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

Authors: Jun Cheng; Hong Zhu; Shouming Zhong; Yuping Zhang
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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 andL2-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,ℜn denotes the n-dimensional Euclidean space and ℜn×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. λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively. The shorthand diag{M1,…,Mn} denotes a block diagonal matrix with diagonal blocks being the matrices M1,…,Mn. 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(2.1)x˙(t)=Aσ(t)x(t)+Aτσ(t)x(t-τ(t))+Bσ(t)ω(t),z(t)=Cσ(t)x(t)+Dσ(t)ω(t),x(t)=ϕ(t),t∈[-τM,0],
where x(t)∈ℛn is the state of the system, ω(t)∈ℛq 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 Ai, Aτi,Bi,Ci, and Di with appropriate dimensions and τ(t) is the unknown time-varying delay satisfying
(2.2)τ0≤τ(t)≤τM,τ˙(t)≤τ,
where τ0,τM, and τ are constants; A,Aτ, and Ci are known real constant matrices; ϕ(t) is a compatible vector-valued initial function on [-τM,0].The switching signalσ(t):ℜ+→ℙ={1,2,…,r} discussed in this paper is time dependent, that is, σ(t):{(t0,σ(t0)),(t1,σ(t1)),…,(tℑ,σ(tℑ))}, where t0 is the initial instant. In this paper, we denote tℑ represent the ℑth switching instant. for convenience, σ_(t) is used to denote the practical switching signal which can be written as
(2.3)σ_(t):{(t0+Δ0,σ(t0)),(t1+Δ1,σ(t1)),…,(tℑ+Δℑ,σ(tℑ))},
where Δℑ<infℑ≥1|tℑ+1-tℑ|, k=0,1,…. Then there exist a matched period time interval [tℑ-1+Δℑ-1) and mismatched period time interval [tℑ,tℑ+Δℑ) because of asynchronous switching. For simplicity, we assume that Δℑ>0, ℑ=0,1,….In this paper, we denoteΛ1 and Λ2 as follows:
(2.4)Λ1={x(t)∈ℜn∣σ(tℑ)=i,σ(tℑ+1)=j,t∈[tℑ+Δℑ,tℑ+1),ℑ=0,1,2,…},Λ2={x(t)∈ℜn∣σ(tℑ+1)=j,t∈[tℑ+1,tℑ+1+Δℑ+1),ℑ=0,1,2,…}.Let(2.5)φ(t)=[x⊺(t),x⊺(t-τ(t)),x⊺(t-τM),x⊺(t-τ0),ω⊺(t)]⊺.This way, system (2.1) can be rewritten as
(2.6)x˙(t)=Υσ(t)·φ(t),z(t)=Cσ(t)x(t)+Dσ(t)ω(t),x(t)=ϕ(t),t∈[-τM,0],
where
(2.7)Υσ(t)=[Aσ(t),Aτσ(t),0,0,Bσ(t)].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 constantsν>0 and ϑ>0 such that
(2.8)∥x(t)∥≤νsup-τM≤ς≤0∥φ(ς)∥ce-ϑt.Definition 2.2 (see [30]).
Forγ>0, the switched system (2.1) is said to have weighted L2-gain, if under zero initial condition ϕ(t),t∈[-τM,0], it holds that
(2.9)∫0∞‍z⊺(s)z(s)ds≤γ2∫0∞‍ω⊺(s)ω(s)ds.Definition 2.3 (see [30]).
For anyT2>T1≥0, let Nσ(T1,T2) denote the switching number of discontinuities of σ(t) during on an intercal (T1,T2). If Nσ(T1,T2)≤N0+(T2-T1)/Ta holds for N0≥0 and Ta>0, then N0 and Ta 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 matrixX∈ℜn×n, and scalars α>0,0≤d1<d2, if there exists a vector function x˙(t):[-d2,0]→ℜn such that the following integration is well defined, then
(2.10)-∫-d2-d1‍x˙(t+θ)⊺eαθXx˙(t+θ)dθ≤αeαd1-eαd2[x(t-d1)x(t-d2)]⊺[X-X-XX][x(t-d1)x(t-d2)].Lemma 2.5 (see [28]).
