# Wirtinger-Type Inequality and the Stability Analysis of Delayed Lur'e System

Authors: Zixin Liu; Jian Yu; Daoyun Xu; Dingtao Peng
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper proposes a new delay-depended stability criterion for a class of delayed Lur'e systems with sector and slope restricted nonlinear perturbation. The proposed method employs an improved Wirtinger-type inequality for constructing a new Lyapunov functional with triple integral items. By using the convex expression of the nonlinear perturbation function, the original nonlinear Lur'e system is transformed into a linear uncertain system. Based on the Lyapunov stable theory, some novel delay-depended stability criteria for the researched system are established in terms of linear matrix inequality technique. Three numerical examples are presented to illustrate the validity of the main results.


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## Body

## 1. Introduction

Lur'e control system is an important nonlinear control system. Since the notion of absolute stability was first time introduced by Lur'e in [1], the problem of the absolute stability of Lur'e control system has been widely studied for several decades (see [2–6]). However, because of the existence of time delays, stochastic disturbances, parameter uncertainties, and so on, the convergence of Lur'e system may often be destroyed. This makes the design or performance for the corresponding closed-loop systems become difficult. Therefore, the stability analysis of delayed Lur'e system becomes very important. Up to now, various stability conditions have been obtained, and many excellent papers and monographs have been available (see [7–12]).Recently, a great deal of effort has been done to the stability analysis of delayed Lur'e system with sector and slope restricted nonlinearities. To enlarge the feasibility region of the stability criteria, by introducing variables in cross-term, Park researched a new bounding technique in [13]. Concerning the descriptor method for delayed system, an extensive work was developed by Fridman and Shaked in [14]. By employing linear matrix inequality and matrix decomposing technique, Cao and Zhong [6] researched the absolute stability problem of Lur’e control systems with multiple time delays and nonlinearities and established some improved delay-dependent criteria. In order to further reduce stability criterion's conservatism, sector bounds and slope bounds are employed to a Lyapunov-Krasovskii functional through convex representation of the nonlinearities so that some new improved criteria were established by Lee and Park in [12] and Yin et al. in [15], respectively.On the other hand, these previous works only focused on the relationship between∫t-τt‍xT(s)Qx(s)ds and (∫t-τt‍x(s)ds)TQ(∫t-τt‍x(s)ds) or between ∫-τ0‍∫t+θt‍xT(s)Qx(s)dsdθ and (∫-τ0‍∫t+θt‍x(s)dsdθ)TQ(∫-τ0‍∫t+θt‍x(s)dsdθ). One natural question is whether there exists a relationship among ∫-τ0‍∫t+θt‍z˙(s)TRz˙(s)dsdθ, ∫-τ0‍∫t+θt‍z(s)dsdθ, ∫t-τt‍z˙T(s)Rz˙(s)ds, z(t-τ), and ∫t-τt‍z(s)ds. This idea motivates this study. By using the Jensen integral inequality, we first establish some improved vector Wirtinger-type inequalities. On the basis of these new established inequalities, a new Lyapunov functional including triple integral items is proposed, and some less conservative delay-dependent stability criteria are derived. Finally, three numerical examples are presented to illustrate the validity of the main results.Notation. The notations are used in our paper except where otherwise specified. R,Rn are real and n-dimension real number sets, respectively; diag(⋯) denotes the block diagonal matrix. I is identity matrix; * represents the elements below the main diagonal of a symmetric block matrix; real matrix P>0(<0) denotes P is a positive-definite (negative-definite) matrix.


