# Delay-Dependent Finite-Time andL2-Gain Analysis for Switched Systems with Time-Varying Delay

Authors: Qiongfen Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

For the switched systems, switching behavior always affects the finite-time stability (FTS) property, which was neglected by most previous studies. This paper is mainly concerned with the problem of delay-dependent finite-time andL2-gain analysis for switched systems with time-varying delay. Several less conservative sufficient conditions related to finite-time stability and boundness of switched system with time-varying delays are proposed; the system trajectory stays within a special bound with the information of  switching signal. At last, a numerical example is also given to illustrate the efficiency of the developed method.


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## Body

## 1. Introduction

During the past decades, the systems with time delays have received much attention since they often fall in various practical systems, for instance, neural networks, networked control systems, engineering systems, biology, economics, and other fields. However, time delay is always frequently the major cause of oscillation and instability; the stability for systems with time delays has been devoted to a lot of effort. Nowadays, the stability of time delay is parted into two classes: delay-independent stability criteria and delay-dependent ones. It should be pointed out that delay-independent criteria turn out to be more conservative, especially for the small-size delays. The problem of delay-dependent stability analysis for time-delay systems has received considerable attention, and lots of important results have also been reported [1–12].In recent years, there has been an increasing interest in analysis of hybrid and switched systems due to their importance both in theory and its applications. As an important class of hybrid systems, switched linear systems comprise a collection of linear subsystems described by differential equations as well as a switching law to specify the switching during these subsystems. A switched system is a type of hybrid system with a combination of the discrete and continuous dynamical systems. These systems are recognized as models for appearance which cannot be denoted by exclusively continuous or discrete processes. Recently, based on the Lyapunov functions and other analysis tools, the corresponding stability and stabilization for both linear and nonlinear and switched systems have been investigated and many good results have been obtained; for a recent survey on this issue and related matters, one can also refer to [13–26].Most of the existing results have concentrated on Lyapunov-based asymptotic stability for the switched systems; the behavior of that is above the infinite time interval. However, in practical engineering, there exists the bad transient characteristics; the system is asymptotically stable without unusable. For the systems which work in a short time interval, for instance, control system, missile system, and communication network system, which mainly care about the behavior over a finite time interval. To deal with this problem, in 1961, Dorato presented the idea of finite-time stability in [27]. Over the past few years, many study efforts have been dedicated to the finite-time stability (FTS) of switched systems due to its wide applications. Comparing with the classical Lyapunov stability, which currently is the focus on a large and growing interdisciplinary area of research. To study the transient behavior of systems, FTS concerns the stability of a system over a finite interval of time and plays an important role. It is important to emphasize the disconnection between classical Lyapunov stability and finite-time stability. The problem about finite-time stabilization has been widely learned in the literature [28–44]. It is worth pointing out that there is a difference between finite-time stability and Lyapunov asymptotic stability, and they are also independent of each other.Recently, some papers are found to be related to finite-time stability for switched systems. For example, based on the technique of average dwell time, the problem of finite-time boundedness for the switched linear system with time delays was investigated in [36]. The issue of finite-time stability and H∞ stabilization for switched nonlinear discrete-time systems was developed in [37]. The problem of finite-time quantized H∞ control problem for a class of discrete-time switched time-delay systems with time-varying exogenous disturbances was also discussed in [38]. But, to the best of the authors' knowledge, the finite-time L2-gain problems for switched systems have not been completely studied. The previous research has some conservatism of stability criterion; it is natural to look for an alternative way to reduce the conservatism of corresponding stability criteria. This idea motivates to our study.The main contribution of this paper is that we present a novel approach for finite-time stability of the given switched system. Moreover, several sufficient conditions ensuring the finite-time stability and boundness are proposed with different information we know about the switching signal. It shows the less conservative results when more information about the switching signal is available. By selecting the appropriate Lyapunov-Krasovskii functional, the sufficient conditions are obtained to guarantee finite-time stability of the systems and the closed-loop system trajectory stays in a special bound. The finite-time stability criteria can also be dealt with in the form of linear matrix inequalities and average dwell time. Finally, a numerical example is given to illustrate the effectiveness of the developed techniques.