Letϱ≥0 and θ>δ>0. If there exists a real-value continuous function x(t)≥0, t≥t0, such that the differential inequality
(2.11)dx(t)dt≤-θx(t)+δsupt-ϱ≤s≤tx(s),t≥t0
holds, then
(2.12)x(t)≤sup-ϱ≤s≤0x(t0+s)e-μ(t-t0),t≥t0,
where μ>0, and satisfies μ-θ+δeμϱ=0.Lemma 2.6 (see [9]).
If a real scalar functionx(t) satisfies the following differential inequality:
(2.13)x˙(t)≤-ςx(t)+ηυ(t),
where ς>0,η>0, then
(2.14)x(t)≤e-ςtx(0)+η∫0t‍e-ςsυ(t-s)ds.


## 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
(3.1)x˙(t)=Υ~σ(t)·ψ(t),x(t)=ϕ(t),t∈[-τM,0],
where
(3.2)Υσ(t)=[Aσ(t),Aτσ(t),0,0],φ(t)=[x⊺(t),x⊺(t-τ(t)),x⊺(t-τM),x⊺(t-τ0)]⊺.Theorem 3.1.
For given scalars0≤τ0≤τM, α>0,β>0, then the system (2.1) is exponentially stable, if there exist positive-definite matrices Pi,Pij,Qki,Qkij(k=1,2,3), and Rli,Rlij(l=1,2) such that the following LMIs hold:
(3.3)Ξ~i=[Ξ~1,1Ξ~1,2-α1-eατMR1i0*Ξ~2,2-αeατ0-eατMR2i-αeατ0-eατMR2i**Ξ~3,30***-e-ατ0Q3i+αeατ0-eατMR2i]<0,(3.4)Ξ_i=[Ξ_1,1Ξ_1,2β1-e-βτMR1ij0*Ξ_2,2βe-βτ0-e-βτMR2ijβe-βτ0-e-βτMR2ij**Ξ_3,30***-eβτ0Q3ij-βe-βτ0-e-βτMR2ij]<0
with
(3.5)Ξ~1,1=Q1i+Q2i+Q3i+PiAi+Ai⊺Pi+αPi+α1-eατMR1i+τMAi⊺R1iAi+(τM-τ0)Ai⊺R2iAi,Ξ~1,2=PiAτi+τMAi⊺R1iAτi+(τM-τ0)Ai⊺R2iAτi,Ξ~2,2=-(1-τ)e-ατMQ1i+2αeατ0-eατMR2i+τMAτi⊺R1iAτi+(τM-τ0)Aτi⊺R2iAτi,Ξ~3,3=-e-ατMQ2i+α1-eατMR1i+αeατ0-eατMR2i,Ξ_1,1=Q1ij+Q2ij+Q3ij+PijAi+Ai⊺Pij-βPij+-β1-e-βτMR1ij+τMAi⊺R1ijAi+(τM-τ0)Ai⊺R2ijAi,Ξ_1,2=PijAτi+τMAi⊺R1ijAτi+(τM-τ0)Ai⊺R2ijAτi,Ξ_2,2=-(1-τ)eβτMQ1ij-2βe-βτ0-e-βτMR2ij+τMAτi⊺R1ijAτi+(τM-τ0)Aτi⊺R2ijAτi,Ξ_3,3=-eβτMQ2ij-β1-e-βτMR1ij-βe-βτ0-e-βτMR2ij.
In this case, for any switching signal with the average dwell-time satisfying(3.6)Ta>Ta*=lnμ2μ1κ,(3.7)T+(t0,t)T-(t0,t)≥β+κα-κ,0≤κ<α.
System (3.1) is exponentially stable with μl≥1(l=1,2) satisfying that
(3.8)Pj≤μ1Pij,Pij≤μ2Pi,Qkj≤μ1Qkij,Qkij≤μ2Qki(k=1,2,3),Rlj≤μ1Rlij,Rlij≤μ2Rli(l=1,2),∀i≠j,i,j∈ℙ.Proof.