## 2. Preliminaries

Consider the following delayed Lur'e system:(1)y˙(t)=Ay(t)+By(t-τ(t))+Cf(σ(t)),σ(t)=HTy(t),∀t≥0,y(s)=φ(s),s∈[-τu,0],
where y(t)∈Rn denotes the state vector; σ(t)∈Rm is the output vector; H=(h1,h2,…,hm)n×m∈Rn×m; A,B,andC are constant known matrices of appropriate dimensions. The delay τ(t) is assumed to satisfy
(2)0<τl≤τ(t)≤τu,τ˙(t)≤τ<1.f(σ(t))∈Rm denotes the nonlinear function in feedback path, which has the following form:
(3)f(σ(t))=[f1(σ1(t)),f2(σ2(t)),…,fm(σm(t))]Tσ(t)=[σ1(t),σ2(t),…,σm(t)]T≜[h1Ty(t),h2Ty(t),…,hmTy(t)]T,
which satisfies a sector condition with fi(·),(i=1,2,…,m) belonging to sector [li-,li+], where li-,li+ are known constant scalars; that is,
(4)li-≤fi(σi(t))σi(t)≤li+,i=1,2,…,m.Notice that the nonlinear functionfi(·) can be written as a convex combination of the sector bounds as follows:
(5)fi(σi(t))=(λi(σi(t))li-+(1-λi(σi(t)))li+)σi(t),2222222222222222222222222222222i=1,2,…,m,
where λi(σi)=(fi(σi(t))-li-σi(t))/(li+-li-)σi(t) satisfying 0≤λi(σi)≤1. Namely, fi(σ(t))=Λi(σi(t))σi(t), where Λi(σi(t)) is an element of a convex hull Co{li-,li+}. Set L=diag{l1,l2,…,lm}, where li=max{|li-|,|li+|}. Obviously, -1≤(λi(σi(t))li-+(1-λi(σi(t)))li+)/li≤1. Define Δi=(λi(σi(t))li-+(1-λi(σi(t)))li+)/li,i=1,2,…,m, Δ=diag(Δ1,Δ2,…,Δm) then nonlinear function f(σ(t)) can be expressed as f(σ(t))=LΔHTy(t), where Δ satisfies ΔTΔ≤I. And the system (1) can be rewritten as the following delayed uncertain system:
(6)y˙(t)=(A+CLΔHT)y(t)+By(t-τ(t)),y(s)=φ(s),s∈[-τu,0].Remark 1.
Different from previous work [6, 9, 10, 12, 15], in this paper, by using the convex expression fi(σ(t))=Λi(σi(t))σi(t), we transform the original nonlinear system (1) into a linear uncertain system (6). As a result, the stability problem of nonlinear Lur'e system (1) can be transformed into the robust stability problem of linear uncertain system (6).LetL-=diag(l1-,l2-,…,lm-) and L+=diag(l1+,l2+,…,lm+). For further discussion, the following lemmas are needed.Lemma 2 (see [16]).
For any positive definite symmetric constant matrixQ and scalar τ>0, such that the following integrations are well defined, then
(7)-∫-τ0‍∫t+θt‍yT(s)Qy(s)dsdθ≤-1τ2(∫-τ0‍∫t+θt‍y(s)dsdθ)TQ(∫-τ0‍∫t+θt‍y(s)dsdθ).Lemma 3 (see [17]).
Given symmetric matrixP1 and any real matrices P2,P3 of appropriate dimensions,
(8)P1+P2ΔP3+P3TΔTP2T<0,
for all Δ∈Θ satisfying ΔTΔ≤I if and only if there exists S∈SΔ such that
(9)[P1+P3TSP3P2P2T-S]<0,
where SΔ≕{diag(s1I,…,skI,S1,…,Sl):Si>0,k,l∈N}.On the basis of Jensen integral inequality, we first give out an improved Wirtinger-type vector inequality as follows.Lemma 4.
Letz(t)∈Rn have continuous derived function z˙(t) on interval [a,b]. Assume that z(a)=0; then for any n×n-matrix R>0, the following inequality holds:
(10)∫ab‍zT(s)Rz(s)ds≤(b-a)22∫ab‍z˙T(s)Rz˙(s)ds.Proof.
Sincez(a)=0, one can get that z(s)=∫as‍z˙(t)dt,
(11)∫ab‍zT(s)Rz(s)ds=∫ab‍(∫as‍z˙(t)dt)TR(∫as‍z˙(t)dt)ds≤∫ab‍(s-a)∫as‍z˙T(t)Rz˙(t)dtds=∫ab‍∫tb‍(s-a)z˙T(t)Rz˙(t)dsdt=∫ab‍z˙T(t)Rz˙(t)((b-a)22-(t-a)22)dt≤(b-a)22∫ab‍z˙T(t)Rz˙(t)dt.
