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

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.

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_𝑡−𝜏𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠 and (∫^𝑡_𝑡−𝜏𝑥(𝑠)𝑑𝑠)^𝑇𝑄(∫^𝑡_𝑡−𝜏𝑥(𝑠)𝑑𝑠) or between ∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠𝑑𝜃 and (∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃)^𝑇𝑄(∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃). One natural question is whether there exists a relationship among ∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑧(̇𝑠)^𝑇𝑅𝑧(̇𝑠)𝑑𝑠𝑑𝜃, ∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃, ∫^𝑡_𝑡−𝜏𝑧̇^𝑇(𝑠)𝑅𝑧(̇𝑠)𝑑𝑠, 𝑧(𝑡−𝜏), and ∫^𝑡_𝑡−𝜏𝑧(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,𝑅^𝑛 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: 𝑦̇(𝑡)(𝑡)((𝑡(𝜎(𝑡,𝜎(𝑡)=𝐴𝑦+𝐵𝑦𝑡−𝜏))+𝐶𝑓))=𝐻^𝑇𝑦(𝑡)𝑦(𝑠)(𝑠),∀𝑡≥0,=𝜑,𝑠∈−𝜏_𝑢,0, (1) where 𝑦(𝑡)∈𝑅^𝑛 denotes the state vector; 𝜎(𝑡)∈𝑅^𝑚 is the output vector; 𝐻=(ℎ₁,ℎ₂,…,ℎ_𝑚)_𝑛×𝑚∈𝑅^𝑛×𝑚; 𝐴,𝐵,and𝐶 are constant known matrices of appropriate dimensions. The delay 𝜏(𝑡) is assumed to satisfy 0<𝜏_𝑙≤𝜏(𝑡)≤𝜏_𝑢,𝜏̇(𝑡)≤𝜏<1. (2)𝑓(𝜎(𝑡))∈𝑅^𝑚 denotes the nonlinear function in feedback path, which has the following form: 𝑓(𝜎(𝑡))=𝑓₁𝜎₁(𝑡),𝑓₂𝜎₂(𝑡),…,𝑓_𝑚𝜎_𝑚(𝑡)^𝑇𝜎(𝑡)=𝜎₁(𝑡),𝜎₂(𝑡),…,𝜎_𝑚(𝑡)^𝑇≜ℎ^𝑇₁𝑦(𝑡),ℎ^𝑇₂𝑦(𝑡),…,ℎ^𝑇_𝑚𝑦(𝑡)^𝑇, (3) which satisfies a sector condition with 𝑓_𝑖(⋅),(𝑖=1,2,…,𝑚) belonging to sector [𝑙⁻_𝑖,𝑙⁺_𝑖], where 𝑙⁻_𝑖,𝑙⁺_𝑖 are known constant scalars; that is, 𝑙⁻_𝑖≤𝑓_𝑖𝜎_𝑖(𝑡)𝜎_𝑖(𝑡)≤𝑙⁺_𝑖,𝑖=1,2,…,𝑚. (4)

Notice that the nonlinear function 𝑓_𝑖(⋅) can be written as a convex combination of the sector bounds as follows: 𝑓_𝑖𝜎_𝑖(𝑡)=𝜆_𝑖𝜎_𝑖(𝑡)𝑙⁻_𝑖+1−𝜆_𝑖𝜎_𝑖(𝑡)𝑙⁺_𝑖𝜎_𝑖(𝑡),𝑖=1,2,…,𝑚, (5) where 𝜆_𝑖(𝜎_𝑖)=(𝑓_𝑖(𝜎_𝑖(𝑡))−𝑙⁻_𝑖𝜎_𝑖(𝑡))/(𝑙⁺_𝑖−𝑙⁻_𝑖)𝜎_𝑖(𝑡) satisfying 0≤𝜆_𝑖(𝜎_𝑖)≤1. Namely, 𝑓_𝑖(𝜎(𝑡))=Λ_𝑖(𝜎_𝑖(𝑡))𝜎_𝑖(𝑡), where Λ_𝑖(𝜎_𝑖(𝑡)) is an element of a convex hull Co{𝑙⁻_𝑖,𝑙⁺_𝑖}. Set 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑚}, where 𝑙_𝑖=max{|𝑙⁻_𝑖|,|𝑙⁺_𝑖|}. Obviously, −1≤(𝜆_𝑖(𝜎_𝑖(𝑡))𝑙⁻_𝑖+(1−𝜆_𝑖(𝜎_𝑖(𝑡)))𝑙⁺_𝑖)/𝑙_𝑖≤1. Define Δ_𝑖=(𝜆_𝑖(𝜎_𝑖(𝑡))𝑙⁻_𝑖+(1−𝜆_𝑖(𝜎_𝑖(𝑡)))𝑙⁺_𝑖)/𝑙_𝑖,𝑖=1,2,…,𝑚, Δ=diag(Δ₁,Δ₂,…,Δ_𝑚) then nonlinear function 𝑓(𝜎(𝑡)) can be expressed as 𝑓(𝜎(𝑡))=𝐿Δ𝐻^𝑇𝑦(𝑡), where Δ satisfies Δ^𝑇Δ≤𝐼. And the system (1) can be rewritten as the following delayed uncertain system: 𝑦̇(𝑡)=𝐴+𝐶𝐿Δ𝐻^𝑇𝑦(𝑡)((𝑡,𝑦(𝑠)(𝑠)+𝐵𝑦𝑡−𝜏))=𝜑,𝑠∈−𝜏_𝑢,0. (6)

Remark 1. Different from previous work [6, 9, 10, 12, 15], in this paper, by using the convex expression 𝑓_𝑖(𝜎(𝑡))=Λ_𝑖(𝜎_𝑖(𝑡))𝜎_𝑖(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).