Whent∈Λ1, we consider the following Lyapunov-Krasovskii functional:
(3.9)Vi(t,x(t))=V1i(t,x(t))+V2i(t,x(t))+V3i(t,x(t)),
where
(3.10)V1i(t,x(t))=x⊺(t)Pix(t),V2i(t,x(t))=∫t-τ(t)t‍e-α(t-s)x⊺(s)Q1ix(s)ds+∫t-τMt‍e-α(t-s)x⊺(s)Q2ix(s)ds+∫t-τ0t‍e-α(t-s)x⊺(s)Q3ix(s)ds,V3i(t,x(t))=∫-τM0‍∫t+θt‍e-α(t-s)x˙⊺(s)R1ix˙(s)dsdθ+∫-τM-τ0‍∫t+θt‍e-α(t-s)x˙⊺(s)R2ix˙(s)dsdθ.
Taking the time derivative ofVi(t,x(t)) for t∈[0,∞) along the trajectory of the system (2.1) turns out to be
(3.11)V˙1i(t,x(t))=2x⊺(t)PiΥ~iψ(t),V˙2i(t,x(t))≤-αV2i(t,x(t))+x⊺(t)(Q1i+Q2i+Q3i)x(t)-(1-τ)e-ατMx⊺(t-τ(t))Q1ix(t-τ(t))-e-ατMx⊺(t-τM)Q2ix(t-τM)-e-ατ0x⊺(t-τ0)Q3ix(t-τ0),V˙3i(t,x(t))≤-αV3i(t,x(t))+τMψ⊺(t)Υ~i⊺R1iΥ~iψ(t)+(τM-τ0)ψ⊺(t)Υ~i⊺R2iΥ~iψ(t)-∫t-τMt‍x˙⊺(s)e-α(t-s)R1ix˙(s)ds-∫t-τMt-τ0‍x˙⊺(s)e-α(t-s)R2ix˙(s)ds.
On the other hand, according to Lemma2.4, we get that
(3.12)-∫t-τMt‍x˙⊺(s)e-α(t-s)R1ix˙(s)ds≤α1-eατM[x(t)x(t-τM)]⊺[R1i-R1i-R1iR1i][x(t)x(t-τM)],-∫t-τMt-τ0‍x˙⊺(s)e-α(t-s)R2ix˙(s)ds≤-∫t-τ(t)t-τ0‍x˙⊺(s)e-α(t-s)R2ix˙(s)ds-∫t-τMt-τ(t)‍x˙⊺(s)e-α(t-s)R2ix˙(s)ds≤αeατ0-eατM[x(t-τ0)x(t-τ(t))]⊺[R2i-R2i-R2iR2i][x(t-τ0)x(t-τ(t))]+αeατ0-eατM[x(t-τ(t))x(t-τM)]⊺[R2i-R2i-R2iR2i][x(t-τ(t))x(t-τM)].
Then we can get(3.13)V˙i(t,x(t))+αVi(t,x(t))≤ψ⊺(t)Ξ~iψ(t),
where
(3.14)Ξ~i=τMΥ~i⊺R1iΥ~i+(τM-τ0)Υ~i⊺R2iΥ~i+[Ξ~1,10PiAτi-α1-eατMR1i0*Ξ~2,20-αeατ0-eατMR2i-αeατ0-eατMR2i**Ξ~3,30***-e-ατ0Q3i+αeατ0-eατMR2i]
with
(3.15)Ξ~1,10=Q1i+Q2i+Q3i+PiAi+Ai⊺Pi+α1-eατMR1i+αPi,Ξ~2,20=-(1-τ)e-ατMQ1i+2αeατ0-eατMR2i.
In view of Schur complement, (3.3) implies that Ξ~i<0. Then we have V˙i(t,x(t))+αVi(t,x(t))<0 for all ψ(t)≠0.
Then during the matched period, by Lemma2.5, Vi(t,x(t)) satisfy
(3.16)Vi(t,x(t))≤e-α(t-t0)Vi(t0,xt0),t0≤t<t1,Vi(t,x(t))≤e-α(t-tℑ-1-Δℑ-1)Vi(tℑ-1+Δℑ-1,xtℑ-1+Δℑ-1),tℑ-1+Δℑ-1≤t<tℑ,ℑ=2,3,….