This completes the proof.On the basis of Lemma4, we further give out some improved Wirtinger-type inequalities as follows.Lemma 5.
Letz(t)∈Rn have continuous derived function z˙(t) on interval [a,b]. Then for any n×n-matrix R>0, scalar τ>0, the following inequality holds:
(
1
)
(12)
-
∫
t
-
τ
t
‍
z
˙
T
(
s
)
R
z
˙
(
s
)
d
s
≤
-
2
τ
3
(
∫
t
-
τ
t
‍
z
(
s
)
d
s
)
T
R
(
∫
t
-
τ
t
‍
z
(
s
)
d
s
)
-
2
τ
z
T
(
t
-
τ
)
R
z
(
t
-
τ
)
+
4
τ
2
(
∫
t
-
τ
t
‍
z
(
s
)
d
s
)
T
R
z
(
t
-
τ
)
;
(
2
)
(13)
-
∫
-
τ
0
‍
∫
t
+
θ
t
‍
z
˙
(
s
)
T
R
z
˙
(
s
)
d
s
d
θ
≤
-
2
τ
4
(
∫
-
τ
0
‍
∫
t
+
θ
t
‍
z
(
s
)
d
s
d
θ
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T
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∫
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∫
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θ
t
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d
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2
τ
2
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-
τ
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‍
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d
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T
R
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t
-
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+
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τ
3
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∫
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t
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θ
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T
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-
τ
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‍
z
(
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d
s
)
.Proof.
Setx(s)=z(s)-z(t-τ),s∈[t-τ,t]. Notice that x(t-τ)=z(t-τ)-z(t-τ)=0; from Lemma 4, we have
(14)∫t-τt‍xT(s)Rx(s)ds=∫t-τt‍(z(s)-z(t-τ))TR(z(s)-z(t-τ))ds≤τ22∫t-τt‍(d(z(s)-z(t-τ))ds)T×R(d(z(s)-z(t-τ))ds)ds=τ22∫t-τt‍z˙(s)TRz˙(s)ds.
Additionally, from Jensen inequality, one can get
(15)∫t-τt‍(z(s)-z(t-τ))TR(z(s)-z(t-τ))ds≥1τ(∫t-τt‍(z(s)-z(t-τ))ds)T×R(∫t-τt‍(z(s)-z(t-τ))ds).
Thus, we have
(16)-∫t-τt‍z˙(s)TRz˙(s)ds≤-2τ3(∫t-τt‍(z(s)-z(t-τ))ds)T×R(∫t-τt‍(z(s)-z(t-τ))ds)=-2τ3(∫t-τt‍z(s)ds)TR(∫t-τt‍z(s)ds)-2τzT(t-τ)Rz(t-τ)+4τ2(∫t-τt‍z(s)ds)TRz(t-τ).
Similarly, let x(s)=z(s)-z(t+θ); from inequality (16), Lemmas 2, and 4, one can obtain
(17)-∫-τ0‍∫t+θt‍z˙(s)TRz˙(s)dsdθ≤-∫-τ0‍∫t+θt‍2(-θ)2x(s)TRx(s)dsdθ≤-1τ2(∫-τ0‍∫t+θt‍2(z(s)-z(t+θ))-θdsdθ)T×R(∫-τ0‍∫t+θt‍2(z(s)-z(t+θ))-θdsdθ)≤-1τ2(∫-τ0‍∫t+θt‍2z(s)τdsdθ-2∫t-τt‍z(s)ds)T×R(∫-τ0‍∫t+θt‍2z(s)τdsdθ-2∫t-τt‍z(s)ds)=-2τ4(∫-τ0‍∫t+θt‍z(s)dsdθ)TR(∫-τ0‍∫t+θt‍z(s)dsdθ)-2τ2(∫t-τt‍z(s)ds)TR(∫t-τt‍z(s)ds)+4τ3(∫-τ0‍∫t+θt‍z(s)dsdθ)TR(∫t-τt‍z(s)ds).
This completes the proof.Remark 6.
