Robust Stability of Switched Delay Systems with Average Dwell Time under Asynchronous Switching

Abstract

The problem of robust stability of switched delay systems with average dwell time under asynchronous switching is investigated. By taking advantage of the average dwell-time method and an integral inequality, two sufficient conditions are developed to guarantee the global exponential stability of the considered switched system. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed techniques.

1. Introduction

In recent years, there has been increasing interest in the analysis and switched systems because of its applications in a variety of areas such as signal processing, signal estimation, pattern recognition, communications, control application, and many practical control systems. Switched linear systems comprise a collection of linear subsystems described by differential or difference equations and a switching law to specify the switching among these subsystems. A switched system is a combination of discrete and continuous dynamical systems. All of these systems arise as models for phenomena which cannot be described by exclusively continuous or exclusively discrete processes. Most recently, on the basis of Lyapunov functions and other analysis tools, the stability or stabilization for linear or nonlinear switched systems has been further investigated, and many valuable results have been obtained, for a recent survey on this topic and related questions has attracted increasing attention [1–10]. The average dwell-time is an effective method for the switched systems which do not exist common Lyapunov function. Time delay commonly exists in engineering. Because of time delay, the system can become unstable or less capable, it is significant to study time delay. There are two kinds of stability for switched systems with time delay: time delay-independent stability and time delay-dependent stability. The time delay-independent stability is obviously conservative to the bounded time delay or small time delay, many results are obtained [11–13]. At present, there has been increasing interest in time-delay switched systems, and many valuable results have been obtained [14–18].

It is worth noting that the aforementioned results are all based on the basic assumption that the switching instants are simultaneous with those of the system. However, in actual operation, there inevitably exists asynchronous switching between the controllers and the practical subsystems, that is to say, the real switching instants of the controller exceed or lag behind those of the system, which will deteriorate performance of the systems. In fact, the necessity of taking into consideration the asynchronous switching is shown in efficient controller design in many mechanical and chemical systems. There are some results presented on control synthesis under asynchronous switching which have been proposed [19–28]. However, to the best of our knowledge, the issue of switched delay systems under asynchronous switching has not fully been investigated, which motivated this study for us.

In this paper, we deal with the problem of robust stability and 𝐿₂-gain of switched delay systems under asynchronous switching. In terms of the average dwell-time method and an integral inequality, two sufficient conditions are developed to guarantee the global exponential stability of the considered switched system. Finally, a numerical example is provided to illustrate the effectiveness and feasibility.

Notations. Throughout this paper, ℜ^𝑛 denotes the 𝑛-dimensional Euclidean space and ℜ^𝑛×𝑛 refers to the set of all 𝑛×𝑚 real matrices. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) mean that the matrix 𝑋, 𝑌 are positive semidefinite, (resp., and positive definite). 𝐼 is the identity matrix with appropriate dimensions. ∗ represents the elements below the main diagonal of a symmetric matrix. The superscripts ⊺ and −1 stand for matrix transposition and matrix inverse, respectively; ‖⋅‖_𝑐 denotes the Euclidean norm. 𝜆_max(𝑃) and 𝜆_min(𝑃) denote the maximum and minimum eigenvalues of matrix 𝑃, respectively. The shorthand diag{𝑀₁,…,𝑀_𝑛} denotes a block diagonal matrix with diagonal blocks being the matrices 𝑀₁,…,𝑀_𝑛. In this paper, if not explicit, matrices are assumed to have compatible dimensions.

2. Preliminaries

In this paper, we consider the following time-delay system described by 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝑥(𝑡−𝜏(𝑡))+𝐵_𝜎(𝑡)𝜔(𝑡),𝑧(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐷_𝜎(𝑡)𝜔(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈−𝜏_𝑀,0, (2.1) where 𝑥(𝑡)∈ℛ^𝑛 is the state of the system, 𝜔(𝑡)∈ℛ^𝑞 is the noise signal. Switching signal 𝜎(𝑡) is a piecewise constant function of time 𝑡, and we take values in a finite set ℙ={1,2,…,𝑟},𝑟>0 denotes the number of subsystems. 𝜎(𝑡)=𝑖∈ℙ means the 𝑖th subsystem is active, and the corresponding subsystem matrices are denoted by known constant matrices 𝐴_𝑖, 𝐴_𝜏𝑖,𝐵_𝑖,𝐶_𝑖, and 𝐷_𝑖 with appropriate dimensions and 𝜏(𝑡) is the unknown time-varying delay satisfying 𝜏₀≤𝜏(𝑡)≤𝜏_𝑀,𝜏̇(𝑡)≤𝜏, (2.2) where 𝜏₀,𝜏_𝑀, and 𝜏 are constants; 𝐴,𝐴_𝜏, and 𝐶_𝑖 are known real constant matrices; 𝜙(𝑡) is a compatible vector-valued initial function on [−𝜏_𝑀,0].

The switching signal 𝜎(𝑡)∶ℜ⁺→ℙ={1,2,…,𝑟} discussed in this paper is time dependent, that is, 𝜎(𝑡)∶{(𝑡₀,𝜎(𝑡₀)),(𝑡₁,𝜎(𝑡₁)),…,(𝑡_ℑ,𝜎(𝑡_ℑ))}, where 𝑡₀ is the initial instant. In this paper, we denote 𝑡_ℑ represent the ℑth switching instant. for convenience, 𝜎(𝑡) is used to denote the practical switching signal which can be written as 𝜎(𝑡)∶𝑡₀+Δ₀,𝜎𝑡₀,𝑡₁+Δ₁,𝜎𝑡₁,…,𝑡_ℑ+Δ_ℑ,𝜎𝑡_ℑ, (2.3) where Δ_ℑ<inf_ℑ≥1|𝑡_ℑ+1−𝑡_ℑ|, 𝑘=0,1,…. Then there exist a matched period time interval [𝑡_ℑ−1+Δ_ℑ−1) and mismatched period time interval [𝑡_ℑ,𝑡_ℑ+Δ_ℑ) because of asynchronous switching. For simplicity, we assume that Δ_ℑ>0, ℑ=0,1,….