Let 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑚) and 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑚). For further discussion, the following lemmas are needed.

Lemma 2 (see [16]). For any positive definite symmetric constant matrix 𝑄 and scalar 𝜏>0, such that the following integrations are well defined, then −⁰_−𝜏^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)(𝑠)1𝑄𝑦𝑑𝑠𝑑𝜃≤−𝜏²⁰_−𝜏^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑄⁰_−𝜏^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃. (7)

Lemma 3 (see [17]). Given symmetric matrix 𝑃₁ and any real matrices 𝑃₂,𝑃₃ of appropriate dimensions, 𝑃₁+𝑃₂Δ𝑃₃+𝑃^𝑇₃Δ^𝑇𝑃^𝑇₂<0, (8) for all Δ∈Θ satisfying Δ^𝑇Δ≤𝐼 if and only if there exists 𝑆∈𝑆_Δ such that 𝑃₁+𝑃^𝑇₃𝑆𝑃₃𝑃₂𝑃^𝑇₂−𝑆<0, (9) where 𝑆_Δ=∶{diag(𝑠₁𝐼,…,𝑠_𝑘𝐼,𝑆₁,…,𝑆_𝑙)∶𝑆_𝑖>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. Let 𝑧(𝑡)∈𝑅^𝑛 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: ^𝑏_𝑎𝑧^𝑇(𝑠)(𝑠)()𝑅𝑧𝑑𝑠≤𝑏−𝑎²2^𝑏_𝑎𝑧̇^𝑇(𝑠)𝑅𝑧̇(𝑠)𝑑𝑠. (10)

Proof. Since 𝑧(𝑎)=0, one can get that 𝑧(𝑠)=∫^𝑠_𝑎𝑧(̇𝑡)𝑑𝑡, ^𝑏_𝑎𝑧^𝑇(𝑠)(𝑠)=𝑅𝑧𝑑𝑠_{𝑏𝑎𝑠𝑎}𝑧̇(𝑡)𝑑𝑡^𝑇𝑅^𝑠_𝑎𝑧̇(𝑡)≤𝑑𝑡𝑑𝑠_𝑏𝑎(𝑠−𝑎)^𝑠_𝑎𝑧̇^𝑇(𝑡)𝑅𝑧̇(𝑡)=𝑑𝑡𝑑𝑠_{𝑏𝑎𝑏𝑡}(𝑠−𝑎)𝑧̇^𝑇(𝑡)𝑅𝑧̇(𝑡)=𝑑𝑠𝑑𝑡_𝑏𝑎𝑧̇^𝑇(𝑡)𝑅𝑧̇(𝑡)(𝑏−𝑎)²2−(𝑡−𝑎)²2≤()𝑑𝑡𝑏−𝑎²2^𝑏_𝑎𝑧̇^𝑇(𝑡)𝑅𝑧̇(𝑡)𝑑𝑡. (11) This completes the proof.

On the basis of Lemma 4, we further give out some improved Wirtinger-type inequalities as follows.

Lemma 5. Let 𝑧(𝑡)∈𝑅^𝑛 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)−^𝑡_𝑡−𝜏𝑧̇^𝑇(𝑠)𝑅𝑧̇(𝑠)2𝑑𝑠≤−𝜏^3𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠^𝑇𝑅^𝑡_𝑡−𝜏𝑧(𝑠)−2𝑑𝑠𝜏𝑧^𝑇()()+4𝑡−𝜏𝑅𝑧𝑡−𝜏𝜏^2𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠^𝑇𝑅𝑧(𝑡−𝜏); (12)
(2)−⁰_−𝜏^𝑡_𝑡+𝜃𝑧(̇𝑠)^𝑇𝑅𝑧̇(𝑠)2𝑑𝑠𝑑𝜃≤−𝜏⁴⁰_−𝜏^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑅⁰_−𝜏^𝑡_𝑡+𝜃𝑧(𝑠)−2𝑑𝑠𝑑𝜃𝜏^2𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠^𝑇𝑅^𝑡_𝑡−𝜏𝑧(𝑠)+4𝑑𝑠𝜏³⁰_−𝜏^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑅^𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠. (13)

Proof. Set 𝑥(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 ^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)(𝑠)=𝑅𝑥𝑑𝑠_{𝑡𝑡−𝜏}(𝑧(𝑠)−𝑧(𝑡−𝜏))^𝑇𝑅(𝑧(𝑠)(≤𝜏−𝑧𝑡−𝜏))𝑑𝑠₂2^𝑡_𝑡−𝜏𝑑(𝑧(𝑠)−𝑧(𝑡−𝜏))𝑑𝑠^𝑇𝑑(𝑧(𝑠)(×𝑅−𝑧𝑡−𝜏))=𝜏𝑑𝑠𝑑𝑠₂2^𝑡_𝑡−𝜏𝑧(̇𝑠)^𝑇𝑅𝑧̇(𝑠)𝑑𝑠. (14) Additionally, from Jensen inequality, one can get ^𝑡_𝑡−𝜏(𝑧(𝑠)−𝑧(𝑡−𝜏))^𝑇𝑅(𝑧(𝑠)(≥1−𝑧𝑡−𝜏))𝑑𝑠𝜏^𝑡_𝑡−𝜏(𝑧(𝑠)−𝑧(𝑡−𝜏))𝑑𝑠^𝑇×𝑅^𝑡_𝑡−𝜏(𝑧(𝑠)−𝑧(𝑡−𝜏))𝑑𝑠. (15) Thus, we have −^𝑡_𝑡−𝜏𝑧(̇𝑠)^𝑇𝑅𝑧̇(𝑠)2𝑑𝑠≤−𝜏^3𝑡_𝑡−𝜏(𝑧(𝑠)−𝑧(𝑡−𝜏))𝑑𝑠^𝑇×𝑅^𝑡_𝑡−𝜏(𝑧(𝑠)(2−𝑧𝑡−𝜏))𝑑𝑠=−𝜏^3𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠^𝑇𝑅^𝑡_𝑡−𝜏𝑧(𝑠)−2𝑑𝑠𝜏𝑧^𝑇()()+4𝑡−𝜏𝑅𝑧𝑡−𝜏𝜏^2𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠^𝑇𝑅𝑧(𝑡−𝜏). (16) Similarly, let 𝑥(𝑠)=𝑧(𝑠)−𝑧(𝑡+𝜃); from inequality (16), Lemmas 2, and 4, one can obtain −⁰_−𝜏^𝑡_𝑡+𝜃𝑧(̇𝑠)^𝑇𝑅𝑧̇(𝑠)𝑑𝑠𝑑𝜃≤−⁰_−𝜏^𝑡_𝑡+𝜃2(−𝜃)²𝑥(𝑠)^𝑇(𝑠)1𝑅𝑥𝑑𝑠𝑑𝜃≤−𝜏²⁰_−𝜏^𝑡_𝑡+𝜃√2(𝑧(𝑠)−𝑧(𝑡+𝜃))−𝜃𝑑𝑠𝑑𝜃^𝑇×𝑅⁰_−𝜏^𝑡_𝑡+𝜃√2(𝑧(𝑠)−𝑧(𝑡+𝜃))1−𝜃𝑑𝑠𝑑𝜃≤−𝜏²⁰_−𝜏^𝑡_𝑡+𝜃√2𝑧(𝑠)𝜏𝑑𝑠𝑑𝜃−√2^𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠^𝑇×𝑅⁰_−𝜏^𝑡_𝑡+𝜃√2𝑧(𝑠)𝜏𝑑𝑠𝑑𝜃−√2^𝑡_𝑡−𝜏𝑧(𝑠)2𝑑𝑠=−𝜏⁴⁰_−𝜏^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑅⁰_−𝜏^𝑡_𝑡+𝜃𝑧(𝑠)−2𝑑𝑠𝑑𝜃𝜏^2𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠^𝑇𝑅^𝑡_𝑡−𝜏𝑧(𝑠)+4𝑑𝑠𝜏³⁰_−𝜏^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑅^𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠. (17) 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 ∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑧(̇𝑠)^𝑇𝑅𝑧(̇𝑠)𝑑𝑠𝑑𝜃, ∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃, ∫^𝑡_𝑡−𝜏𝑧̇^𝑇(𝑠)𝑅𝑧(̇𝑠)𝑑𝑠, 𝑧(𝑡−𝜏), and ∫^𝑡_𝑡−𝜏𝑧(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 scalars 𝜏_𝑙>0, 𝜏_𝑢>0, 𝜏<1, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑚), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑚), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices 𝐷₁=diag{𝑑₁,𝑑₂,…,𝑑_𝑚}, 𝐷₂=diag{𝜆₁,𝜆₂,…,𝜆_𝑚}, 𝐷₃=diag{𝛼₁,𝛼₂,…,𝛼_𝑚}, symmetric positive definite matrices 𝑄₁, 𝑄₂, 𝑄₃, 𝑄₄, and arbitrary matrices 𝑀₁,𝑀₂ of appropriate dimensions such that the following condition holds: Ξ+Ψ^𝑇𝑆ΨΦ∗−𝑆<0, (18) where 𝑆∈𝑆_Δ,Ξ=(Ξ_𝑖𝑗),Φ,Ψ∈𝑅^7𝑚×7𝑚, 𝑖,𝑗=1,2,…,7, Ξ₁₁=𝜏²_𝑙+𝜏²_𝑢𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇+𝑀₁𝐴+𝐴^𝑇𝑀^𝑇₁+𝑄₄,Ξ₁₂=𝑀₁Ξ𝐵,₁₃=−𝑀₁+𝐴^𝑇𝑀^𝑇₂+𝑄₁−𝐿⁻𝐻𝐻^𝑇𝐷₁+𝐿⁺𝐻𝐻^𝑇𝐷₂,Ξ₂₂=−(1−𝜏)𝑄₄,Ξ₂₃=𝐵^𝑇𝑀^𝑇₂,Ξ₃₃=−𝑀₂−𝑀^𝑇₂+𝜏²_𝑙2𝑄₂+𝜏²_𝑢2𝑄₃,Ξ₄₄2=−𝜏²_𝑙𝑄₂,Ξ₄₆=4𝜏³_𝑙𝑄₂,Ξ₅₅2=−𝜏²_𝑙𝑄₃,Ξ₅₇=4𝜏³_𝑢𝑄₃,Ξ₆₆2=−𝜏⁴_𝑙𝑄₂−2𝜏²_𝑙𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇,Ξ₇₇2=−𝜏⁴_𝑢𝑄₃−2𝜏²_𝑢𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇,Φ=diag𝑀₁𝐶𝐿,0,𝐻,0,0,0,0,Ψ=𝐻,0,𝑀₂𝐶𝐿+𝐷₁−𝐷₂𝑂.𝐻𝐿,0,0,0,0 (19)

