T]ζ (k). (35)

By using Schur Complement, we know Π1 + Π2 + ΠT2 + Π3 +c4E(Z4 + Z5)−1ET + c5FZ−16 F

T+ c5GZ−16 G

T < 0 from matrixinequality (14). Thus, 1V2(k) ≤ 0 from inequality (35) and hence(15) holds. The proof is completed. �

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Dr. Xi-Ming Sun was born in 1973. He received the M.S.degree in applied mathematics from Bohai University,China, in 2003, and the Ph.D. degree in Control Theoryand Control Engineering from Northeastern University,China, in 2006. Since August, 2006, he has been a researchfellow in University of Glamorgan, UK. He is currentlyan associate professor in Dalian University of Technology,China. His current research interests are switched systems,time-delay systems and networked control systems.

Guo-Ping Liu is a chair of control engineering in the Uni-versity of Glamorgan in the UK. He received his B.Eng.and M.Eng. degrees in Electrical and Electronic Engineer-ing from the Central South University of Technology (nowthe Central South University) in China in 1982 and 1985,respectively, and his Ph.D. degree in control engineeringfrom UMIST (now the University of Manchester) in the UKin 1992. He did the postdoctoral research in the Universityof York in 1992–1993. He worked as a research fellow inthe University of Sheffield in 1994. During 1996–2000, hewas a senior engineer in GEC-Alsthom and ALSTOM, and

then a principal engineer and a project leader in ABB ALSTOM Power. He was a se-nior lecturer in the University of Nottingham in 2000–2003. He has been a profes-sor in the University of Glamorgan since 2004, a visiting professor of the ChineseAcademy of Sciences since 2000, and a visiting professor of the Central South Uni-versity since 1994. He is the editor-in-chief of the International Journal of Automa-tion and Computing, a senior IEEE member and the general chair of the 2007 IEEEInternational Conference on Networking, Sensing and Control.Prof. Liu was awarded the Alexander von Humboldt Research Fellowship in

1992. He received the best paper prize for applications at the UKACC InternationalConference on Control in 1998. His paper was shortlisted for the best applicationprize at the 14th IFAC World Congress in 1999. He has worked more than 50 aca-demic research and industrial technology projects. He has more than 300 publica-tions on control systems. He authored or co-authored 6 books. His main researchareas include networked control systems, modelling and control of fuel cells, ad-vanced control of industrial systems, nonlinear system identification and control,and multiobjective optimisation and control.

Dr. David Rees received the B.Sc. (Honours) Degreein Electrical Engineering from the University of Wales,College Swansea, UK in 1967 and the Ph.D. Degree fromthe Council of National Academic Awards, UK in 1976.He was Associate Head of the School of Electronics andthe School’s Research Director. He has published over 240Journal and Conference publications. He was joint editorof ‘Industrial Digital Control Systems’, published by PeterPeregrinus in 1988, and has contributed to numerousmonographs, the latest is three chapters in ‘DynamicModelling of Gas turbines — Identification, Simulation,

Condition Monitoring and Optimal Control’, published by Springer, 2004. He is aFellow of the Institution of Electrical Engineers and a past Chairman of the IEEcontrol Applications Professional Group. He was the recipient of an IEE PremiumAward in 1996. His current research interests include nonlinear modelling andsystem identification, and networked control systems.

Professor Wei Wang was born in Shenyang, China, in1955. He obtained the Bachelor, Master Degree andPh.D. in Industrial Automation from Northeastern Univer-sity, China, in 1982, 1986 and 1988, respectively. He ispresently professor and director of Research Center of In-formation and Control, Dalian University of Technology,China. Previously he was a post-doctor at the Division ofEngineering Cybernetics, Norwegian Science and Technol-ogy University (1990–1992), professor and vice director ofResearch Center of Automation, Northeastern University,China (1995–1999), vice director of the National Engineer-

ing Research Center of Metallurgical Automation (1995–1999), and a research fel-low at the Department of Engineering Science, University of Oxford (1998–1999).His research interests are in adaptive control, predictive control, robotics, com-

puter integrated manufacturing systems, and computer control of industrial pro-cess. He is the author of the two books in Chinese entitled ‘‘generalized predictivecontrol theory and its applications’’ (1997) and ‘‘multi-model control and its ap-plications’’ (2002) published both by Science Publishing House, China. He has pub-lished over 200 papers in international and domestic journals.Hewas awarded the National Distinguished Young Fund of the National Natural

Science Foundations of China in 1998. He also awarded the Science and TechnologyAwards twice from the Chinese Ministry of Education in 1995 and 1999 and theScience and Technology Award from the Ministry of Chinese Metallurgical Indus-try in 1999. He was the NOC Vice Chairman of IFAC Symposium on Low Cost Au-tomation in 1998, the IPC Vice Chairman of IFACWorkshop onMining, Mineral andMetal Processing in 2003, the IPC Chairman of the 6th World Congress on Intelli-gent Control and Automation in 2006 and the IPC Chairman of IFAC Symposium onCost Oriented Automation in 2007. He is currently the chairman of IFAC TechnicalCommittee (4.4) of Cost Oriented Automation (2005–2008) and a member of IFACTechnical Committee of Mining, Mineral and Metal Processing (1999–2008).