In this paper, we denote Λ₁ and Λ₂ as follows: Λ₁=𝑥(𝑡)∈ℜ^𝑛∣𝜎𝑡_ℑ=𝑖,𝜎𝑡_ℑ+1=𝑗,𝑡∈𝑡_ℑ+Δ_ℑ,𝑡_ℑ+1,Λ,ℑ=0,1,2,…₂=𝑥(𝑡)∈ℜ^𝑛∣𝜎𝑡_ℑ+1=𝑗,𝑡∈𝑡_ℑ+1,𝑡_ℑ+1+Δ_ℑ+1,ℑ=0,1,2,…. (2.4)

Let 𝜑(𝑡)=𝑥^⊺(𝑡),𝑥^⊺(𝑡−𝜏(𝑡)),𝑥^⊺𝑡−𝜏_𝑀,𝑥^⊺𝑡−𝜏₀,𝜔^⊺(𝑡)^⊺. (2.5)

This way, system (2.1) can be rewritten as 𝑥̇(𝑡)=Υ_𝜎(𝑡)(𝑡),𝑧(𝑡)⋅𝜑=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐷_𝜎(𝑡)𝜔(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈−𝜏_𝑀,0, (2.6) where Υ_𝜎(𝑡)=𝐴_𝜎(𝑡),𝐴_{𝜏𝜎(𝑡)},0,0,𝐵_𝜎(𝑡). (2.7)

First of all, we will give some definitions and lemmas about system (2.6) which plays an important role in the derivation of our result.

Definition 2.1 (see [29]). The unforced system is said to be exponential stable if there exist constants 𝜈>0 and 𝜗>0 such that ‖𝑥(𝑡)‖≤𝜈sup_{−𝜏𝑀≤𝜍≤0}𝜑(𝜍)_𝑐𝑒^−𝜗𝑡. (2.8)

Definition 2.2 (see [30]). For 𝛾>0, the switched system (2.1) is said to have weighted 𝐿₂-gain, if under zero initial condition 𝜙(𝑡),𝑡∈[−𝜏_𝑀,0], it holds that ^∞₀𝑧^⊺(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾^2∞₀𝜔^⊺(𝑠)𝜔(𝑠)𝑑𝑠. (2.9)

Definition 2.3 (see [30]). For any 𝑇₂>𝑇₁≥0, let 𝑁_𝜎(𝑇₁,𝑇₂) denote the switching number of discontinuities of 𝜎(𝑡) during on an intercal (𝑇₁,𝑇₂). If 𝑁_𝜎(𝑇₁,𝑇₂)≤𝑁₀+(𝑇₂−𝑇₁)/𝑇_𝑎 holds for 𝑁₀≥0 and 𝑇_𝑎>0, then 𝑁₀ and 𝑇_𝑎 are called chattering bound and average dwell time, respectively. Here we assume 𝑁₀=0 for simplicity as commonly used in the literature.

Lemma 2.4 (see [29]). For any given symmetric positive definite matrix 𝑋∈ℜ^𝑛×𝑛, and scalars 𝛼>0,0≤𝑑₁<𝑑₂, if there exists a vector function 𝑥̇(𝑡)∶[−𝑑₂,0]→ℜ^𝑛 such that the following integration is well defined, then −^−𝑑1_−𝑑2𝑥̇(𝑡+𝜃)^⊺𝑒^𝛼𝜃𝑋𝑥̇()≤𝛼𝑡+𝜃𝑑𝜃𝑒^𝛼𝑑1−𝑒^𝛼𝑑2𝑥(𝑡−𝑑₁)𝑥(𝑡−𝑑₂)^⊺𝑋−𝑋𝑥−𝑋𝑋𝑡−𝑑₁𝑥𝑡−𝑑₂. (2.10)

Lemma 2.5 (see [28]). Let 𝜚≥0 and 𝜃>𝛿>0. If there exists a real-value continuous function 𝑥(𝑡)≥0, 𝑡≥𝑡₀, such that the differential inequality d𝑥(𝑡)(𝑡)d𝑡≤−𝜃𝑥+𝛿sup_{𝑡−𝜚≤𝑠≤𝑡}𝑥(𝑠),𝑡≥𝑡₀ (2.11) holds, then 𝑥(𝑡)≤sup_{−𝜚≤𝑠≤0}𝑥𝑡₀+𝑠𝑒^{−𝜇(𝑡−𝑡0)},𝑡≥𝑡₀, (2.12) where 𝜇>0, and satisfies 𝜇−𝜃+𝛿𝑒^𝜇𝜚=0.

Lemma 2.6 (see [9]). If a real scalar function 𝑥(𝑡) satisfies the following differential inequality: 𝑥̇(𝑡)≤−𝜍𝑥(𝑡)+𝜂𝜐(𝑡), (2.13) where 𝜍>0,𝜂>0, then 𝑥(𝑡)≤𝑒^−𝜍𝑡𝑥(0)+𝜂^𝑡₀𝑒^−𝜍𝑠𝜐(𝑡−𝑠)𝑑𝑠. (2.14)

3. Main Results

The following theorem presents a sufficient stability condition for system (2.1). We first present conditions 𝜔(𝑡)=0, the corresponding closed-loop system is given by 𝑥̇(𝑡)=Υ_𝜎(𝑡)(𝑡),𝑥(𝑡)(𝑡)⋅𝜓=𝜙,𝑡∈−𝜏_𝑀,0, (3.1) where Υ_𝜎(𝑡)=𝐴_𝜎(𝑡),𝐴_{𝜏𝜎(𝑡)},0,0,𝜑(𝑡)=𝑥^⊺(𝑡),𝑥^⊺(𝑡−𝜏(𝑡)),𝑥^⊺𝑡−𝜏_𝑀,𝑥^⊺𝑡−𝜏₀^⊺. (3.2)