Proof. Choose a new class of Lyapunov functional candidate as follows: 𝑉𝑦(𝑡)=𝑉₁𝑦(𝑡)+𝑉₂𝑦(𝑡)+𝑉₃𝑦(𝑡)+𝑉₄𝑦(𝑡), (20) where 𝑉₁𝑦(𝑡)=𝑦^𝑇(𝑡)𝑄₁𝑦(𝑡)+2^𝑚_𝑖=1^ℎ𝑇𝑖₀^𝑦(𝑡)𝜆_𝑖𝑓_𝑖(𝑠)−𝑙⁻_𝑖𝑠+𝑑𝑠_{ℎ𝑇𝑖0𝑦(𝑡)}𝑑_𝑖𝑙⁺_𝑖𝑠−𝑓_𝑖(𝑠);𝑉𝑑𝑠₂𝑦(𝑡)=2^𝑚_𝑖=1⁰_−𝜏𝑙⁰_𝜃^𝑡_𝑡+𝜇𝛼_𝑖𝜎_𝑖(𝑠)𝑓_𝑖𝜎_𝑖(𝑠)−𝑙⁻_𝑖𝜎_𝑖(𝑠)×𝑑𝑠𝑑𝜇𝑑𝜃+2^𝑚_𝑖=1⁰_−𝜏𝑙⁰_𝜃^𝑡_𝑡+𝜇𝛼_𝑖𝜎_𝑖(𝑠)𝑙⁺_𝑖𝜎_𝑖(𝑠)−𝑓_𝑖𝜎_𝑖(𝑠)×𝑑𝑠𝑑𝜇𝑑𝜃+2^𝑚_𝑖=1⁰_−𝜏𝑢⁰_𝜃^𝑡_𝑡+𝜇𝛼_𝑖𝜎_𝑖(𝑠)𝑓_𝑖𝜎_𝑖(𝑠)−𝑙⁻_𝑖𝜎_𝑖(𝑠)×𝑑𝑠𝑑𝜇𝑑𝜃+2^𝑚_𝑖=1⁰_−𝜏𝑢⁰_𝜃^𝑡_𝑡+𝜇𝛼_𝑖𝜎_𝑖(𝑠)𝑙⁺_𝑖𝜎_𝑖(𝑠)−𝑓_𝑖𝜎_𝑖(𝑠);𝑉×𝑑𝑠𝑑𝜇𝑑𝜃₃𝑦(𝑡)=⁰_−𝜏𝑙⁰_𝜃^𝑡_𝑡+𝜇𝑦̇^𝑇(𝑠)𝑄₂𝑦̇(𝑠)+𝑑𝑠𝑑𝜇𝑑𝜃_{0−𝜏𝑢0𝜃𝑡𝑡+𝜇}𝑦̇^𝑇(𝑠)𝑄₃𝑦̇(𝑠)𝑉𝑑𝑠𝑑𝜇𝑑𝜃;₄𝑦(𝑡)=^𝑡_{(𝑡)𝑡−𝜏}𝑦^𝑇(𝑠)𝑄₄𝑦(𝑠)𝑑𝑠. (21)
The time derivative of 𝑉(𝑦(𝑡)) along the trajectory of system (1) is given as 𝑉̇𝑦(𝑡)=𝑉̇₁𝑦(𝑡)+𝑉̇₂𝑦(𝑡)+𝑉̇₃𝑦(𝑡)+𝑉̇₄𝑦(𝑡), (22) where 𝑉̇₁𝑦(𝑡)=2𝑦^𝑇(𝑡)𝑄₁𝑦̇(𝑡)+2𝑓^𝑇(𝜎(𝑡))𝐻^𝑇−𝑦^𝑇(𝑡)𝐿⁻𝐻𝐻^𝑇𝐷₁𝑦̇(𝑡)+2𝑦^𝑇(𝑡)𝐿⁺𝐻𝐻^𝑇−𝑓^𝑇(𝜎(𝑡))𝐻^𝑇𝐷₂𝑦̇(𝑡),=2𝑦^𝑇(𝑡)𝑄₁−𝐿⁻𝐻𝐻^𝑇𝐷₁+𝐿⁺𝐻𝐻^𝑇𝐷₂𝑦̇(𝑡)+2𝑦^𝑇(𝑡)𝐻Δ𝐿𝐻^𝑇𝐷₁−𝐷₂𝑦̇(𝑡). (23)
Consider 𝑉̇₂𝑦(𝑡)=2^𝑚_𝑖=1⁰_−𝜏𝑙⁰_𝜃𝛼_𝑖𝜎_𝑖(𝑡)𝑓_𝑖𝜎_𝑖(𝑡)−𝑙⁻_𝑖𝜎_𝑖(𝑡−)𝑑𝜇𝑑𝜃_{0−𝜏𝑙0𝜃}𝛼_𝑖𝜎_𝑖𝑡+𝜇𝑓_𝑖𝜎_𝑖𝑡+𝜇−𝑙⁻_𝑖𝜎_𝑖𝑡+𝜇𝑑𝜇𝑑𝜃+2^𝑚_𝑖=1⁰_−𝜏𝑙⁰_𝜃𝛼_𝑖𝜎_𝑖(𝑡)𝑙⁺_𝑖𝜎_𝑖(𝑡)−𝑓_𝑖𝜎_𝑖(𝑡)−𝑑𝜇𝑑𝜃_{0−𝜏𝑙0𝜃}𝛼_𝑖𝜎_𝑖𝑡+𝜇𝑙⁺_𝑖𝜎_𝑖𝑡+𝜇−𝑓_𝑖𝜎_𝑖𝑡+𝜇𝑑𝜇𝑑𝜃+2^𝑚_𝑖=1⁰_−𝜏𝑢⁰_𝜃𝛼_𝑖𝜎_𝑖(𝑡)𝑓_𝑖𝜎_𝑖(𝑡)−𝑙⁻_𝑖𝜎_𝑖(𝑡−)𝑑𝜇𝑑𝜃_{0−𝜏𝑢0𝜃}𝛼_𝑖𝜎_𝑖𝑡+𝜇𝑓_𝑖𝜎_𝑖𝑡+𝜇−𝑙⁻_𝑖𝜎_𝑖𝑡+𝜇𝑑𝜇𝑑𝜃+2^𝑚_𝑖=1⁰_−𝜏𝑢⁰_𝜃𝛼_𝑖𝜎_𝑖(𝑡)𝑙⁺_𝑖𝜎_𝑖(𝑡)−𝑓_𝑖𝜎_𝑖(𝑡)−𝑑𝜇𝑑𝜃_{0−𝜏𝑢0𝜃}𝛼_𝑖𝜎_𝑖𝑡+𝜇𝑙⁺_𝑖𝜎_𝑖𝑡+𝜇−𝑓_𝑖𝜎_𝑖𝑡+𝜇𝑑𝜇𝑑𝜃=2^𝑚_𝑖=1𝛼_𝑖𝜎_𝑖(𝑡)𝑓_𝑖𝜎_𝑖(𝑡)−𝑙⁻_𝑖𝜎_𝑖(𝑡𝜏)^2𝑙2−⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝛼_𝑖𝜎_𝑖(𝑠)𝑓_𝑖𝜎_𝑖(𝑠)−𝑙⁻_𝑖𝜎_𝑖(𝑠)𝑑𝑠𝑑𝜃+2^𝑚_𝑖=1𝛼_𝑖𝜎_𝑖(𝑡)𝑙⁺_𝑖𝜎_𝑖(𝑡)−𝑓_𝑖𝜎_𝑖(𝑡)𝜏²_𝑙2−⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝛼_𝑖𝑦_𝑖(𝑠)𝑙⁺_𝑖𝑦_𝑖(𝑠)−𝑓_𝑖𝜎_𝑖(𝑠)𝑑𝑠𝑑𝜃+2^𝑚_𝑖=1𝛼_𝑖𝜎_𝑖(𝑡)𝑓_𝑖𝜎_𝑖(𝑡)−𝑙⁻_𝑖𝜎_𝑖(𝑡𝜏)^2𝑢2−⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝛼_𝑖𝜎_𝑖(𝑠)𝑓_𝑖𝜎_𝑖(𝑠)−𝑙⁻_𝑖𝜎_𝑖(𝑠)𝑑𝑠𝑑𝜃+2^𝑚_𝑖=1𝛼_𝑖𝜎_𝑖(𝑡)𝑙⁺_𝑖𝜎_𝑖(𝑡)−𝑓_𝑖𝜎_𝑖(𝑡)𝜏²_𝑢2−⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝛼_𝑖𝜎_𝑖(𝑠)𝑙⁺_𝑖𝜎_𝑖(𝑠)−𝑓_𝑖𝜎_𝑖(𝑠)𝑑𝑠𝑑𝜃=2^𝑚_𝑖=1𝛼_𝑖𝜏²_𝑙2𝜎_𝑖(𝑡)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑡)+^𝑛_𝑖=1𝛼_𝑖𝜏²_𝑢2𝜎_𝑖(𝑡)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑡)−⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝛼_𝑖𝜎_𝑖(𝑠)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑠)−𝑑𝑠𝑑𝜃_{0−𝜏𝑢𝑡𝑡+𝜃}𝛼_𝑖𝜎_𝑖(𝑠)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑠)𝑑𝑠𝑑𝜃. (24)
Notice that ^𝑚_𝑖=1𝛼_𝑖𝜏²_𝑙𝜎_𝑖(𝑡)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑡)=𝜏²_𝑙𝑦^𝑇(𝑡)𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇𝑦(𝑡),_𝑚𝑖=1𝛼_𝑖𝜏²_𝑢𝜎_𝑖(𝑡)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑡)=𝜏²_𝑢𝑦^𝑇(𝑡)𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇𝑦(𝑡). (25) From Lemma 2, we have 2^𝑚_𝑖=1−⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝛼_𝑖𝜎_𝑖(𝑠)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑠)𝑑𝑠𝑑𝜃=−2⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇𝑦(𝑠)2𝑑𝑠𝑑𝜃≤−𝜏²_𝑙⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝐻𝐷₃×𝐿⁺−𝐿⁻𝐻^𝑇0_−𝜏𝑙^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃. (26)
Similarly, 2^𝑚_𝑖=1−⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝛼_𝑖𝜎_𝑖(𝑠)𝑙⁺_𝑖−𝑙⁻_𝑖𝜎_𝑖(𝑠)2𝑑𝑠𝑑𝜃≤−𝜏²_𝑢⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝐻𝐷₃×𝐿⁺−𝐿⁻𝐻^𝑇0_−𝜏𝑢^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃. (27)
Namely, 𝑉̇₂𝑦(𝑡)≤𝑦^𝑇(𝑡)𝜏²_𝑙+𝜏²_𝑢𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇𝑦(𝑡)−2𝜏²_𝑙⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇×⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦(𝑠)−2𝑑𝑠𝑑𝜃𝜏²_𝑢⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇×⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝑦(𝑠);𝑉̇𝑑𝑠𝑑𝜃₃𝑦(𝑡)=⁰_−𝜏𝑙⁰_𝜃𝑦̇^𝑇(𝑡)𝑄₂𝑦̇(𝑡)−𝑦̇^𝑇𝑡+𝜇𝑄₂𝑦̇𝑡+𝜇𝑑𝜇𝑑𝜃+⁰_−𝜏𝑢⁰_𝜃𝑦̇^𝑇(𝑡)𝑄₃𝑦̇(𝑡)−𝑦̇^𝑇𝑡+𝜇𝑄₃𝑦̇=𝜏𝑡+𝜇𝑑𝜇𝑑𝜃_2𝑙2𝑦̇^𝑇(𝑡)𝑄₂𝑦̇(𝑡)+𝜏²_𝑢2𝑦̇^𝑇(𝑡)𝑄₃𝑦̇(𝑡)−⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦̇^𝑇(𝑠)𝑄₂𝑦̇(𝑠)𝑑𝑠𝑑𝜃−⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝑦̇^𝑇(𝑠)𝑄₃𝑦̇(𝑠)≤𝜏𝑑𝑠𝑑𝜃_2𝑙2𝑦̇^𝑇(𝑡)𝑄₂𝑦̇(𝑡)+𝜏²_𝑢2𝑦̇^𝑇(𝑡)𝑄₃𝑦̇(𝑡)−2𝜏⁴_𝑙⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑄₂⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦(𝑠)−2𝑑𝑠𝑑𝜃𝜏²_𝑙^𝑡_{𝑡−𝜏𝑙}𝑦(𝑠)𝑑𝑠^𝑇𝑄₂^𝑡_{𝑡−𝜏𝑙}𝑦(𝑠)+4𝑑𝑠𝜏³_𝑙⁰_−𝜏𝑙^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑄₂^𝑡_{𝑡−𝜏𝑙}𝑦(𝑠)−2𝑑𝑠𝜏⁴_𝑢⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑄₃⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝑦(𝑠)−2𝑑𝑠𝑑𝜃𝜏²_𝑢^𝑡_{𝑡−𝜏𝑢}𝑦(𝑠)𝑑𝑠^𝑇𝑄₃^𝑡_{𝑡−𝜏𝑢}𝑦(𝑠)+4𝑑𝑠𝜏³_𝑢⁰_−𝜏𝑢^𝑡_𝑡+𝜃𝑦(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑄₃^𝑡_{𝑡−𝜏𝑢}𝑦(𝑠),𝑉̇𝑑𝑠₄𝑦(𝑡)=𝑦^𝑇(𝑡)𝑄₄𝑦(𝑡)−(1−𝜏̇(𝑡))𝑦^𝑇×(𝑡−𝜏(𝑡))𝑄₄𝑦(𝑡−𝜏(𝑡))≤𝑦^𝑇(𝑡)𝑄₄𝑦(𝑡)−(1−𝜏)𝑦^𝑇(𝑡−𝜏(𝑡))𝑄₄𝑦(𝑡−𝜏(𝑡)). (28) Additionally, for arbitrary matrices 𝑀₁, 𝑀₂ of appropriate dimensions, we have 2𝑦(𝑡)^𝑇𝑀₁−𝑦̇(𝑡)+𝐴+𝐶𝐿Δ𝐻^𝑇𝑦(𝑡)((𝑡2𝑦̇+𝐵𝑦𝑡−𝜏))=0,_𝑇(𝑡)𝑀₂−𝑦̇(𝑡)+𝐴+𝐶𝐿Δ𝐻^𝑇𝑦(𝑡)+𝐵𝑦(𝑡−𝜏(𝑡))=0. (29) Hence, 𝑉̇(𝑦(𝑡))≤𝜉^𝑇(𝑡)(Ξ+ΦΔΨ+Ψ^𝑇Δ^𝑇Φ^𝑇)𝜉(𝑡), where 𝜉(𝑡)=[𝑦^𝑇(𝑡),𝑦^𝑇(𝑡−𝜏(𝑡)),𝑦̇^𝑇(𝑡),∫^𝑡_{𝑡−𝜏𝑙}𝑦^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−𝜏𝑢}𝑦^𝑇(𝑠)𝑑𝑠,∫⁰_−𝜏𝑙∫^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑑𝑠𝑑𝜃,∫⁰_−𝜏𝑢∫^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑑𝑠𝑑𝜃]^𝑇, and Δ=Δ^𝑇=diag(Δ,0,Δ,0,0,0,0). Since Δ^𝑇Δ≤𝐼, thus Δ^𝑇Δ≤𝐼. From Lemma 3, if the conditions in Theorem 7 are satisfied, then, Ξ+ΦΔΨ+Ψ^𝑇Δ^𝑇Φ^𝑇<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 Lemma 5, we establish the relationship among ∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑧(̇𝑠)^𝑇𝑅𝑧(̇𝑠)𝑑𝑠𝑑𝜃, ∫⁰_−𝜏∫^𝑡_𝑡+𝜃𝑧(𝑠)𝑑𝑠𝑑𝜃, and ∫^𝑡_𝑡−𝜏𝑧(𝑠)𝑑𝑠. On the basis of this new relationship, in Theorem 7, items ∫^𝑡_{𝑡−𝜏𝑙}𝑦^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−𝜏𝑢}𝑦^𝑇(𝑠)𝑑𝑠,∫⁰_−𝜏𝑙∫^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑑𝑠𝑑𝜃, and ∫⁰_−𝜏𝑢∫^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑑𝑠𝑑𝜃 are introduced as the state of 𝜉(𝑡); this may reduce criterion's conservatism.

