# Robust Reliable Stabilization of Switched Nonlinear Systems with Time-Varying Delays and Delayed Switching

Authors: Zhengrong Xiang; Qingwei Chen
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

This paper is concerned with the problem of robust reliable stabilization of switched nonlinear systems with time-varying delays and delayed switching is investigated. The parameter uncertainties are allowed to be norm-bounded. The switching instants of the controller experience delays with respect to those of the system. The purpose of this problem is to design a reliable state feedback controller such that, for all admissible parameter uncertainties and actuator failure, the system state of the closed-loop system is exponentially stable. We show that the addressed problem can be solved by means of algebraic matrix inequalities. The explicit expression of the desired robust controllers is derived in terms of linear matrix inequalities (LMIs).


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## Body

## 1. Introduction

A switched system is composed of a family of continuous-time or discrete-time subsystems and a rule specifying the switching among them. Switched systems have received increasing attentions in the past few years, since many real-word systems such as mechanical systems, automotive industry, aircraft, and air traffic control systems, chemical processes can be modelled as switched systems (see [1–3]). A large number of results have been reported for such systems (see [4–8]).On the other hand, time delay systems have continuously been receiveing considerable attention over the past decades. The main reason is that many kinds of engineering systems, for instance, long-distance transportation systems, hydraulic pressure systems, network control systems, and so on, include time delay phenomena in their dynamics. Many valuable results have been obtained for switched systems with time delay (see [9–15]). On the other hand, the actuators may be subjected to failures in real practice, therefore, it is of practical importance to design a control system which can tolerate faults of actuators. Several design approaches to the reliable controller have been proposed for linear and nonlinear systems (see [16–19]), and these results have been extended to switched systems (see [20–22]).Recently, the asynchronous switching control problem of switched systems has stirred renewed research interests, and a varity of switched systems have been investigated by different approaches [23–26]. However, to the best of the authors’ knowledge, the issue of reliable stabilization of switched nonlinear systems with time-varying delay under asynchronous switching has not been fully investigated, which motivated the present study.In this paper, we are interested in designing the robust reliable controller for uncertain switched nonlinear system with time-varying delays and delayed switching. The remainder of the paper is organized as follows. In Section2, problem formulation is presented and the failure model of actuator in switched system is introduced briefly. In addition, some necessary lemmas are given. In Section 3, based on the average dwell-time approach, controller design for switched nonlinear system with time-varying delays and delayed switching is developed, and sufficient conditions for the existence of the controller are formulated in terms of a set of matrix inequalities. Concluding remarks are given in Section 4.Notation 1.
Throughout this paper, the superscript “T” denotes the transpose, ∥·∥ denotes the Euclidean norm. λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively, I is an identity matrix with appropriate dimension. diag{ai} denotes diagonal matrix with the diagonal elements ai,i=1,2,…,n. The asterisk * in a matrix is used to denote term that is induced by symmetry. The set of positive integers is represented by Z+.