Theorem 3.1. For given scalars 0≤𝜏₀≤𝜏_𝑀, 𝛼>0,𝛽>0, then the system (2.1) is exponentially stable, if there exist positive-definite matrices 𝑃_𝑖,𝑃_𝑖𝑗,𝑄_𝑘𝑖,𝑄_𝑘𝑖𝑗(𝑘=1,2,3), and 𝑅_𝑙𝑖,𝑅_𝑙𝑖𝑗(𝑙=1,2) such that the following LMIs hold: Ξ_𝑖=Ξ_1,1Ξ_1,2−𝛼1−𝑒^𝛼𝜏𝑀𝑅_1𝑖0∗Ξ_2,2−𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖−𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖∗∗Ξ_3,30∗∗∗−𝑒^−𝛼𝜏0𝑄_3𝑖+𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖Ξ<0,_𝑖=Ξ_1,1Ξ_1,2𝛽1−𝑒^{−𝛽𝜏𝑀}𝑅_1𝑖𝑗0∗Ξ_2,2𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗∗∗Ξ_3,30∗∗∗−𝑒^𝛽𝜏0𝑄_3𝑖𝑗−𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗<0 (3.3)(3.4) with Ξ_1,1=𝑄_1𝑖+𝑄_2𝑖+𝑄_3𝑖+𝑃_𝑖𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖+𝛼𝑃_𝑖+𝛼1−𝑒^𝛼𝜏𝑀𝑅_1𝑖+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖𝐴_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝐴_𝑖,Ξ_1,2=𝑃_𝑖𝐴_𝜏𝑖+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖𝐴_𝜏𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝐴_𝜏𝑖,Ξ_2,2=−(1−𝜏)𝑒^{−𝛼𝜏𝑀}𝑄_1𝑖𝛼+2𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖+𝜏_𝑀𝐴^⊺_𝜏𝑖𝑅_1𝑖𝐴_𝜏𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝜏𝑖𝑅_2𝑖𝐴_𝜏𝑖,Ξ_3,3=−𝑒^{−𝛼𝜏𝑀}𝑄_2𝑖+𝛼1−𝑒^𝛼𝜏𝑀𝑅_1𝑖+𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖,Ξ_1,1=𝑄_1𝑖𝑗+𝑄_2𝑖𝑗+𝑄_3𝑖𝑗+𝑃_𝑖𝑗𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖𝑗−𝛽𝑃_𝑖𝑗+−𝛽1−𝑒^{−𝛽𝜏𝑀}𝑅_1𝑖𝑗+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖𝑗𝐴_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝑗𝐴_𝑖,Ξ_1,2=𝑃_𝑖𝑗𝐴_𝜏𝑖+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖𝑗𝐴_𝜏𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝑗𝐴_𝜏𝑖,Ξ_2,2=−(1−𝜏)𝑒^𝛽𝜏𝑀𝑄_1𝑖𝑗𝛽−2𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗+𝜏_𝑀𝐴^⊺_𝜏𝑖𝑅_1𝑖𝑗𝐴_𝜏𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝜏𝑖𝑅_2𝑖𝑗𝐴_𝜏𝑖,Ξ_3,3=−𝑒^𝛽𝜏𝑀𝑄_2𝑖𝑗−𝛽1−𝑒^{−𝛽𝜏𝑀}𝑅_1𝑖𝑗−𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗. (3.5)
In this case, for any switching signal with the average dwell-time satisfying 𝑇_𝑎>𝑇^∗_𝑎=ln𝜇₂𝜇₁𝜅,𝑇⁺𝑡₀,𝑡𝑇⁻𝑡₀≥,𝑡𝛽+𝜅𝛼−𝜅,0≤𝜅<𝛼. (3.6)(3.7) System (3.1) is exponentially stable with 𝜇_𝑙≥1(𝑙=1,2) satisfying that 𝑃_𝑗≤𝜇₁𝑃_𝑖𝑗,𝑃_𝑖𝑗≤𝜇₂𝑃_𝑖,𝑄_𝑘𝑗≤𝜇₁𝑄_𝑘𝑖𝑗,𝑄_𝑘𝑖𝑗≤𝜇₂𝑄_𝑘𝑖(),𝑅𝑘=1,2,3_𝑙𝑗≤𝜇₁𝑅_𝑙𝑖𝑗,𝑅_𝑙𝑖𝑗≤𝜇₂𝑅_𝑙𝑖(𝑙=1,2),∀𝑖≠𝑗,𝑖,𝑗∈ℙ. (3.8)