For further use of the information of nonlinear function 𝑓(𝜎(𝑡)), let us define 𝑊(𝑡)=(𝑤_𝑖𝑗(𝑡))_𝑚×𝑛=Λ(𝑡)𝐻^𝑇, where Λ(𝑡)=diag(Λ₁(𝜎₁(𝑡)), Λ₂(𝜎₂(𝑡)),…,Λ_𝑚(𝜎_𝑚(𝑡))). Since Λ_𝑖(𝜎_𝑖(𝑡))=𝜆_𝑖(𝜎_𝑖(𝑡))𝑙⁻_𝑖+(1−𝜆_𝑖(𝜎_𝑖(𝑡)))𝑙⁺_𝑖∈[𝑙⁻_𝑖,𝑙⁺_𝑖],∀𝑖∈𝑁, there must exist 𝑊=(𝑤_𝑖𝑗)_𝑚×𝑛,𝑊=(𝑤_𝑖𝑗)_𝑚×𝑛 such that 𝑤_𝑖𝑗≤𝑤_𝑖𝑗(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)=(𝑤_𝑖𝑗(𝑡))_𝑚×𝑛=Λ(𝑡)𝐻^𝑇 can be rewritten as Λ(𝑡)𝐻^𝑇=𝐻+𝐸Σ𝐹, where 𝐸=ℎ₁₁𝑒₁,…,ℎ_1𝑚𝑒₁,ℎ_𝑚1𝑒_𝑚,…,ℎ_𝑚𝑚𝑒_𝑚∈𝑅^𝑚×𝑚2,𝐹=ℎ₁₁𝑒₁,…,ℎ_1𝑚𝑒_𝑚,ℎ_𝑚1𝑒₁,…,ℎ_𝑚𝑚𝑒_𝑚∈𝑅^𝑚×𝑚2, (30)𝑒_𝑖(𝑖=1,2,…,𝑚) denotes the 𝑖th column vectors of the 𝑚×𝑚 identity matrix, and Σ=diag(𝜀₁₁,…,𝜀_1𝑚,…,𝜀_𝑚1,…,𝜀_𝑚𝑚,), where 𝜀_𝑖𝑗(𝑖,𝑗=1,2,…,𝑚) satisfies |𝜀_𝑖𝑗|≤1. This means that system (1) can be rewritten as 𝑦̇(𝑡)=𝐻𝑦(𝑡)((𝑡,𝑦(𝑠)(𝑠)𝐴+𝐶+𝐶𝐸Σ𝐹+𝐵𝑦𝑡−𝜏))=𝜑,𝑠∈−𝜏_𝑢,0, (31) where Σ satisfies Σ^𝑇Σ=ΣΣ^𝑇≤𝐼. From Theorem 7, we can get the following result.