Compared with traditional Wirtinger-type integral inequality, Jensen integral inequality [18], and double integral Jensen inequality [16], Lemma 5 gives out a transitional form among ∫-τ0‍∫t+θt‍z˙(s)TRz˙(s)dsdθ, ∫-τ0‍∫t+θt‍z(s)dsdθ, ∫t-τt‍z˙T(s)Rz˙(s)ds, z(t-τ), and ∫t-τt‍z(s)ds. These inequality relationships can be used as a handy tool to deal with the stability problem.


## 3.  Main Results

In this section, we attempt to establish some new practically computable stability criteria for system (1). By constructing a new Lyapunov functional including triple integral items, we obtain the following stability result.Theorem 7.
For given scalarsτl>0, τu>0, τ<1, L-=diag(l1-,l2-,…,lm-), L+=diag(l1+,l2+,…,lm+), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices D1=diag{d1,d2,…,dm}, D2=diag{λ1,λ2,…,λm}, D3=diag{α1,α2,…,αm}, symmetric positive definite matrices Q1, Q2, Q3, Q4, and arbitrary matrices M1,M2 of appropriate dimensions such that the following condition holds:
(18)[Ξ+ΨTSΨΦ*-S]<0,
where S∈SΔ,Ξ=(Ξij),Φ,Ψ∈R7m×7m, i,j=1,2,…,7,
(19)Ξ11=(τl2+τu2)HD3(L+-L-)HT+M1A+ATM1T+Q4,Ξ12=M1B,Ξ13=-M1+ATM2T+Q1-L-HHTD1+L+HHTD2,Ξ22=-(1-τ)Q4,Ξ23=BTM2T,Ξ33=-M2-M2T+τl22Q2+τu22Q3,Ξ44=-2τl2Q2,Ξ46=4τl3Q2,Ξ55=-2τl2Q3,Ξ57=4τu3Q3,Ξ66=-2τl4Q2-2τl2(HD3(L+-L-)HT),Ξ77=-2τu4Q3-2τu2(HD3(L+-L-)HT),Φ=diag{M1CL,0,H,0,0,0,0},Ψ=(H,0,M2CL+(D1-D2)HL,0,0,0,0O).Proof.
Choose a new class of Lyapunov functional candidate as follows:(20)V(y(t))=V1(y(t))+V2(y(t))+V3(y(t))+V4(y(t)),
where
(21)V1(y(t))=yT(t)Q1y(t)+2∑i=1m‍{∫0hiTy(t)‍λi(fi(s)-li-s)ds+∫0hiTy(t)‍di(li+s-fi(s))ds};V2(y(t))=2∑i=1m‍{∫-τl0‍∫θ0‍∫t+μt‍αiσi(s)[fi(σi(s))-li-σi(s)]×dsdμdθ∫t+μt‍}+2∑i=1m‍{∫-τl0‍∫θ0‍∫t+μt‍αiσi(s)[li+σi(s)-fi(σi(s))]×dsdμdθ∫t+μt‍}+2∑i=1m‍{∫-τu0‍∫θ0‍∫t+μt‍αiσi(s)[fi(σi(s))-li-σi(s)]×dsdμdθ∫t+μt‍}+2∑i=1m‍{∫-τu0‍∫θ0‍∫t+μt‍αiσi(s)[li+σi(s)-fi(σi(s))]×dsdμdθ∫t+μt‍};V3(y(t))=∫-τl0‍∫θ0‍∫t+μt‍y˙T(s)Q2y˙(s)dsdμdθ+∫-τu0‍∫θ0‍∫t+μt‍y˙T(s)Q3y˙(s)dsdμdθ;V4(y(t))=∫t-τ(t)t‍yT(s)Q4y(s)ds.
The time derivative ofV(y(t)) along the trajectory of system (1) is given as
(22)V˙(y(t))=V˙1(y(t))+V˙2(y(t))+V˙3(y(t))+V˙4(y(t)),
where
(23)V˙1(y(t))=2yT(t)Q1y˙(t)+2[fT(σ(t))HT222222222222222222222-yT(t)L-HHT]D1y˙(t)+2[yT(t)L+HHT-fT(σ(t))HT]D2y˙(t),=2yT(t)(Q1-L-HHTD1+L+HHTD2)y˙(t)+2yT(t)HΔLHT(D1-D2)y˙(t).