Proof. When 𝑡∈Λ₁, we consider the following Lyapunov-Krasovskii functional: 𝑉_𝑖(𝑡,𝑥(𝑡))=𝑉_1𝑖(𝑡,𝑥(𝑡))+𝑉_2𝑖(𝑡,𝑥(𝑡))+𝑉_3𝑖(𝑡,𝑥(𝑡)), (3.9) where 𝑉_1𝑖(𝑡,𝑥(𝑡))=𝑥^⊺(𝑡)𝑃_𝑖𝑥(𝑡),𝑉_2𝑖(𝑡,𝑥(𝑡))=^𝑡_{𝑡−𝜏(𝑡)}𝑒^{−𝛼(𝑡−𝑠)}𝑥^⊺(𝑠)𝑄_1𝑖𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝜏𝑀}𝑒^{−𝛼(𝑡−𝑠)}𝑥^⊺(𝑠)𝑄_2𝑖𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏0}𝑒^{−𝛼(𝑡−𝑠)}𝑥^⊺(𝑠)𝑄_3𝑖𝑥(𝑠)𝑉𝑑𝑠,_3𝑖(𝑡,𝑥(𝑡))=⁰_−𝜏𝑀^𝑡_𝑡+𝜃𝑒^{−𝛼(𝑡−𝑠)}𝑥̇^⊺(𝑠)𝑅_1𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^−𝜏0_−𝜏𝑀^𝑡_𝑡+𝜃𝑒^{−𝛼(𝑡−𝑠)}𝑥̇^⊺(𝑠)𝑅_2𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (3.10)
Taking the time derivative of 𝑉_𝑖(𝑡,𝑥(𝑡)) for 𝑡∈[0,∞) along the trajectory of the system (2.1) turns out to be 𝑉̇_1𝑖(𝑡,𝑥(𝑡))=2𝑥^⊺(𝑡)𝑃_𝑖Υ_𝑖𝜓(𝑡),𝑉̇_2𝑖(𝑡,𝑥(𝑡))≤−𝛼𝑉_2𝑖(𝑡,𝑥(𝑡))+𝑥^⊺(𝑡)𝑄_1𝑖+𝑄_2𝑖+𝑄_3𝑖𝑥(𝑡)−(1−𝜏)𝑒^{−𝛼𝜏𝑀}𝑥^⊺(𝑡−𝜏(𝑡))𝑄_1𝑖𝑥(𝑡−𝜏(𝑡))−𝑒^{−𝛼𝜏𝑀}𝑥^⊺𝑡−𝜏_𝑀𝑄_2𝑖𝑥𝑡−𝜏_𝑀−𝑒^−𝛼𝜏0𝑥^⊺𝑡−𝜏₀𝑄_3𝑖𝑥𝑡−𝜏₀,𝑉̇_3𝑖(𝑡,𝑥(𝑡))≤−𝛼𝑉_3𝑖(𝑡,𝑥(𝑡))+𝜏_𝑀𝜓^⊺(𝑡)Υ^⊺_𝑖𝑅_1𝑖Υ_𝑖𝜓(𝑡)+𝜏_𝑀−𝜏₀𝜓^⊺(𝑡)Υ^⊺_𝑖𝑅_2𝑖Υ_𝑖𝜓(𝑡)−^𝑡_{𝑡−𝜏𝑀}𝑥̇^⊺(𝑠)𝑒^{−𝛼(𝑡−𝑠)}𝑅_1𝑖𝑥̇(𝑠)𝑑𝑠−^𝑡−𝜏0_{𝑡−𝜏𝑀}𝑥̇^⊺(𝑠)𝑒^{−𝛼(𝑡−𝑠)}𝑅_2𝑖𝑥̇(𝑠)𝑑𝑠. (3.11)
On the other hand, according to Lemma 2.4, we get that −^𝑡_{𝑡−𝜏𝑀}𝑥̇^⊺(𝑠)𝑒^{−𝛼(𝑡−𝑠)}𝑅_1𝑖𝑥̇(𝑠)𝛼𝑑𝑠≤1−𝑒^𝛼𝜏𝑀𝑥(𝑡)𝑥𝑡−𝜏_𝑀^⊺𝑅_1𝑖−𝑅_1𝑖−𝑅_1𝑖𝑅_1𝑖𝑥(𝑡)𝑥𝑡−𝜏_𝑀,−^𝑡−𝜏0_{𝑡−𝜏𝑀}𝑥̇^⊺(𝑠)𝑒^{−𝛼(𝑡−𝑠)}𝑅_2𝑖𝑥̇(𝑠)𝑑𝑠≤−^𝑡−𝜏0_{𝑡−𝜏(𝑡)}𝑥̇^⊺(𝑠)𝑒^{−𝛼(𝑡−𝑠)}𝑅_2𝑖𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑀}𝑥̇^⊺(𝑠)𝑒^{−𝛼(𝑡−𝑠)}𝑅_2𝑖𝑥̇(𝑠)≤𝛼𝑑𝑠𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑥(𝑡−𝜏₀)𝑥(𝑡−𝜏(𝑡))^⊺𝑅_2𝑖−𝑅_2𝑖−𝑅_2𝑖𝑅_2𝑖𝑥𝑡−𝜏₀𝑥((𝑡+𝛼𝑡−𝜏))𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑥(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏_𝑀)^⊺𝑅_2𝑖−𝑅_2𝑖−𝑅_2𝑖𝑅_2𝑖𝑥((𝑡𝑥𝑡−𝜏))𝑡−𝜏_𝑀. (3.12)
Then we can get 𝑉̇_𝑖(𝑡,𝑥(𝑡))+𝛼𝑉_𝑖(𝑡,𝑥(𝑡))≤𝜓^⊺(t)Ξ_𝑖𝜓(𝑡), (3.13) where Ξ_𝑖=𝜏_𝑀Υ^⊺_𝑖𝑅_1𝑖Υ_𝑖+𝜏_𝑀−𝜏₀Υ^⊺_𝑖𝑅_2𝑖Υ_𝑖+Ξ⁰_1,1𝑃_𝑖𝐴_𝜏𝑖−𝛼1−𝑒^𝛼𝜏𝑀𝑅_1𝑖0∗Ξ⁰_2,2−𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖−𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖∗∗Ξ_3,30∗∗∗−𝑒^−𝛼𝜏0𝑄_3𝑖+𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖 (3.14) with Ξ⁰_1,1=𝑄_1𝑖+𝑄_2𝑖+𝑄_3𝑖+𝑃_𝑖𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖+𝛼1−𝑒^𝛼𝜏𝑀𝑅_1𝑖+𝛼𝑃_𝑖,Ξ⁰_2,2=−(1−𝜏)𝑒^{−𝛼𝜏𝑀}𝑄_1𝑖𝛼+2𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖. (3.15)
In view of Schur complement, (3.3) implies that Ξ_𝑖<0. Then we have 𝑉̇_𝑖(𝑡,𝑥(𝑡))+𝛼𝑉_𝑖(𝑡,𝑥(𝑡))<0 for all 𝜓(𝑡)≠0.
Then during the matched period, by Lemma 2.5, 𝑉_𝑖(𝑡,𝑥(𝑡)) satisfy 𝑉_𝑖(𝑡,𝑥(𝑡))≤𝑒^{−𝛼(𝑡−𝑡0)}𝑉_𝑖𝑡₀,𝑥_𝑡0,𝑡₀≤𝑡<𝑡₁,𝑉_𝑖(𝑡,𝑥(𝑡))≤𝑒^{−𝛼(𝑡−𝑡ℑ−1−Δℑ−1)}𝑉_𝑖𝑡_ℑ−1+Δ_ℑ−1,𝑥_{𝑡ℑ−1+Δℑ−1},𝑡_ℑ−1+Δ_ℑ−1≤𝑡<𝑡_ℑ,ℑ=2,3,…. (3.16)