Theorem 9. For given scalars 𝜏_𝑙>0,𝜏_𝑢>0,𝜏<1, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑚), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑚), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices 𝐷₁=diag{𝑑₁,𝑑₂,…,𝑑_𝑚}, 𝐷₂=diag{𝜆₁,𝜆₂,…,𝜆_𝑚}, 𝐷₃=diag{𝛼₁,𝛼₂,…,𝛼_𝑚}, symmetric positive definite matrices 𝑄₁, 𝑄₂, 𝑄₃, 𝑄₄, and arbitrary matrices 𝑀₁,𝑀₂ of appropriate dimensions such that the following condition holds: Ξ^′+Ψ^′𝑇𝑆Ψ^′Φ^′∗−𝑆<0, (32) where 𝑆∈𝑆_Δ,Ξ^′=(Ξ^′_𝑖𝑗),Φ^′,Ψ^′∈𝑅^7𝑚×7𝑚, 𝑖,𝑗=1,2,…,7, Ξ^′₁₁=𝜏²_𝑙+𝜏²_𝑢𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇+𝑀₁𝐻+𝐻𝐴+𝐶𝐴+𝐶^𝑇𝑀^𝑇₁+𝑄₄,Ξ^′₁₂=𝑀₁Ξ𝐵,_′13=−𝑀₁+𝐴+𝐶𝐻^𝑇𝑀^𝑇₂+𝑄₁+𝐻^𝑇𝐻^𝑇−𝐿⁻𝐻𝐻^𝑇𝐷₁+𝐿⁺𝐻𝐻^𝑇−𝐻^𝑇𝐻^𝑇𝐷₂,Ξ^′₂₂=−(1−𝜏)𝑄₄,Ξ^′₂₃=𝐵^𝑇𝑀^𝑇₂,Ξ^′₃₃=−𝑀₂−𝑀^𝑇₂+𝜏²_𝑙2𝑄₂+𝜏²_𝑢2𝑄₃,Ξ^′₄₄2=−𝜏²_𝑙𝑄₂,Ξ^′₄₆=4𝜏³_𝑙𝑄₂,Ξ^′₅₅2=−𝜏²_𝑙𝑄₃,Ξ^′₅₇=4𝜏³_𝑢𝑄₃,Ξ^′₆₆2=−𝜏⁴_𝑙𝑄₂−2𝜏²_𝑙𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇,Ξ^′₇₇2=−𝜏⁴_𝑢𝑄₃−2𝜏²_𝑢𝐻𝐷₃𝐿⁺−𝐿⁻𝐻^𝑇,Φ^′=diag𝑀₁𝐶𝐸,0,𝐹^𝑇,Ψ,0,0,0,0_′=𝐹^𝑇,0,𝑀₂𝐶𝐸+𝐷₁−𝐷₂𝑂.𝐻𝐸,0,0,0,0 (33)