## 2. Preliminaries

In this paper, the following switched systems are described as follows:(1)x˙(t)=Aσ(t)x(t)+Bσ(t)x(t-τ(t))+Gσ(t)ω(t),z(t)=Cσ(t)x(t)+Dσ(t)ω(t),t≥0,x~(t)=ϕ(t),t∈[-h,0],
where x(t)∈ℛn is the state vector of the system, z(t)∈ℛm is the controlled output, and ω(t)∈L2q[0,∞] is the external disturbance vector and satisfies the constraint:
(2)∫0T‍ω⊺(t)ω(t)≤d,d≥0.Aσ(t), Bσ(t), Gσ(t), Cσ(t), and Dσ(t) are the constant real matrices with appropriate dimension. τ(t) represents the mode-dependent time-varying state delay in the switched system and satisfies the following conditions:
(3)0<τ(t)≤h<∞,(4)τ˙(t)≤τ,
where h is the upper bound of the time-varying delay τ(t) and τ is the variation rate of the time-varying delay τ(t). ϕ(t) is the differentiable vector-valued initial function on [-h,0]. σ(t):[0,∞)→𝒩={1,2,…,N} is the right continuous piecewise constant switching signal to be designed.Corresponding to the switching signalσ(t), we get the follwing switching sequence:
(5)∑‍={x0;(i0,t0),…,(ik,tk),…,∣ik∈𝒩,k=0,1,…},
where t0 is the initial time when tk∈[tk,tk+1), x(t0) is the initial state and ikth subsystem is active. Therefore, the trajectory x(t) of the switched system (1) is called the trajectory of the ikth subsystem. As assumed before, we get rid of Zeno behavior for all types related to switching signal. Throughout this paper, we assumed that the state of the switched system (1) does not jump at the switching instants; that is, the trajectory x(t) is continuous everywhere.Firstly, we will give the definitions and lemmas about switched system (1), which plays an major role in the derivation of our results.Definition 1.
Switch system (1) is said to be finite-time bounded with respect to (c1,c2,T,S,d), if condition (6) holds:
(6)max-τ≤t0≤0{x⊺(t0)Sx(t0),x˙⊺(t0)Sx˙(t0)}≤c1⟹x⊺(t)Sx(t)<c2,∀t∈[0,T],∀ω(t):∫0T‍ω⊺(s)ω(s)ds≤d,
where c2>c1≥0 and S>0.Definition 2.
Forγ>0, d>0, T>0, η>0, Λ>0, and c2>c1>0, system (3) is said to be finite-time stable with a weighted L2 performance γ with respect to (c1,c2,T,S,d), if the following condition holds:
(7)∫0T‍[ηs-lnλ1c2Λc1+dγ2(1/η)(1-eηT)]z⊺(s)z(s)ds≤γ2e-ηT∫0T‍ω⊺(s)ω(s)ds,
and under zero initial condition, it holds for all nonzero ω: ∫0T‍ω⊺(s)ω(s)ds≤d.Definition 3 (see [21]).
For anyT2>T1≥0, let Nσ(T1,T2) denote the switching number of σ(t) during (T1,T2). If Nσ(T1,T2)≤N0+(T2-T1)/τa holds for N0≥0 and τa>0, then N0 and τa are called chattering bound and average dwell time, respectively. Here we assume N0=0 for simplicity as supported by reference.Lemma 4 (see [9]).
Letfi:ℛm→ℛ(i=1,2,…,N) have positive values in an open subset 𝒟 of ℛm. Then, the reciprocally convex combination of fi over 𝒟 satisfies
(8)min{βi∣βi>0,∑iβi=1}∑i‍1βifi(t)=∑i‍fi(t)+maxgi,j(t)∑i≠j‍gi,j(t)subjectto{gi,j:ℛm⟶ℛ,gj,i(t)=gi,j(t),[fi(t)gi,j(t)gi,j(t)fj(t)]≥0}.Lemma 5 (see [11]).
For any constant matrixZ∈ℛn×n, Z=Z⊺>0, scalars h>0, such that the following integrations are well defined; then
(9)-h∫t-ht‍x⊺(s)Zx(s)ds≤-[∫t-ht‍x(s)ds]⊺Z[∫t-ht‍x(s)ds],-h22∫-h0‍∫t+θt‍x⊺(s)Zx(s)dsdθ≤-[∫-h0‍∫t+θt‍x(s)dsdθ]⊺Z[∫-h0‍∫t+θt‍x(s)dsdθ].Lemma 6 (Schur’s complement).
Given constant matricesX, Y, Z, where X=X⊺ and 0<Y=Y⊺, then X+Z⊺Y-1Z<0 if and only if
(10)[XZ⊺*-Y]<0or[-YZ*X]<0.