When 𝑡∈Λ₂, we consider the following Lyapunov-Krasovskii functional: 𝑉_𝑖𝑗(𝑡,𝑥(𝑡))=𝑉_1𝑖𝑗(𝑡,𝑥(𝑡))+𝑉_2𝑖𝑗(𝑡,𝑥(𝑡))+𝑉_3𝑖𝑗(𝑡,𝑥(𝑡)), (3.17) where 𝑉_1𝑖𝑗(𝑡,𝑥(𝑡))=𝑥^⊺(𝑡)𝑃_𝑖𝑗𝑥(𝑡),𝑉_2𝑖𝑗(𝑡,𝑥(𝑡))=^𝑡_{𝑡−𝜏(𝑡)}𝑒^{𝛽(𝑡−𝑠)}𝑥^⊺(𝑠)𝑄_1𝑖𝑗𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝜏𝑀}𝑒^{𝛽(𝑡−𝑠)}𝑥^⊺(𝑠)𝑄_2𝑖𝑗𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏0𝑒^{𝛽(𝑡−𝑠)}𝑥^⊺(𝑠)𝑄_3𝑖𝑗𝑥(𝑠)𝑉𝑑𝑠,_3𝑖𝑗(𝑡,𝑥(𝑡))=⁰_−𝜏𝑀^𝑡_𝑡+𝜃𝑒^{𝛽(𝑡−𝑠)}𝑥̇^⊺(𝑠)𝑅_1𝑖𝑗𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^−𝜏0_−𝜏𝑀^𝑡_𝑡+𝜃𝑒^{𝛽(𝑡−𝑠)}𝑥̇^⊺(𝑠)𝑅_2𝑖𝑗𝑥̇(𝑠)𝑑𝑠𝑑𝜃, (3.18) where 𝛽>0.
Similarly we have that 𝑉̇_𝑖𝑗(𝑡,𝑥(𝑡))−𝛽𝑉_𝑖𝑗(𝑡,𝑥(𝑡))≤𝜓^⊺(𝑡)Ξ_𝑖𝜓(𝑡), (3.19) where Ξ_1𝑖=𝜏_𝑀Υ^⊺_𝑖𝑅_1𝑖𝑗Υ_𝑖+𝜏_𝑀−𝜏₀Υ^⊺_𝑖𝑅_2𝑖𝑗Υ_𝑖+Ξ⁰_1,1𝑃_𝑖𝑗𝐴_𝜏𝑖𝛽1−𝑒^{−𝛽𝜏𝑀}𝑅_1𝑖𝑗0∗Ξ⁰_2,2𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗∗∗Ξ_3,30∗∗∗−𝑒^𝛽𝜏0𝑄_3𝑖𝑗−𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗 (3.20) with Ξ⁰_1,1=𝑄_1𝑖𝑗+𝑄_2𝑖𝑗+𝑄_3𝑖𝑗+𝑃_𝑖𝑗𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖𝑗−𝛽1−𝑒^{−𝛽𝜏𝑀}𝑅_1𝑖𝑗−𝛽𝑃_𝑖𝑗,Ξ⁰_2,2=−(1−𝜏)𝑒^𝛽𝜏𝑀𝑄_1𝑖𝑗𝛽−2𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗. (3.21)
Through Lemma 2.5, we also have 𝑉_𝑖𝑗(𝑡,𝑥(𝑡))≤𝑒^{𝛽(𝑡−𝑡ℑ)}𝑉_𝑖𝑗𝑡_ℑ,𝑥_𝑡ℑ,𝑡_ℑ≤𝑡≤𝑡_ℑ+Δ_ℑ,ℑ=1,2,…. (3.22)
When 𝑡∈[𝑡_ℑ,𝑡_ℑ+Δ_ℑ),ℑ=1,2,…, we have the relationship 𝑁_𝜎(𝑡)(𝑡₀,𝑡)<𝑁_𝜎(𝑡)(𝑡₀,𝑡), and it follows that 𝑉(𝑡,𝑥(𝑡))=𝑉_{𝜎(𝑡ℑ−1+Δℑ−1)𝜎(𝑡ℑ)}(𝑡)≤𝑉_{𝜎(𝑡ℑ−1+Δℑ−1)𝜎(𝑡ℑ)}𝑡_ℑ𝑒^{𝛽(𝑡−𝑡ℑ)}≤𝜇₂𝑉_{𝜎(𝑡ℑ−1)}𝑡⁻_ℑ𝑒^{𝛽(𝑡−𝑡ℑ)}≤𝜇₂𝑉_{𝜎(𝑡ℑ−1)}𝑡_ℑ−1+Δ_ℑ−1𝑒^{𝛽(𝑡−𝑡ℑ)−𝛼(𝑡ℑ−𝑡ℑ−1−Δℑ−1)}≤𝜇₂𝜇₁𝑉_{𝜎(𝑡ℑ−2+Δℑ−2)}𝑡⁻_ℑ𝑒^{𝛽(𝑡−𝑡ℑ)−𝛼(𝑡ℑ−𝑡ℑ−1−Δℑ−1)}≤𝜇≤⋯₂𝜇₁^{𝑁𝜎(𝑡)(𝑡0,𝑡)}𝜇₁⁻¹𝑉_𝜎(𝑡0)𝑡₀𝑒^{𝛽𝑇−(𝑡0,𝑡)−𝛼𝑇+(𝑡0,𝑡)}. (3.23)
From (3.7), we can obtain 𝑇⁻𝑡₀,𝑡𝛽−𝑇⁺𝑡₀,𝑡𝛼≤−𝜅𝑡−𝑡₀. (3.24)
Then through (3.23) and (3.24), we can easily get 𝑉(𝑡,𝑥(𝑡))≤𝜇₁⁻¹𝜇₂𝜇₁^{(𝑡−𝑡0)/𝑇𝑎}𝑉_𝜎(𝑡0)𝑒^{−𝜅(𝑡−𝑡0)}≤𝜇₁⁻¹𝑉_𝜎(𝑡0)𝑒^{−(𝜅−(ln𝜇2𝜇1)/𝑇𝑎)(𝑡−𝑡0)}. (3.25)
By (3.9) and (3.25), then we have 𝜆_min𝑃_𝑖‖𝑥(𝑡)‖²≤𝑉(𝑡,𝑥(𝑡))≤𝜇₁⁻¹max_{∀𝑖∈ℙ}𝜆_max𝑃_𝑖+²_𝑘=1𝜏_𝑀max_{∀𝑖∈ℙ}𝜆_max𝑄_𝑘𝑖+𝜏₀max_{∀𝑖∈ℙ}𝜆_max𝑄_3𝑖+𝜏²_𝑀2max_{∀𝑖∈ℙ}𝜆_max𝑅_1𝑖+𝜏²_𝑀−𝜏²₀2max_{∀𝑖∈ℙ}𝜆_max𝑅_2𝑖𝑥_𝑡0²𝑒^{−(𝜅−(ln𝜇2𝜇1)/𝑇𝑎)(𝑡−𝑡0)}, (3.26) which means ‖𝑥(𝑡)‖≤𝜇₁^−1/2𝑎₁𝑏₁𝑥_𝑡0𝑒^{−(1/2)(𝜅−ln𝜇2𝜇1/𝑇𝑎)(𝑡−𝑡0)}, (3.27) where 𝑎₁=max_{∀𝑖∈ℙ}𝜆_max𝑃_𝑖+²_𝑘=1𝜏_𝑀max_{∀𝑖∈ℙ}𝜆_max𝑄_𝑘𝑖+𝜏₀max_{∀𝑖∈ℙ}𝜆_max𝑄_3𝑖+𝜏²_𝑀2max_{∀𝑖∈ℙ}𝜆_max𝑅_1𝑖+𝜏²_𝑀−𝜏²₀2max_{∀𝑖∈ℙ}𝜆_max𝑅_2𝑖,𝑏₁=𝜆_min𝑃_𝑖. (3.28)
Similarly, when 𝑡∈Λ₁, we can also have ‖𝑥(𝑡)‖≤𝜇₁^−1/2𝑎₂𝑏₂𝑥_𝑡0𝑒^{−(1/2)(𝜅−ln𝜇2𝜇1/𝑇𝑎)(𝑡−𝑡0)}, (3.29) where 𝑎₂=max_{∀𝑖,𝑗∈ℙ}𝜆_maxP_𝑖𝑗+²_𝑘=1𝜏_𝑀max_{∀𝑖,𝑗∈ℙ}𝜆_max𝑄_𝑘𝑖𝑗+𝜏₀max_{∀𝑖,𝑗∈ℙ}𝜆_max𝑄_3𝑖𝑗+𝜏²_𝑀2max_{∀𝑖,𝑗∈ℙ}𝜆_max𝑅_1𝑖𝑗+𝜏²_𝑀−𝜏²₀2max_{∀𝑖,𝑗∈ℙ}𝜆_max𝑅_2𝑖𝑗,𝑏₂=𝜆_min𝑃_𝑖𝑗. (3.30)
For convenience, let 𝑎=max{𝑎₁,𝑎₂}, 𝑏=min_{𝑖≠𝑗,𝑖,𝑗∈ℙ}𝜆_min(𝑃_𝑖,𝑃_𝑖𝑗), through (3.27) and (3.29), we have 𝜂_𝑡≤𝜇₁^−1/2𝑎𝑏𝜂_𝑡0𝑒^{−(1/2)(𝜅−ln𝜇2𝜇1/𝑇𝑎)(𝑡−𝑡0)}. (3.31)