Remark 10. Due to the existence of items Ψ^𝑇𝑆Ψ 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 scalars 𝜏_𝑙>0,𝜏_𝑢>0,𝜏<1, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑚), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑚), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices 𝐷₁=diag{𝑑₁,𝑑₂,…,𝑑_𝑚}, 𝐷₂=diag{𝜆₁,𝜆₂,…,𝜆_𝑚}, 𝐷₃=diag{𝛼₁,𝛼₂,…,𝛼_𝑚}, 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: 𝛿ΞΦ𝛿Ψ^𝑇∗−𝐼0∗∗−𝐼<0. (34)

Corollary 12. For given scalars 𝜏_𝑙>0,𝜏_𝑢>0,𝜏<1, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑚), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑚), system (1) is globally asymptotically stable if there exist positive definite diagonal matrices 𝐷₁=diag{𝑑₁,𝑑₂,…,𝑑_𝑚}, 𝐷₂=diag{𝜆₁,𝜆₂,…,𝜆_𝑚}, 𝐷₃=diag{𝛼₁,𝛼₂,…,𝛼_𝑚}, 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: 𝛿Ξ^′Φ^′𝛿Ψ^′𝑇∗−𝐼0∗∗−𝐼<0. (35)

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 ,.𝐴=−20−1−2,𝐵=−0.2−0.50.5−0.2𝐶=−0.2−0.3,𝐻=0.60.8 (36) 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=^{−0.05−0.2}_0.1−0.1,𝐵₁₂=^{−0.15−0.3}_0.4−0.1 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=^−0.20_0−0.3,𝐻=^0.60_00.8𝑓(𝜎(𝑡))=[𝑓₁(𝜎₁(𝑡)),𝑓₂(𝜎₂(𝑡))]^𝑇, 𝑓_𝑖(𝜎_𝑖(𝑡))=0.5(|𝜎_𝑖(𝑡)+1|−|𝜎_𝑖(𝑡)−1|),𝑖=1,2. Obviously, nonlinear function 𝑓(𝜎(𝑡)) satisfies 𝜎(𝑡)𝑓(𝜎(𝑡))≥0, 𝑓(0)=0, and 0≤𝑓_𝑖(𝜎_𝑖(𝑡))/𝜎_𝑖(𝑡)≤0.5. If 𝜏_𝑙=4, 𝜏_𝑢=8,𝜏̇(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 ,.𝐴=−1.200.8−1,𝐵=−100−1𝐶=−10.6−0.6−1,𝐻=1001 (37)𝑙⁻₁=𝑙⁻₂=0, 𝑙⁺₁=1, 𝑙⁺₂=3.

This system has been investigated in [25, 26], and the maximum value of 𝜏_𝑢 for the stability of system (1) are 𝜏_𝑢^max=0.5805, and 𝜏_𝑢^max=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,𝜏_𝑙=0.01; by using the results obtained in Corollaries 11 and 12, one can get the maximum values of 𝜏_𝑢^max 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 ,;𝐴=−200−0.9,𝐵=−10−1−1𝐶=0.20.3,𝐻=0.60.8 (38)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 are 𝑙⁻₁=0.2,𝑙⁺₁=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.

Acknowledgments

This work was supported by China Postdoctoral Science Foundation Grant (2012 M521718) and Soft Science Research Project in Guizhou Province ([2011]LKC2004).