Consider(24)V˙2(y(t))=2∑i=1m‍{∫-τl0‍∫θ0‍αiσi(t)[fi(σi(t))-li-σi(t)]dμdθ2222222-∫-τl0‍∫θ0‍αiσi(t+μ)[fi(σi(t+μ))22222-li-σi(t+μ)]dμdθ∫-τl0‍∫θ0‍}+2∑i=1m‍{∫-τl0‍∫θ0‍αiσi(t)[li+σi(t)-fi(σi(t))]dμdθ-∫-τl0‍∫θ0‍αiσi(t+μ)[li+σi(t+μ)-fi(σi(t+μ))]dμdθ∫-τl0‍}+2∑i=1m‍{∫-τu0‍∫θ0‍αiσi(t)[fi(σi(t))-li-σi(t)]dμdθ-∫-τu0‍∫θ0‍αiσi(t+μ)[fi(σi(t+μ))-li-σi(t+μ)]dμdθ∫-τu0‍}+2∑i=1m‍{∫-τu0‍∫θ0‍αiσi(t)[li+σi(t)-fi(σi(t))]dμdθ-∫-τu0‍∫θ0‍αiσi(t+μ)[li+σi(t+μ)-fi(σi(t+μ))]dμdθ∫-τu0‍}=2∑i=1m‍{αiσi(t)[fi(σi(t))-li-σi(t)]τl22-∫-τl0‍∫t+θt‍αiσi(s)[fi(σi(s))-li-σi(s)]dsdθτl22}+2∑i=1m‍{αiσi(t)[li+σi(t)-fi(σi(t))]τl22-∫-τl0‍∫t+θt‍αiyi(s)[li+yi(s)-fi(σi(s))]dsdθτl22}+2∑i=1m‍{αiσi(t)[fi(σi(t))-li-σi(t)]τu22-∫-τu0‍∫t+θt‍αiσi(s)[fi(σi(s))-li-σi(s)]dsdθτl22}+2∑i=1m‍{αiσi(t)[li+σi(t)-fi(σi(t))]τu22-∫-τu0‍∫t+θt‍αiσi(s)[li+σi(s)-fi(σi(s))]dsdθτl22}=2∑i=1m‍{αiτl22σi(t)[li+-li-]σi(t)+∑i=1n‍αiτu22σi(t)[li+-li-]σi(t)-∫-τl0‍∫t+θt‍αiσi(s)[li+-li-]σi(s)dsdθ-∫-τu0‍∫t+θt‍αiσi(s)[li+-li-]σi(s)dsdθαiτl22}.
Notice that(25)∑i=1m‍αiτl2σi(t)[li+-li-]σi(t)=τl2yT(t)HD3[L+-L-]HTy(t),∑i=1m‍αiτu2σi(t)[li+-li-]σi(t)=τu2yT(t)HD3[L+-L-]HTy(t).
From Lemma 2, we have
(26)2∑i=1m‍{-∫-τl0‍∫t+θt‍αiσi(s)[li+-li-]σi(s)dsdθ}=-2∫-τl0‍∫t+θt‍yT(s)HD3[L+-L-]HTy(s)dsdθ≤-2τl2(∫-τl0‍∫t+θt‍y(s)dsdθ)THD3×[L+-L-]HT(∫-τl0‍∫t+θt‍y(s)dsdθ).
Similarly,(27)2∑i=1m‍{-∫-τu0‍∫t+θt‍αiσi(s)[li+-li-]σi(s)dsdθ}≤-2τu2(∫-τu0‍∫t+θt‍y(s)dsdθ)THD3×[L+-L-]HT(∫-τu0‍∫t+θt‍y(s)dsdθ).