Theorem 3.2. For given scalars 0≤𝜏₀≤𝜏_𝑀,𝛼>0,𝛽>0, then the system (2.1) is exponentially stable with 𝐿₂-gain, if there exist positive-definite matrices 𝑃_𝑖,𝑃_𝑖𝑗,𝑄_𝑘𝑖,𝑄_𝑘𝑖𝑗(𝑘=1,2,3), and 𝑅_𝑙𝑖, 𝑅_𝑙𝑖𝑗(𝑙=1,2) such that the following LMIs hold: Σ_1𝑖=Σ_1,1Ξ_1,2−𝛼1−𝑒^𝛼𝜏𝑀𝑅_1𝑖0Σ_1,5∗Ξ_2,2−𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖−𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖Σ_2,5∗∗Ξ_3,300∗∗∗−𝑒^−𝛼𝜏0𝑄_3𝑖+𝛼𝑒^𝛼𝜏0−𝑒^𝛼𝜏𝑀𝑅_2𝑖0∗∗∗∗Σ_5,5Σ_1𝑖,=Σ_1,1Ξ_1,2𝛽1−𝑒^{−𝛽𝜏𝑀}𝑅_1𝑖𝑗0Σ_1,5∗Ξ_2,2𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗Σ_2,5∗∗Ξ_3,300∗∗∗−𝑒^𝛽𝜏0𝑄_3𝑖𝑗−𝛽𝑒^−𝛽𝜏0−𝑒^{−𝛽𝜏𝑀}𝑅_2𝑖𝑗0∗∗∗∗Σ_5,5<0 (3.32)(3.33) with Σ_1,1=𝑄_1𝑖+𝑄_2𝑖+𝑄_3𝑖+𝑃_𝑖𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖+𝛼1−𝑒^𝛼𝜏𝑀𝑅_1𝑖+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖𝐴_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝐴_𝑖+𝐶^⊺_𝑖𝐶_𝑖,Σ_1,5=𝑃_𝑖𝐵_𝑖+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖𝐵_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝐵_𝑖+𝐶^⊺_𝑖𝐷_𝑖,Σ_2,5=𝜏_𝑀𝐴^⊺_𝜏𝑖𝑅_1𝑖𝐵_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝜏𝑖𝑅_2𝑖𝐵_𝑖,Σ_5,5=𝜏_𝑀𝐵^⊺_𝑖𝑅_1𝑖𝐵_𝑖+𝜏_𝑀−𝜏₀𝐵^⊺_𝑖𝑅_2𝑖𝐵_𝑖+𝐷^⊺_𝑖𝐷_𝑖−𝛾²Σ𝐼,_1,1=𝑄_1𝑖𝑗+𝑄_2𝑖𝑗+𝑄_3𝑖𝑗+𝑃_𝑖𝑗𝐴_𝑖+𝐴^⊺_𝑖𝑃_𝑖𝑗−𝛽1−𝑒^{−𝛽𝜏𝑀}𝑅_1𝑖𝑗+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖𝑗𝐴_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝑗𝐴_𝑖+𝐶^⊺_𝑖𝐶_𝑖,Σ_1,5=𝑃_𝑖𝑗𝐵_𝑖+𝜏_𝑀𝐴^⊺_𝑖𝑅_1𝑖j𝐵_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝑖𝑅_2𝑖𝑗𝐵_𝑖+𝐶^⊺_𝑖𝐷_𝑖,Σ_2,5=𝜏_𝑀𝐴^⊺_𝜏𝑖𝑅_1𝑖𝑗𝐵_𝑖+𝜏_𝑀−𝜏₀𝐴^⊺_𝜏𝑖𝑅_2𝑖𝑗𝐵_𝑖,Σ_5,5=𝜏_𝑀𝐵^⊺_𝑖𝑅_1𝑖𝑗𝐵_𝑖+𝜏_𝑀−𝜏₀𝐵^⊺_𝑖𝑅_2𝑖𝑗𝐵_𝑖+𝐷^⊺_𝑖𝐷_𝑖−𝛾²𝐼. (3.34)