Whent∈Λ2, we consider the following Lyapunov-Krasovskii functional:
(3.17)Vij(t,x(t))=V1ij(t,x(t))+V2ij(t,x(t))+V3ij(t,x(t)),
where
(3.18)V1ij(t,x(t))=x⊺(t)Pijx(t),V2ij(t,x(t))=∫t-τ(t)t‍eβ(t-s)x⊺(s)Q1ijx(s)ds+∫t-τMt‍eβ(t-s)x⊺(s)Q2ijx(s)ds+∫t-τ0t‍eβ(t-s)x⊺(s)Q3ijx(s)ds,V3ij(t,x(t))=∫-τM0‍∫t+θt‍eβ(t-s)x˙⊺(s)R1ijx˙(s)dsdθ+∫-τM-τ0‍∫t+θt‍eβ(t-s)x˙⊺(s)R2ijx˙(s)dsdθ,
where β>0.
Similarly we have that(3.19)V˙ij(t,x(t))-βVij(t,x(t))≤ψ⊺(t)Ξ_iψ(t),
where
(3.20)Ξ_1i=τMΥ~i⊺R1ijΥ~i+(τM-τ0)Υ~i⊺R2ijΥ~i+[Ξ_1,10PijAτiβ1-e-βτMR1ij0*Ξ_2,20βe-βτ0-e-βτMR2ijβe-βτ0-e-βτMR2ij**Ξ_3,30***-eβτ0Q3ij-βe-βτ0-e-βτMR2ij]
with
(3.21)Ξ_1,10=Q1ij+Q2ij+Q3ij+PijAi+Ai⊺Pij-β1-e-βτMR1ij-βPij,Ξ_2,20=-(1-τ)eβτMQ1ij-2βe-βτ0-e-βτMR2ij.
Through Lemma2.5, we also have
(3.22)Vij(t,x(t))≤eβ(t-tℑ)Vij(tℑ,xtℑ),tℑ≤t≤tℑ+Δℑ,ℑ=1,2,….
Whent∈[tℑ,tℑ+Δℑ),ℑ=1,2,…, we have the relationship Nσ_(t)(t0,t)<Nσ(t)(t0,t), and it follows that
(3.23)V(t,x(t))=Vσ_(tℑ-1+Δℑ-1)σ(tℑ)(t)≤Vσ_(tℑ-1+Δℑ-1)σ(tℑ)(tℑ)eβ(t-tℑ)≤μ2Vσ(tℑ-1)(tℑ-)eβ(t-tℑ)≤μ2Vσ(tℑ-1)(tℑ-1+Δℑ-1)eβ(t-tℑ)-α(tℑ-tℑ-1-Δℑ-1)≤μ2μ1Vσ_(tℑ-2+Δℑ-2)(tℑ-)eβ(t-tℑ)-α(tℑ-tℑ-1-Δℑ-1)≤⋯≤(μ2μ1)Nσ(t)(t0,t)μ1-1Vσ(t0)(t0)eβT-(t0,t)-αT+(t0,t).
From (3.7), we can obtain
(3.24)T-(t0,t)β-T+(t0,t)α≤-κ(t-t0).
Then through (3.23) and (3.24), we can easily get
(3.25)V(t,x(t))≤μ1-1(μ2μ1)(t-t0)/TaVσ(t0)e-κ(t-t0)≤μ1-1Vσ(t0)e-(κ-(lnμ2μ1)/Ta)(t-t0).
By (3.9) and (3.25), then we have
(3.26)λmin(Pi)∥x(t)∥2≤V(t,x(t))≤μ1-1[max∀i∈ℙλmax(Pi)+∑k=12‍τMmax∀i∈ℙλmax(Qki)+τ0max∀i∈ℙλmax(Q3i)+τM22max∀i∈ℙλmax(R1i)+(τM2-τ02)2max∀i∈ℙλmax(R2i)]∥xt0∥2e-(κ-(lnμ2μ1)/Ta)(t-t0),
which means
(3.27)∥x(t)∥≤μ1-1/2a1b1∥xt0∥e-(1/2)(κ-lnμ2μ1/Ta)(t-t0),
where
(3.28)a1=max∀i∈ℙλmax(Pi)+∑k=12‍τMmax∀i∈ℙλmax(Qki)+τ0max∀i∈ℙλmax(Q3i)+τM22max∀i∈ℙλmax(R1i)+(τM2-τ02)2max∀i∈ℙλmax(R2i),b1=λmin(Pi).