Namely,(28)V˙2(y(t))≤yT(t)[(τl2+τu2)HD3(L+-L-)HT]y(t)-2τl2(∫-τl0‍∫t+θt‍y(s)dsdθ)T[HD3(L+-L-)HT]×(∫-τl0‍∫t+θt‍y(s)dsdθ)-2τu2(∫-τu0‍∫t+θt‍y(s)dsdθ)T[HD3(L+-L-)HT]×(∫-τu0‍∫t+θt‍y(s)dsdθ);V˙3(y(t))=∫-τl0‍∫θ0‍[y˙T(t)Q2y˙(t)-y˙T(t+μ)Q2y˙(t+μ)]dμdθ+∫-τu0‍∫θ0‍[y˙T(t)Q3y˙(t)-y˙T(t+μ)Q3y˙(t+μ)]dμdθ=τl22y˙T(t)Q2y˙(t)+τu22y˙T(t)Q3y˙(t)-∫-τl0‍∫t+θt‍y˙T(s)Q2y˙(s)dsdθ-∫-τu0‍∫t+θt‍y˙T(s)Q3y˙(s)dsdθ≤τl22y˙T(t)Q2y˙(t)+τu22y˙T(t)Q3y˙(t)-2τl4(∫-τl0‍∫t+θt‍y(s)dsdθ)TQ2(∫-τl0‍∫t+θt‍y(s)dsdθ)-2τl2(∫t-τlt‍y(s)ds)TQ2(∫t-τlt‍y(s)ds)+4τl3(∫-τl0‍∫t+θt‍y(s)dsdθ)TQ2(∫t-τlt‍y(s)ds)-2τu4(∫-τu0‍∫t+θt‍y(s)dsdθ)TQ3(∫-τu0‍∫t+θt‍y(s)dsdθ)-2τu2(∫t-τut‍y(s)ds)TQ3(∫t-τut‍y(s)ds)+4τu3(∫-τu0‍∫t+θt‍y(s)dsdθ)TQ3(∫t-τut‍y(s)ds),V˙4(y(t))=yT(t)Q4y(t)-(1-τ˙(t))yT×(t-τ(t))Q4y(t-τ(t))≤yT(t)Q4y(t)-(1-τ)yT(t-τ(t))Q4y(t-τ(t)).
Additionally, for arbitrary matrices M1, M2 of appropriate dimensions, we have
(29)2y(t)TM1[-y˙(t)+(A+CLΔHT)y(t)+By(t-τ(t))]=0,2y˙T(t)M2[-y˙(t)+(A+CLΔHT)y(t)+By(t-τ(t))]=0.
Hence, V˙(y(t))≤ξT(t)(Ξ+ΦΔ~Ψ+ΨTΔ~TΦT)ξ(t), where ξ(t)=[yT(t),yT(t-τ(t)),y˙T(t),∫t-τlt‍yT(s)ds,∫t-τut‍yT(s)ds,∫-τl0‍∫t+θt‍yT(s)dsdθ,∫-τu0‍∫t+θt‍yT(s)dsdθ]T, and Δ~=Δ~T=diag(Δ,0,Δ,0,0,0,0). Since ΔTΔ≤I, thus Δ~TΔ~≤I. From Lemma 3, if the conditions in Theorem 7 are satisfied, then, Ξ+ΦΔ~Ψ+ΨTΔ~TΦT<0; by Lyapunov stable theory, the delayed Lur'e system (1) is asymptotically stable, which completes the proof.Remark 8.