Proof. For all nonzero 𝜔(𝑡)∈𝐿₂[0,∞) and a scalar 𝛾>0, then we establish system (2.1) with 𝐿₂-gain performance ‖𝑧(𝑡)‖₂≤𝛾‖𝜔(𝑡)‖. For convenience, denoting Π(𝑡)=𝑧^⊺(𝑡)𝑧(𝑡)−𝛾²𝜔^⊺(𝑡)𝜔(𝑡).
When 𝑡∈[𝑡₀,𝑡₁)∪[𝑡_ℑ−1+Δ_ℑ−1,𝑡_ℑ),𝑘=2,3,…, by the system (2.1), we can obtain 𝑉̇(𝑡,𝑥(𝑡))+𝛼𝑉_𝑖(𝑡,𝑥(𝑡))+Π(𝑡)=𝜑^⊺(𝑡)Σ_𝑖𝜑(𝑡). (3.35)
From (3.32), we can easily get 𝑉̇_𝑖𝑡,𝜂_𝑡+𝛼𝑉_𝑖𝑡,𝜂_𝑡+Π(𝑡)<0. (3.36)
Integrate this inequality during [𝑡₀,𝑡], it is known that 𝑉_𝑖(𝑡,𝑥(𝑡))≤𝑒^{−𝛼(𝑡−𝑡0)}𝑉_𝑖𝑡₀,𝑥_𝑡0−^𝑡_𝑡−𝑡0𝑒^{−𝛼(𝑡−𝑠)}Π(𝑠)𝑑𝑠,𝑡₀≤𝑡<𝑡₁,𝑉_𝑖(𝑡,𝑥(𝑡))≤𝑒^{−𝛼(𝑡−𝑡ℑ−1−Δℑ−1)}𝑉_𝑖𝑡_ℑ−1+Δ_ℑ−1,𝑥_{𝑡ℑ−1+Δℑ−1}−^𝑡_{𝑡ℑ−1+Δℑ−1}𝑒^{−𝛼(𝑡−𝑠)}Π(𝑠)𝑑𝑠,𝑡_ℑ−1+Δ_ℑ−1≤𝑡<𝑡_ℑ,ℑ=2,3,…. (3.37)
When 𝑡∈[𝑡_ℑ,𝑡_ℑ+Δ_ℑ),ℑ=2,3,…, by the system (2.1), by the same way, we can obtain 𝑉̇(𝑡,𝑥(𝑡))−𝛽𝑉_𝑖(𝑡,𝑥(𝑡))+Π(𝑡)=𝜑^⊺(𝑡)Σ_𝑖𝜑(𝑡)<0. (3.38)
Then we have 𝑉_𝑖𝑗(𝑡,𝑥(𝑡))≤𝑉_𝑖𝑗𝑡_ℑ,𝑥_𝑡ℑ𝑒^{𝛽(𝑡−𝑡ℑ)}−^𝑡_𝑡ℑ𝑒^{𝛽(𝑡−𝑠)}Π(𝑠)𝑑𝑠,𝑡_ℑ≤𝑡_ℑ+Δ_ℑ,ℑ=1,2,…. (3.39)
When 𝑡∈[𝑡_ℑ,𝑡_ℑ+Δ_ℑ),ℑ=1,2,…, it follows that 𝑉(𝑡,𝑥(𝑡))=𝑉_{𝜎(𝑡ℑ−1+Δℑ−1)𝜎(𝑡ℑ)}𝑡_ℑ𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}−^𝑡_𝑡ℑ𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}Π(𝑠)𝑑𝑠≤𝜇₂𝑉_{𝜎(𝑡ℑ−1)}𝑡⁻_ℑ𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}−^𝑡_𝑡ℑ𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}Π(𝑠)𝑑𝑠≤𝜇₂𝜇₁𝑉_{𝜎(𝑡ℑ−2+Δℑ−2)}𝜎𝑡_ℑ−1𝑡_ℑ−1𝑒^{𝛽𝑇−(𝑡ℑ−1,𝑡)−𝛼𝑇+(𝑡ℑ−1,𝑡)}−𝜇₂𝜇₁^{𝑡ℑ−1−Δℑ−1}_𝑡ℑ−1𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}Π(𝑠)𝑑𝑠−𝜇₂^𝑡ℑ_{𝑡ℑ−1+Δℑ−1}𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}Π(𝑠)𝑑𝑠−^𝑡_𝑡ℑ𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}Π(𝑠)𝑑𝑠≤⋯≤−^𝑡_𝑡0𝜇^{𝑁𝜎(𝑡)}₂^(𝑠,𝑡)𝜇^{𝑁𝜎(𝑡)}₁^(𝑠,𝑡)𝑒^{𝛽𝑇−(𝑠,𝑡)−𝛼𝑇+(𝑠,𝑡)}Π(𝑠)𝑑𝑠+𝜇^{𝑁𝜎(𝑡)(𝑡0}₂^,𝑡)𝜇^{𝑁𝜎(𝑡)(𝑡0}₁^,𝑡)𝑉_𝜎(𝑡0)𝑡₀𝑒^{𝛽𝑇−(𝑡ℑ,𝑡)−𝛼𝑇+(𝑡ℑ,𝑡)}. (3.40)
Under the zero initial condition, Let 𝑡₀=0, (3.40) implies ^𝑡₀𝜇^{𝑁𝜎(𝑡)}₂^(𝑠,𝑡)𝜇^{𝑁𝜎(𝑡)}₁^(𝑠,𝑡)𝑒^{𝛽𝑇−(𝑠,𝑡)−𝛼𝑇+(𝑠,𝑡)}Λ(𝑠)𝑑𝑠≤0. (3.41)
Integrate (3.41) during [0,∞), then we can obtain ^∞₀^𝑡₀𝑒^{[𝑁𝜎(𝑡)(0,𝑠)ln𝜇2+𝑁𝜎(𝑡)(0,𝑠)ln𝜇1]}𝑒^{𝛽𝑇−(𝑠,𝑡)−𝛼𝑇+(𝑠,𝑡)}Π(𝑠)<𝑑𝑠𝑑𝑡_∞0𝑒^{[𝑁𝜎(𝑡)(0,𝑠)ln𝜇2+𝑁𝜎(𝑡)(0,𝑠)ln𝜇1]}Π(𝑠)^∞_𝑠𝑒^{−𝜅(𝑡−𝑠)}𝑑𝑡𝑑𝑠≤^∞₀Π(𝑠)𝑑𝑠≤0. (3.42)
When 𝑡∈[𝑡₀,𝑡₁)∪[𝑡_ℑ−1+Δ_ℑ−1,𝑡_ℑ),ℑ=2,3,…, by the same mathematical operations, we have ∫^∞_𝑡0Π(𝑠)𝑑𝑠<0.
From which we can get ‖𝑧(𝑡)‖₂≤‖𝜔(𝑡)‖₂. This proof is completed.