## 3. Finite-Time Boundedness Analysis

Theorem 7.
System (3) is said to be finite-time stability with respect to (c1,c2,R,d,T) if there exist symmetric positive matrices Pi, Qsi  s=1,2, Xki  (k=1,2), Yi and matrices Mi, Ni, Vi, scalars α≥0, μ≥1, λl>0  (l=1,2,…,8), d>0, h>0, Λ>0, τ>0, r(τ); such that ∀i,j∈𝒩, we have the following linear matrix inequalities:(11)Ξ1i=[Ξ11iΞ12iWiAi⊺Vi⊺hAi⊺Mi⊺+Yi+Yi⊺Yi+Yi⊺PiGi*Ξ22i-Wi+X2ihBi⊺Vi⊺hBi⊺Mi⊺00**-Q1i-X2ih0000***Ξ44i-hMi⊺0ViGi****-hX1i-Yi-Yi⊺0hMiGi*****-Yi-Yi⊺0******-ηHi]<0,Ξ2i=[Ξ11iΞ12iWiAi⊺Vi⊺Yi+Yi⊺hAi⊺Ni⊺+Yi+Yi⊺PiGi*Ξ22i-Wi+X2ihBi⊺Vi⊺0hBi⊺Ni⊺0**-Q1i-X2ih0000***Ξ44i0-hNi⊺ViGi****-Yi-Yi⊺00*****-hX1i-Yi-Yi⊺hNiGi******-ηHi]<0,(12)Pi<μPj,Q1i<μQ1j,Q2i<μQ2j,X1i<μX1j,X2i<μX2j,Yi<μYj,(13)λ1c2e-ηT>Λc1+dλ8(1-e-ηT).
with the average dwell time of the switching signal σ satisfying
(14)τa>τa*=Tlnμln(λ1c2)-ln[Λc1+dλ8(1-e-ηT)]-ηT,
where
(15)Ξ11i=δPi+PiAi+Ai⊺Pi+eδh(Q1i+Q2i)+hX1i-X2ih-2Yi-2Y2i⊺,Ξ12i=PiBi+X2ih-Wi,Ξ22i=r(τ)Q1i-X2ih+Wi+Wi⊺-X2i⊺h,Ξ44i=hX2i+eδh-δh-1δ2Yi-Vi-Vi⊺,r(τ)={-(1-τ)eδh,ifτ>1;-(1-τ),ifτ≤1.
Consider
(16)Λ=λ2+heδh(λ3+λ4)+h2eδh(λ5+λ6)+12λ7h3eδh,(17)λ1=mini∈𝒩{λmin(P¯i)},λ2=maxi∈𝒩{λmax(P¯i)},(18)λ3=maxi∈𝒩{λmax(Q¯1i)},λ4=maxi∈𝒩{λmax(Q¯2i)},(19)λ5=maxi∈𝒩{λmax(X¯1i)},λ6=maxi∈𝒩{λmax(X¯2i)},(20)λ7=maxi∈𝒩{λmax(Y¯i)},λ8=maxi∈𝒩{λmax(Hi)}.Proof.
Definext(s)=x(t+s). We consider the following Lyapunov-Krasovskii functional:
(21)Vσ(t)(xt,t)=∑l=14‍Vlσ(t)(xt,t),V1σ(t)(xt,t)=x⊺(t)eδtPσ(t)x(t),V2σ(t)(xt,t)=∫t-ht‍eδ(s+h)x⊺(s)Q1σ(t)x(s)ds+∫t-τ(t)t‍eδ(s+h)x⊺(s)Q2σ(t)x(s)ds,V3σ(t)(xt,t)=∫-h0‍∫t+θt‍eδ(s-θ)x⊺(s)X1σ(t)x(s)dsdθ+∫-h0‍∫t+θt‍eδ(s-θ)x˙⊺(s)X2σ(t)x˙(s)dsdθ,V4σ(t)(xt,t)=∫-h0‍∫θ0‍∫t+υt‍eδ(s-θ)x˙⊺(s)Yσ(t)x˙(s)dsdυdθ.
Taking the time derivative ofVσ(t)(xt,t) along the trajectory of the system (1), one has