Different from previous work, in the proof of Lemma5, we establish the relationship among ∫-τ0‍∫t+θt‍z˙(s)TRz˙(s)dsdθ, ∫-τ0‍∫t+θt‍z(s)dsdθ, and ∫t-τt‍z(s)ds. On the basis of this new relationship, in Theorem 7, items ∫t-τlt‍yT(s)ds,∫t-τut‍yT(s)ds,∫-τl0‍∫t+θt‍yT(s)dsdθ, and ∫-τu0‍∫t+θt‍yT(s)dsdθ are introduced as the state of ξ(t); this may reduce criterion's conservatism.For further use of the information of nonlinear functionf(σ(t)), let us define W(t)=(wij(t))m×n=Λ(t)HT, where Λ(t)=diag(Λ1(σ1(t)), Λ2(σ2(t)),…,Λm(σm(t))). Since Λi(σi(t))=λi(σi(t))li-+(1-λi(σi(t)))li+∈[li-,li+],∀i∈N, there must exist W_=(w_ij)m×n,W¯=(w¯ij)m×n such that w_ij≤wij(t)≤w¯ij,∀t∈[t0,+∞),i,j∈N. Set H~=(W_+W¯)/2=(h~ij)m×n, H^=(W¯-W_)/2=(h^ij)m×n. From the result obtained in [19], W(t)=(wij(t))m×n=Λ(t)HT can be rewritten as Λ(t)HT=H~+EΣF, where
(30)E=[h^11e1,…,h^1me1,h^m1em,…,h^mmem]∈Rm×m2,F=[h^11e1,…,h^1mem,h^m1e1,…,h^mmem]∈Rm×m2,ei(i=1,2,…,m) denotes the ith column vectors of the m×m identity matrix, and Σ=diag(ε11,…,ε1m,…,εm1,…,εmm,), where εij(i,j=1,2,…,m) satisfies |εij|≤1. This means that system (1) can be rewritten as
(31)y˙(t)=[(A+CH~)+CEΣF]y(t)+By(t-τ(t)),y(s)=φ(s),s∈[-τu,0],
where Σ satisfies ΣTΣ=ΣΣT≤I. From Theorem 7, we can get the following result.Theorem 9.
For given scalarsτl>0,τu>0,τ<1, L-=diag(l1-,l2-,…,lm-), L+=diag(l1+,l2+,…,lm+), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices D1=diag{d1,d2,…,dm}, D2=diag{λ1,λ2,…,λm}, D3=diag{α1,α2,…,αm}, symmetric positive definite matrices Q1, Q2, Q3, Q4, and arbitrary matrices M1,M2 of appropriate dimensions such that the following condition holds:
(32)[Ξ′+Ψ′TSΨ′Φ′*-S]<0,
where S∈SΔ,Ξ′=(Ξij′),Φ′,Ψ′∈R7m×7m, i,j=1,2,…,7,
(33)Ξ11′=(τl2+τu2)HD3(L+-L-)HT+M1(A+CH~)+(A+CH~)TM1T+Q4,Ξ12′=M1B,Ξ13′=-M1+(A+CH~)TM2T+Q1+(H~THT-L-HHT)D1+(L+HHT-H~THT)D2,Ξ22′=-(1-τ)Q4,Ξ23′=BTM2T,Ξ33′=-M2-M2T+τl22Q2+τu22Q3,Ξ44′=-2τl2Q2,Ξ46′=4τl3Q2,Ξ55′=-2τl2Q3,Ξ57′=4τu3Q3,Ξ66′=-2τl4Q2-2τl2(HD3(L+-L-)HT),Ξ77′=-2τu4Q3-2τu2(HD3(L+-L-)HT),Φ′=diag{M1CE,0,FT,0,0,0,0},Ψ′=(FT,0,M2CE+(D1-D2)HE,0,0,0,0O).Remark 10.
Due to the existence of itemsΨTSΨ and Ψ′TSΨ′, the results established in Theorems 7 and 9 are not LMI criteria. In order to overcome this flaw, by using the lemma derived in [20], we further establish the following more practicable stable rules.Corollary 11.
For given scalarsτl>0,τu>0,τ<1, L-=diag(l1-,l2-,…,lm-), L+=diag(l1+,l2+,…,lm+), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices D1=diag{d1,d2,…,dm}, D2=diag{λ1,λ2,…,λm}, D3=diag{α1,α2,…,αm}, symmetric positive definite matrices Q1, Q2, Q3, Q4, positive scalar δ>0, and arbitrary matrices M1,M2 of appropriate dimensions such that the following condition holds:
(34)[δΞΦδΨT*-I0**-I]<0.Corollary 12.
For given scalarsτl>0,τu>0,τ<1, L-=diag(l1-,l2-,…,lm-), L+=diag(l1+,l2+,…,lm+), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices D1=diag{d1,d2,…,dm}, D2=diag{λ1,λ2,…,λm}, D3=diag{α1,α2,…,αm}, symmetric positive definite matrices Q1, Q2, Q3, Q4, positive scaler δ>0, and arbitrary matrices M1,M2 of appropriate dimensions such that the following condition holds:
(35)[δΞ′Φ′δΨ′T*-I0**-I]<0.