Remark 3.3. If 𝜇₁=𝜇₂=0, which implies that 𝑃_𝑖=𝑃_𝑖𝑗=𝑃,𝑄_𝑘𝑖=𝑄_𝑘𝑖𝑗=𝑄_𝑘,𝑅_𝑙𝑖=𝑅_𝑙𝑖𝑗=𝑅_𝑙,𝑖,𝑗∈ℙ, by (3.3)-(3.4) and (3.32)-(3.33), we have 𝑇_𝑎=0, then it requires a common Lyapunov functional for all subsystems, and the switching signals can be arbitrary. If 𝜇_𝑘→∞(𝑘=1,2), we get from (3.3)-(3.4) and (3.32)-(3.33) that there is no switching, that is, switching signal will have a great dwell-time on the average.

4. Illustrative Example

In this section, a numerical example is given to illustrate the effectiveness of the obtained results.

Example 4.1. Consider the system (2.1) with parameters as follows: 𝐴₁=−1.50.30−2.4,𝐴₂=−1.40.80.6−2.4,𝐴_𝜏1=,𝐴−2.2−0.30−1.6_𝜏2=1.200−1.8,𝐶₁=−0.40.20.20.1,𝐶₂=0.20.30.10.35,𝐵₁=,𝐵−0.350.28₂=−0.660.25,𝐷₁=−0.35−0.15,𝐷₂=.0.57−0.48 (4.1)
𝜏₀ is fixed and assumed to be 0.2. The initial condition is assumed to be 𝑥(0)=[9,−9]^⊺, 𝜔(𝑡)=0.5sin𝑡. Then by solving the LMIs in Theorem 3.2, different 𝑇_𝑎 and 𝜏_𝑀 for different 𝜇₁𝜇₂, 𝛼, and 𝛽 can be obtained in Table 1. It can be seen that, for the given 𝜏₀, the upper bounds of the time delay 𝜏_𝑀 and the minimal average dwell-time 𝑇_𝑎 are dependent on 𝛼, 𝛽, and 𝜇₁𝜇₂. Then the simulation result of the system is shown in Figures 1, 2, and 3. The switching signal 𝜎(𝑡) with average dwell-time 𝑇_𝑎 is shown in Figure 1. Figures 2–3 indicate that the state response of the switched system without asynchronous switching and with asynchronous switching, respectively.

Table 1 

Different 𝑇_𝑎 and 𝜏_𝑀 for 𝛼=0.25, 𝛽=0.2.

Figure 1 

Switching signal with ADT.

Figure 2 

The state response without asynchronous switching.

Figure 3 

The state response with asynchronous switching.

5. Conclusions

In terms of the LMI approach, the problem of robust stabilization of switched delay systems with average dwell-time under asynchronous switching has been considered. Two sufficient conditions are developed to guarantee the global exponential stability of the considered switched system. At last, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed techniques.

Acknowledgment

This work was supported by the Fundamental Research Funds for the Central Universities (103.1.2E022050205).