(22)V˙1i(xt,t)=x⊺(t)eδt(δPi+PiAi+Ai⊺Pi)x(t)+2eδtx⊺(t)PiBix(t-τ(t))+2eδtx⊺(t)PiGiω(t),V˙2i(xt,t)=eδtx⊺(t)eδh(Q1i+Q2i)x(t)-eδtx⊺(t-h)Q1ix(t-h)-(1-τ˙(t))eδtx⊺(t-τ(t))Q2ix(t-τ(t)).
Since(23)0≤τ(t)≤h,
we define
(24)r(τ)={-(1-τ)eδh,ifτ>1;-(1-τ),ifτ≤1.
Then,(25)V˙2i(xt,t)=eδtx⊺(t)eδh(Q1i+Q2i)x(t)-eδtx⊺(t-h)Q1ix(t-h)+eδtx⊺(t-τ(t))r(τ)Q2ix(t-τ(t)),V˙3i(xt,t)=eδtx⊺(t)hX1ix(t)+eδtx˙⊺(t)hX2ix˙(t)-∫t-ht‍eδ(s+h)x⊺(s)X1ix(s)ds-∫t-ht‍eδ(s+h)x˙⊺(s)X2ix˙(s)ds≤eδtx⊺(t)hX1ix(t)+eδtx˙⊺(t)hX2ix˙(t)-eδt∫t-τ(t)t‍x⊺(s)X1ix(s)ds-eδt∫t-ht-τ(t)‍x⊺(s)X1ix(s)ds-eδt∫t-ht‍x˙⊺(s)X2ix˙(s)ds.
By Lemma5, one can obtain
(26)-∫t-τ(t)t‍x⊺(s)X1ix(s)ds≤-τ(t)U1⊺X1iU1,-∫t-ht-τ(t)‍x⊺(s)X1ix(s)ds≤-(h-τ(t))U2⊺X1iU2,
where
(27)U1=1τ(t)∫t-τ(t)t‍x(s)ds,U2=1h-τ(t)∫t-ht-τ(t)‍x(s)ds,limτ(t)→01τ(t)∫t-τ(t)t‍x(s)ds=x(t),limτ(t)→h1h-τ(t)∫t-ht-τ(t)‍x(s)ds=x(t-h).
From the Leibniz-Newton formula, the following equation is true for any matricesMi, Ni, and Vi with appropriate dimensions:
(28)(2τ(t)U1⊺Mi+2(h-τ(t))U2⊺Ni+2x˙⊺(t)Vi)×[-x˙(t)+Aix(t)+Bix(t-τ(t))+Giω(t)]=0.
From Lemma4, it yields
(29)-∫t-ht‍x˙⊺(s)X2ix˙(s)ds=-∫t-τ(t)t‍x˙⊺X2ix˙(s)ds-∫t-ht-τ(t)‍x˙⊺X2ix˙(s)ds≤-hh-τ(t)[∫t-ht-τ(t)‍x˙ds]⊺X2ih[∫t-ht-τ(t)‍x˙ds]-hτ(t)[∫t-τ(t)t‍x˙ds]⊺X2ih[∫t-τ(t)t‍x˙ds]≤-[x(t)-x(t-τ(t))x(t-τ(t))-x(t-h)]⊺×[X2ihWi*X2ih][x(t)-x(t-τ(t))x(t-τ(t))-x(t-h)]V˙4i(xt,t)=eδtx˙⊺(t)Yix˙(t)∫-h0‍∫θ0‍e-δυdυdθ-eδt∫-h0‍∫t+θt‍x˙⊺(s)Yix˙(s)dsdθ=eδtx˙⊺(t)eδh-δh-1δ2Yix˙(t)-eδt∫-h0‍∫t+θt‍x˙⊺(s)Yix˙(s)dsdθ.
By using Lemma 5, we have
(30)-∫-h0‍∫t+θt‍x˙(s)Yix˙(s)dsdθ=-∫-τ(t)0‍∫t+θt‍x˙(s)Yix˙(s)dsdθ-∫-h-τ(t)‍∫t+θt‍x˙(s)Yix˙(s)dsdθ≤-2τ2(t)[∫-τ(t)0‍∫t+θt‍x˙(s)dsdθ]⊺×Yi[∫-τ(t)0‍∫t+θt‍x˙(s)dsdθ]-2(h-τ(t))2[∫-h-τ(t)‍∫t+θt‍x˙(s)dsdθ]⊺×Yi[∫-h-τ(t)‍∫t+θt‍x˙(s)dsdθ]=-2τ2(t)[τ(t)x(t)-∫t-τ(t)t‍x(s)ds]⊺×Yi[τ(t)x(t)-∫t-τ(t)t‍x(s)ds]-2(h-τ(t))2[(h-τ(t)x(t))-∫t-ht-τ(t)‍x(s)ds]⊺×Yi[(h-τ(t)x(t))-∫t-ht-τ(t)‍x(s)ds]=-2[x(t)-1τ(t)∫t-τ(t)t‍x(s)ds]⊺×Yi[x(t)-1τ(t)∫t-τ(t)t‍x(s)ds]-2[x(t)-1h-τ(t)∫t-ht-τ(t)‍x(s)ds]⊺×Yi[x(t)-1h-τ(t)∫t-ht-τ(t)‍x(s)ds]=-2[x(t)-U1]⊺Yi[x(t)-U1]-2[x(t)-U2]⊺Yi[x(t)-U2].