## 4. Numerical Example

In order to show the effectiveness of the technique proposed in this paper, we revisit the example in [21], and compare our criteria with existing delay-dependent criteria.Example 1.
In order to compare with preview results easily, consider the delayed system (1) with parameters given by
(36)A=[-20-1-2],B=[-0.2-0.50.5-0.2],C=[-0.2-0.3],H=[0.60.8].
Time-varying delay τ(t)=τ~ is a constant.This system has been investigated in [19, 22–24]. And the maximum values of τ~max for the stability of system (1) are τ~max=0.3053, τ~max=0.3230, τ~max=0.9278, and τ~max=2.055, respectively. In order to deduce the conservatism of those criteria established in [19, 22–24], Tian et al. derived a new improved result by decomposing matrix B as B=B11+B12 in [21], where B11=[-0.05-0.20.1-0.1],B12=[-0.15-0.30.4-0.1] and gave out the maximum delay bound as τ~max=3.7272. However, one can see that, when time-varying delay τ(t) is not a constant, then the results established in [19, 21–24] are invalid. If C=[-0.200-0.3],H=[0.6000.8]f(σ(t))=[f1(σ1(t)),f2(σ2(t))]T, fi(σi(t))=0.5(|σi(t)+1|-|σi(t)-1|),i=1,2. Obviously, nonlinear function f(σ(t)) satisfies σ(t)f(σ(t))≥0, f(0)=0, and 0≤fi(σi(t))/σi(t)≤0.5. If τl=4, τu=8,τ˙(t)=0.9. Let δ=0.1; one can get that the results obtained in Corollaries 11 and 12 are feasibility. This mean that the results obtained in this paper are more general and less conservative than those in [19, 21–24].Example 2.
Consider the system described in (1) with parameters given by
(37)A=[-1.200.8-1],B=[-100-1],C=[-10.6-0.6-1],H=[1001].l1-=l2-=0, l1+=1, l2+=3.This system has been investigated in [25, 26], and the maximum value of τu for the stability of system (1) are τumax=0.5805, and τumax=0.6780, respectively. In [27], by using the sector bounds and slope bounds to the Lyapunov-Krasovskii functional through convex representation of the nonlinearities, Choi et al. improved the upper bound of τ(t) to 1.1131. Let δ=0.1,τl=0.01; by using the results obtained in Corollaries 11 and 12, one can get the maximum values of τumax are 1.2103 and 1.2314, respectively. This means that the results obtained in this paper are less conservative than those in [25–27].Example 3.
Consider the system described in (1) with parameters given by
(38)A=[-200-0.9],B=[-10-1-1],C=[0.20.3],H=[0.60.8];f(σ(t))=(0.35+0.15sin(t))σ(t). Time-varying delay τ(t)=τ~ is a constant.Obviously, the bounds of the sector nonlinearity arel1-=0.2,l1+=0.5. For this system, the maximum values of τ~max for the stability of system (1) are τ~max=2.4859 and τ~max=2.5049 in [5, 26], respectively. By using various convex optimization techniques, the results obtained in [5, 26] were further improved by Lee and Park in [12], and τ~max=2.5361. From Corollary 11, we find the maximum allowable time delay bound can be 2.5812, which means that the result established in Corollary 11 is less conservative than the ones obtained in [5, 26].


## 5. Conclusions

Combined with Lyapunov stable theory and double integral inequality, this paper researches a class of delayed Lur'e systems with interval time-varying delays. Different from previous work on this topic, this paper first establishes some new vector Wirtinger-type inequalities. Then, by using the property of convex function, the original nonlinear Lur'e system is transformed into a linear uncertain system. At last, by constructing a new Lyapunov functional including triple integral items, some new less conservative delay-dependent stability criteria are established. Numerical examples show that the new criteria derived in this paper are less conservative than some previous results obtained in the references cited therein.



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