Similarly, whent∈Λ1, we can also have
(3.29)∥x(t)∥≤μ1-1/2a2b2∥xt0∥e-(1/2)(κ-lnμ2μ1/Ta)(t-t0),
where
(3.30)a2=max∀i,j∈ℙλmax(Pij)+∑k=12‍τMmax∀i,j∈ℙλmax(Qkij)+τ0max∀i,j∈ℙλmax(Q3ij)+τM22max∀i,j∈ℙλmax(R1ij)+(τM2-τ02)2max∀i,j∈ℙλmax(R2ij),b2=λmin(Pij).
For convenience, leta=max{a1,a2}, b=mini≠j,i,j∈ℙλmin(Pi,Pij), through (3.27) and (3.29), we have
(3.31)∥ηt∥≤μ1-1/2ab∥ηt0∥e-(1/2)(κ-lnμ2μ1/Ta)(t-t0).Theorem 3.2.
For given scalars0≤τ0≤τM,α>0,β>0, then the system (2.1) is exponentially stable with L2-gain, if there exist positive-definite matrices Pi,Pij,Qki,Qkij(k=1,2,3), and Rli, Rlij(l=1,2) such that the following LMIs hold:
(3.32)Σ~1i=[Σ~1,1Ξ~1,2-α1-eατMR1i0Σ~1,5*Ξ~2,2-αeατ0-eατMR2i-αeατ0-eατMR2iΣ~2,5**Ξ~3,300***-e-ατ0Q3i+αeατ0-eατMR2i0****Σ~5,5],(3.33)Σ_1i=[Σ_1,1Ξ_1,2β1-e-βτMR1ij0Σ_1,5*Ξ_2,2βe-βτ0-e-βτMR2ijβe-βτ0-e-βτMR2ijΣ_2,5**Ξ_3,300***-eβτ0Q3ij-βe-βτ0-e-βτMR2ij0****Σ_5,5]<0
with
(3.34)Σ~1,1=Q1i+Q2i+Q3i+PiAi+Ai⊺Pi+α1-eατMR1i+τMAi⊺R1iAi+(τM-τ0)Ai⊺R2iAi+Ci⊺Ci,Σ~1,5=PiBi+τMAi⊺R1iBi+(τM-τ0)Ai⊺R2iBi+Ci⊺Di,Σ~2,5=τMAτi⊺R1iBi+(τM-τ0)Aτi⊺R2iBi,Σ~5,5=τMBi⊺R1iBi+(τM-τ0)Bi⊺R2iBi+Di⊺Di-γ2I,Σ_1,1=Q1ij+Q2ij+Q3ij+PijAi+Ai⊺Pij-β1-e-βτMR1ij+τMAi⊺R1ijAi+(τM-τ0)Ai⊺R2ijAi+Ci⊺Ci,Σ_1,5=PijBi+τMAi⊺R1ijBi+(τM-τ0)Ai⊺R2ijBi+Ci⊺Di,Σ_2,5=τMAτi⊺R1ijBi+(τM-τ0)Aτi⊺R2ijBi,Σ_5,5=τMBi⊺R1ijBi+(τM-τ0)Bi⊺R2ijBi+Di⊺Di-γ2I.Proof.
For all nonzeroω(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).
Whent∈[t0,t1)∪[tℑ-1+Δℑ-1,tℑ),k=2,3,…, by the system (2.1), we can obtain
(3.35)V˙(t,x(t))+αVi(t,x(t))+Π(t)=φ⊺(t)Σ~iφ(t).
From (3.32), we can easily get
(3.36)V˙i(t,ηt)+αVi(t,ηt)+Π(t)<0.
Integrate this inequality during[t0,t], it is known that
(3.37)Vi(t,x(t))≤e-α(t-t0)Vi(t0,xt0)-∫t-t0t‍e-α(t-s)Π(s)ds,t0≤t<t1,Vi(t,x(t))≤e-α(t-tℑ-1-Δℑ-1)Vi(tℑ-1+Δℑ-1,xtℑ-1+Δℑ-1)-∫tℑ-1+Δℑ-1t‍e-α(t-s)Π(s)ds,tℑ-1+Δℑ-1≤t<tℑ,ℑ=2,3,….