Therefore, for a givenη>0 and from (21)–(30), one can obtain that
(31)V˙i(xt,t)-ηω⊺Hiω(t)≤eδtξ⊺(t)Ξ¯iξ(t),
where(32)ξ⊺(t)=[x⊺(t),x⊺(t-τ(t)),x⊺(t-h),x˙⊺(t),U1⊺,U2⊺,ω⊺(t)].Ξ¯i=[Ξ11iΞ12iWiAi⊺Vi⊺τ(t)Ai⊺Mi⊺+Yi+Yi⊺(h-τ(t))Ai⊺Ni⊺+Yi+Yi⊺PiGi*Ξ22i-Wi+X2ihBi⊺Vi⊺τ(t)Bi⊺Mi⊺(h-τ(t))Bi⊺Ni⊺0**-Q1i-X2ih0000***Ξ44i-τ(t)Mi⊺-(h-τ(t))Ni⊺ViGi****-τ(t)X1i-Yi-Yi⊺0τ(t)MiGi*****-(h-τ(t))X1i-Yi-Yi⊺(h-τ(t))NiGi******-ηHi].
The LMIs (11) lead to τ(t)→h and to τ(t)→0, respectively. It is easy to see that Ξ1i results from Ξ¯i∣τ(t)=h and Ξ2i results from Ξ¯i∣τ(t)=0. Thus, we obtain
(33)V˙i(xt,t)-ηVi(xt,t)<ηω⊺(t)Hiω(t).
Notice that(34)ddt(e-ηtVi(xt,t))<ηe-ηtω⊺(t)Hiω(t).
Integrating (34) from tk to t, we can get that
(35)Vi(xt,t)<eη(t-tk)Vi(xtk,tk)+η∫tkt‍eη(t-s)ω⊺(s)Hiω(s)ds.
Note that (12) and μ≥1, it yields
(36)Vσ(tk)(xtk,tk)≤μVσ(tk-1)(xtk,tk).
Then, we can easily have(37)Vσ(tk-1)(xtk,tk)<eη(tk-tk-1)Vσ(tk-1)(xtk-1,tk-1)+η∫tk-1tk‍eη(tk-s)ω⊺(s)Hiω(s)ds.
Thus, from (35)–(37), it yields
(38)Vσ(t)(xt,t)≤eη(t-tk)Vσ(tk)(xtk,tk)+η∫tkt‍eη(t-s)ω⊺(s)Hiω(s)ds≤μeη(t-tk)Vσ(tk-1)(xtk,tk)+η∫tkt‍eη(t-s)ω⊺(s)Hiω(s)ds≤μeη(t-tk-1)Vσ(tk-1)(xtk-1,tk-1)+ημ∫tk-1tk‍eη(t-s)ω⊺(s)Hiω(s)ds+η∫tkt‍eη(t-s)ω⊺(s)Hiω(s)ds≤μ2eη(t-tk-2)Vσ(tk-2)(xtk-2,tk-2)+ημ2∫tk-2tk-1‍eη(t-s)ω⊺(s)Hiω(s)ds+ημ∫tk-1tk‍eη(t-s)ω⊺(s)Hiω(s)ds+η∫tkt‍eη(t-s)ω⊺(s)Hiω(s)ds≤⋯≤μNσ(0,t)eηtVσ(0)(x0,0)+ημNσ(0,t)∫0t1‍eη(t-s)ω⊺(s)Hiω(s)ds+ημNσ(t1,t)∫t1t2‍eη(t-s)ω⊺(s)Hiω(s)ds+⋯+η∫tkt‍eη(t-s)ω⊺(s)Hiω(s)ds=μNσ(0,t)eηtVσ(0)(x0,0)+η∫0t‍eη(t-s)μNσ(s,t)ω⊺(s)Hiω(s)ds≤μNσ(0,t)eηTVσ(0)(x0,0)+ημNσ(0,t)dλmax(Hi)eαT∫0t‍e-ηsds≤μNσ(0,T)eηT×{Vσ(0)(x0,0)+dλmax(Hi)η∫0T‍e-ηsds}≤μT/τaeηT{Vσ(0)(x0,0)+dλmax(Hi)(1-e-ηT)}≤μNσ(0,T)eηT×{Vσ(0)(x0,0)+dλmax(Hi)η∫0T‍e-ηsds}=μT/τaeηT{Vσ(0)(x0,0)+dλ8(1-e-ηT)}.
DefineP¯i=R-1/2PiR-1/2, Q¯si=R-1/2QsiR-1/2(s=1,2), X¯li=R-1/2XliR-1/2(l=1,2),Y¯i=R-1/2YiR-1/2.
Note that(39)Vσ(0)(x0,0)=maxi∈𝒩λmax(P¯i)x⊺(0)Rx(0)+(maxi∈𝒩λmax(Q¯1i)+maxi∈𝒩λmax(Q¯2i))eδh×∫-h0‍eδsx⊺(s)Rx(s)ds+maxi∈𝒩λmax(X¯1i)eδh∫-h0‍∫θ0‍e-δθx⊺(s)Rx(s)dsdθ+maxi∈𝒩λmax(X¯2i)eδh∫-h0‍∫θ0‍e-δθx˙⊺(s)Rx˙(s)dsdθ+maxi∈𝒩λmax(X¯2i)eδh×∫-h0‍∫θ0‍∫υ0‍e-δθx˙⊺(s)Rx˙(s)dsdθdυ≤{12maxi∈𝒩λmax(P¯i)+heδh(maxi∈𝒩λmax(Q¯1i)+maxi∈𝒩λmax(Q¯2i))+h2eδh(maxi∈𝒩λmax(X¯1i)+maxi∈𝒩λmax(X¯2i))+12h3eδhmaxi∈𝒩λmax(Y¯i)}×sup-h≤s≤0{x⊺(s)Rx(s),x˙⊺(s)Rx˙(s)}≤(12λ2+heδh(λ3+λ4)+h2eδh(λ5+λ6)+12λ7h3eδh)×sup-h≤s≤0{x⊺(s)Rx(s),x˙⊺(s)Rx˙(s)}≤(12λ2+heδh(λ3+λ4)+h2eδh(λ5+λ6)+12λ7h3eδh)c1=Λc1.
Thus,(40)Vσ(t)(xt,t)≤μT/τaeαT{Λc1+dλ8(1-e-ηT)}=e(α+lnμ/τa)T{Λc1+dλ8(1-e-ηT)}.
On the other hand,(41)Vσ(t)(xt,t)≥λmin(P¯i)x⊺(t)Rx(t)=λ1x⊺(t)Rx(t).
From (40) and (41), one obtains
(42)x⊺(t)Rx(t)≤Λc1+dλ8(1-e-ηT)λ1e(η+lnμ/τa)T.
Whenμ=1, which is the trivial case, from (13), x⊺(t)Rx(t)<c2e-ηTeηT=c2. When μ≥1, from (13), ln(λ1c2)-ln[Λc1+dλ8(1-e-ηT)]-αT>0, we have
(43)Tτa<ln(λ1c2)-ln[Λc1+dλ8(1-e-ηT)]-ηTlnμ=ln(λ1c2e-ηT/(Λc1+dλ8(1-e-ηT)))lnμ.
Substituting (43) into (42) yields
(44)x⊺(t)Rx(t)<c2.
The proof is completed.Remark 8.
It should be mentioned that to reduce the conservatism,-∫-τ(t)0‍∫t+θt‍x˙⊺(s)Zix˙dsdθ and -∫-h-τ(t)‍∫t+θt‍x˙⊺(s)Zix˙dsdθ are bounded with -2[x(t)-U1]⊺Zi[x(t)-U1] and -2[x(t)-U2]⊺ Zi[x(t)-U2], respectively.Remark 9.
One can clearly see that criteria given in Theorem7 are delay-dependent. It is well known that the delay-dependent criteria are less conservative than delay-independent criteria when the delay is small or belongs to a given interval. Interval time-varying delay is a time delay that varies in an interval in which the lower bound is not restricted to zero.Remark 10.
In Theorem7, the free weight matrix technology is employed, which reduces the conservativeness of the stability condition.