Whent∈[tℑ,tℑ+Δℑ),ℑ=2,3,…, by the system (2.1), by the same way, we can obtain
(3.38)V˙(t,x(t))-βVi(t,x(t))+Π(t)=φ⊺(t)Σ_iφ(t)<0.
Then we have(3.39)Vij(t,x(t))≤Vij(tℑ,xtℑ)eβ(t-tℑ)-∫tℑt‍eβ(t-s)Π(s)ds,tℑ≤tℑ+Δℑ,ℑ=1,2,….
Whent∈[tℑ,tℑ+Δℑ),ℑ=1,2,…, it follows that
(3.40)V(t,x(t))=Vσ_(tℑ-1+Δℑ-1)σ(tℑ)(tℑ)eβT-(tℑ,t)-αT+(tℑ,t)-∫tℑt‍eβT-(tℑ,t)-αT+(tℑ,t)Π(s)ds≤μ2Vσ(tℑ-1)(tℑ-)eβT-(tℑ,t)-αT+(tℑ,t)-∫tℑt‍eβT-(tℑ,t)-αT+(tℑ,t)Π(s)ds≤μ2μ1Vσ_(tℑ-2+Δℑ-2)σ(tℑ-1)(tℑ-1)eβT-(tℑ-1,t)-αT+(tℑ-1,t)-μ2μ1∫tℑ-1tℑ-1-Δℑ-1‍eβT-(tℑ,t)-αT+(tℑ,t)Π(s)ds-μ2∫tℑ-1+Δℑ-1tℑ‍eβT-(tℑ,t)-αT+(tℑ,t)Π(s)ds-∫tℑt‍eβT-(tℑ,t)-αT+(tℑ,t)Π(s)ds≤⋯≤-∫t0t‍μ2Nσ(t)(s,t)μ1Nσ_(t)(s,t)eβT-(s,t)-αT+(s,t)Π(s)ds+μ2Nσ(t)(t0,t)μ1Nσ_(t)(t0,t)Vσ(t0)(t0)eβT-(tℑ,t)-αT+(tℑ,t).
Under the zero initial condition, Lett0=0, (3.40) implies
(3.41)∫0t‍μ2Nσ(t)(s,t)μ1Nσ_(t)(s,t)eβT-(s,t)-αT+(s,t)Λ(s)ds≤0.
Integrate (3.41) during [0,∞), then we can obtain
(3.42)∫0∞‍∫0t‍e[Nσ(t)(0,s)lnμ2+Nσ_(t)(0,s)lnμ1]eβT-(s,t)-αT+(s,t)Π(s)dsdt<∫0∞‍e[Nσ(t)(0,s)lnμ2+Nσ_(t)(0,s)lnμ1]Π(s)(∫s∞‍e-κ(t-s)dt)ds≤∫0∞‍Π(s)ds≤0.
Whent∈[t0,t1)∪[tℑ-1+Δℑ-1,tℑ),ℑ=2,3,…, by the same mathematical operations, we have ∫t0∞‍Π(s)ds<0.
From which we can get∥z~(t)∥2≤∥ω(t)∥2. This proof is completed.Remark 3.3.
Ifμ1=μ2=0, which implies that Pi=Pij=P,Qki=Qkij=Qk,Rli=Rlij=Rl,i,j∈ℙ, by (3.3)-(3.4) and (3.32)-(3.33), we have Ta=0, then it requires a common Lyapunov functional for all subsystems, and the switching signals can be arbitrary. If μk→∞(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:
(4.1)A1=[-1.50.30-2.4],A2=[-1.40.80.6-2.4],Aτ1=[-2.2-0.30-1.6],Aτ2=[1.200-1.8],C1=[-0.40.20.20.1],C2=[0.20.30.10.35],B1=[-0.350.28],B2=[-0.660.25],D1=[-0.35-0.15],D2=[0.57-0.48].
τ
0 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 Ta and τM 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 τM and the minimal average dwell-time Ta 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 Ta 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
DifferentTa and τM for α=0.25, β=0.2.
μ
1
μ
2
1.0
1.5
2.0
T
a
0.0521
2.1022
5.2454
τ
M
1.2036
1.5415
1.7154Figure 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.



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*Source: 956370-2012-10-11.xml*