## 4. Finite-Time WeightedL2-Gain Analysis

Theorem 11.
System (3) is finite-time bounded with respect to (c1,c2,R,d,T) if there exist symmetric positive matrices Pi, Qsi  s=1,2, Xki(k=1,2), Yi and matrices Mi, Ni, Vi, scalars α≥0, μ≥1, λl>0(l=1,2,…,7), d>0, h>0, Λ>0, γ>0, τ>0, r(τ), such that ∀i,j∈𝒩; we have the following linear matrix inequalities:(45)Σ1i=[Ξ11iΞ12iWiAi⊺Vi⊺hAi⊺Mi⊺+Yi+Yi⊺Yi+Yi⊺PiGiCi⊺*Ξ22i-Wi+X2ihBi⊺Vi⊺hBi⊺Mi⊺000**-Q1i-X2ih00000***Ξ44i-hMi⊺0ViGi0****-hX1i-Yi-Yi⊺0hMiGi0*****-Yi-Yi⊺00******-γ2IDi⊺*******-I]<0,Σ2i=[Ξ11iΞ12iWiAi⊺Vi⊺Yi+Yi⊺hAi⊺Ni⊺+Yi+Yi⊺PiGiCi⊺*Ξ22i-Wi+X2ihBi⊺Vi⊺0hBi⊺Ni⊺00**-Q1i-X2ih00000***Ξ44i0-hNi⊺ViGi0****-Yi-Yi⊺000*****-hX1i-Yi-Yi⊺hNiGi0******-γ2IDi⊺*******-I]<0,(46)Pi<μPj,Q1i<μQ1j,Q2i<μQ2j,X1i<μX1j,X2i<μX2j,Yi<μYj,λ1c2e-ηT>Λc1+dγ21η(1-e-ηT),
with the average dwell time of the switching signal σ satisfying
(47)τa>τa*=Tlnμln(λ1c2)-ln[Λc1+dγ2(1/η)(1-e-ηT)]-ηT.Proof.
Choose the same Lyapunov-Krasovskii functional as in Theorem7; after some mathematical manipulation and Schur complement, we can get
(48)V˙σ(t)(xt,t)+e⊺(t)e(t)-γ2ω⊺(t)ω(t)=ξ⊺(t)Σsiξ(t),(s=1,2).
Define(49)J(t)=e⊺(t)e(t)-γ2ω⊺(t)ω(t).
From (45), we obtain
(50)V˙σ(t)(xt,t)-ηVσ(t)(xt,t)+J(t)<0.
Whent∈[tk,tk+1], where tk is the kth switching instant,
(51)Vσ(t)(xt,t)<eη(t-tk)Vσ(tk)(xtk,tk)-∫tkt‍eη(t-s)J(s)ds.
Notice thatx(tk)=x(tk-); then one obtains
(52)Vσ(tk)(x(tk),tk)≤μVσ(tk-)(x(tk),tk).
For anyt∈[0,T], one has
(53)Vσ(t)(xt,t)≤eη(t-tk)Vσ(tk)(xtk,tk)+η∫tkt‍eη(t-s)J(s)ds≤μeη(t-tk-1)Vσ(tk-1)(xtk-1,tk-1)+ημ∫tk-1tk‍eη(t-s)J(s)ds+η∫tkt‍eη(t-s)J(s)ds+ημ∫tk-1tk‍eη(t-s)ω⊺(s)Hiω(s)ds+η∫tkt‍eη(t-s)J(s)ds≤⋯≤μNσ(0,t)eηtVσ(0)(x0,0)+ημNσ(0,t)∫0t1‍eη(t-s)J(s)ds+ημNσ(t1,t)∫t1t2‍eη(t-s)J(s)ds+⋯+η∫tkt‍eη(t-s)J(s)ds≤μNσ(0,T)eηTVσ(0)(x0,0)+η∫0T‍eη(T-s)μNσ(s,T)J(s)ds.
Under zero initial condition, we have(54)∫0T‍e-ηsμNσ(s,T)J(s)ds<0,
which implies that
(55)∫0T‍e-αsμNσ(s,T)z⊺(s)z(s)ds<∫0T‍μNσ(s,T)γ2ω⊺(s)ω(s)ds.
Multiplying both sides of (55) by μ-Nσ(0,T) yields
(56)∫0T‍e-αsμ-Nσ(0,s)z⊺(s)z(s)ds<∫0T‍μ-Nσ(0,s)γ2ω⊺(s)ω(s)ds.
It is easy to deduce from (47) that
(57)Nσ(0,s)≤sτa≤ln(λ1c2/(Λc1+dγ2(1/η)(1-eηT)))-ηslnμ.
Sinceμ≥1, we have
(58)∫0Tμln((ηs-ln(λ1c2/(Λc1+dγ2(1/η)(1-eηT))))/lnμ)‍×z⊺(s)z(s)ds≤∫0T‍μ-Nσ(0,s)e-ηsz⊺(s)z(s)ds≤∫0T‍μ-Nσ(0,s)e-ηsγ2ω⊺(s)ω(s)ds≤e-ηT∫0T‍γ2ω⊺(s)ω(s)ds.
Therefore, we can obtain(59)∫0T[ηs-lnλ1c2Λc1+dγ2(1/η)(1-eηT)]z⊺(s)z(s)ds‍≤γ2e-ηT∫0Tω⊺(s)ω(s)ds.
According to Definition2, we know that Theorem 11 holds. This completes the proof.Remark 12.
It can be seen that Theorem11 presents a more general result than [33–42] which contain the decay η; such a condition is derived by advantage of the simultaneous usage of the Lyapunov functional and the state variable transformation.Remark 13.
In many real applications, the minimum value ofγmin2 is of interest. In Theorem 11, with a fixed μ and η, γmin can be obtained through following procedure:
(60)minγ2s.t. LMIs((37)-(39)).


## 5. Illustrative Example

Example 1.
Consider the system (1) with the following data:
(61)A1=[-1.71.701.3-10.70.71-0.6],A2=[1-100.70-0.61.70-1.7],B1=[1.5-1.70.1-1.31-0.3-0.710.6],B2=[-100.11.3-0.10.61.50.11.8],G1=G2=[100010001],x(t)=[0.700],ω(t)=[0.03sin(t)0.02cos(2t)0.015(sin(t+1)+cos(t-2))],t∈[-τ,0].
The values of c1, c2, T, d and matrix R are given as follows:
(62)c1=0.5,T=10,R=I,d=0.01,δ=0.042,η=0.075.Through Theorem7, when c2=50, we can see that the admissible maximum h computed by [36] is 0.2, and the maximal value of τ in this paper is 0.262. It is obvious that our method shows less conservatism result than that in [36]. Moreover, when τ=0.2, the optimal bound with minimum value of c2min relies on the parameter η. One can obtain clearly that Theorem 7 in our paper can indeed provide much smaller admissible c2min than the stability criteria in [36] which shows the less conservative result in this paper.


## 6. Conclusions

In this paper, we have examined the problems of delay-dependent finite-time andL2-gain analysis for switched systems with time-varying delay by allowing new Lyapunov-Krasovskii functional and average dwell time. A numerical example has also been given to demonstrate the effectiveness of the proposed approach. It is possible to extend the main results obtained in this paper to real systems which contain some practical constrains, for example, the failures of controllers. Details will be reported in our follow-up work in the future.



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