## 2. Problem Formulation and Preliminaries

Consider the following uncertain nonlinear switched system with actuator fault(2.1)ẋ(t)=Âσ(t)x(t)+Âdσ(t)x(t-dσ(t)(t))+Bσ(t)uf(t)+fσ(t)(x(t),t),x(t)=φ(t),t∈[t0-d,t0],
where x(t)∈Rn is the state vector, uf(t)∈Rl is the control input of actuator fault, φ(t) is a continuous vector-valued function. The function σ(t):[t0,∞)→N̲={1,2,…,N} is the switching signal which is deterministic, piecewise constant and right continuous, that is, σ:{(t0,σ(t0)),(t1,σ(t1)),…}, k∈Z+, where t0≥0 is the initial time, and tk denotes the kth switching instant. dσ(t)(t) denotes the time-varying state delay satisfying 0<dσ(t)(t)≤d,  ḋσ(t)(t)≤τ for constants d and τ. Moreover, σ(t)=i means that the ith subsystem is activated. N denotes the number of subsystems. fi(x(t),t) (i∈N̲) are nonlinear functions satisfying
(2.2)∥fi(x(t),t)∥≤∥Uix(t)∥,
where Ui are known real constant matrices.Âi,Âdi for i∈N̲ are uncertain real-valued matrices with appropriate dimensions, and have the following form:
(2.3)[ÂiÂdi]=[AiAdi]+HiFi(t)[E1iE2i],
where Ai,Adi,Bi,H1i,E1i,E2i are known real constant matrices with proper dimensions, and H1i,E1i,E2i denote the structure of the uncertainties, Fi(t) are unknown time-varying matrices which satisfy
(2.4)FiT(t)Fi(t)≤I.The control input of actuator faultuf(t) can be described as
(2.5)uf(t)=Mσ(t)u(t),
where u(t)=Kσ(t)x(t) is the switching controller which will be designed, Mi(i∈N̲) are the actuator fault matrices with the following form:
(2.6)Mi=diag{mi1,mi2,…,mil},0≤m̲ik≤mik≤m̅ik,m̅ik≥1,k=1,2,…,l.
For simplicity, we introduce the following notation
(2.7)Mi0=diag{m̃i1,m̃i2,…,m̃il},Ji=diag{ji1,ji2,…,jil},Li=diag{li1,li2,…,lil},
where m̃ik=1/2(m̅ik+m̲ik), jik=(m̅ik-m̲ik)/(m̅ik+m̲ik), lik=(mik-m̃ik)/m̃ik.From (2.6)-(2.7), we have
(2.8)Mi=Mi0(I+Li),|Li|≤Ji≤I,
where |Li|=diag{|li1|,|li2|,…,|lil|}.Remark 2.1.
mik=1  means normal operation of the kth actuator signal of the ith subsystem. When mik=0, it covers the case of the complete failure of the kth actuator signal of the ith subsystem. When m̲ik>0 and mik≠1, it corresponds to the case of partial failure of the kth actuator signal of the ith subsystem.The delayed switching of the controller can be shown in Figure1.Figure 1
Diagram of the delayed switching.We can see from Figure1 that the controller Ki operates the ith subsystem in [tk-1+Δk-1,tk), and operates the jth subsystem in [tk,tk+Δk).Letσ′(t) denote the switching signal of the controller, the switching instants of the controller can be described as
(2.9)t1+Δ1,t2+Δ2,…,tk+Δk,…,k∈Z+,
where Δk<infk∈Z+(tk+1-tk), Δk represents the delayed period, and it is said to be mismatched period.Remark 2.2.
Mismatched periodΔk<infk∈Z+(tk+1-tk) guarantees that there always exists a period that the controller and the system operate synchronously, and this period is said to be matched period in the later section.Due to the delayed switching, the real input of actuator fault can be written as(2.10)uf(t)=Mσ′(t)Kσ′(t)x(t).Under switching controller (2.10), the resulting closed-loop system is given by
(2.11)ẋ(t)=(Aσ(t)+Bσ(t)Mσ′(t)Kσ′(t))x(t)+Adσ(t)x(t-dσ(t)(t))+fσ(t)(x(t),t),x(t)=φ(t),t∈[t0-d,t0].System (2.1) without uncertainties and actuator fault can be written as
(2.12)ẋ(t)=Aσ(t)x(t)+Adσ(t)x(t-dσ(t)(t))+Bσ(t)u(t)+fσ(t)(x(t),t),x(t)=φ(t),t∈[t0-d,t0].Definition 2.3 (see [13]).
If there exists switching signalσ(t), such that the trajectory of system (2.1) satisfies ∥x(t)∥≤α∥x(t0)∥he-β(t-t0), then system (2.1) is said to be exponentially stable with convergent rate β, where α≥1, β>0, t≥t0, ∥x(t)∥h=sup-d≤θ≤0{∥x(t+θ)∥,∥ẋ(t+θ)∥}.Definition 2.4 (see [27]).
For anyT2>T1≥0, let Nσ(T1,T2) denote the switching number of σ(t) on an interval (T1,T2). If
(2.13)Nσ(T1,T2)≤N0+T2-T1τa
hold for given N0≥0, τa>0, then the constant τa is called the average dwell time and N0 is the chatter bound.The following lemmas play an important role in the later development.Lemma 2.5 (see [28]).
For given vectorsa,  b and the positive matrix  X>0, there exists the matrix  M with appropriate dimension, such that
(2.14)-2aTb≤infX>0,M{[ab]T[XXMMTX(MTX+I)X-1(XM+I)][ab]}.Lemma 2.6 (see [29]).
For matricesX,Y  of appropriate dimension and  Q>0, we have
(2.15)XTY+YTX≤XTQX+YTQ-1Y.Lemma 2.7 (see [30]).
LetU,V,W, and  X be real matrices of appropriate dimensions with  X satisfying  X=XT, then for all  VTV≤I,
(2.16)X+UVW+WTVTUT<0
if and only if there exists a scalar  ε>0 such that
(2.17)X+εUUT+ε-1WTW<0.Lemma 2.8 (see [31]).
For matricesR1,R2 with appropriate dimension, there exists a positive scalar  β>0, such that
(2.18)R1Σ(t)R2+R2TΣT(t)R1T≤βR1UR1T+β-1R2TUR2
hold, where  Σ(t) is time-varying diagonal matrix,  U is known real-value matrix satisfying |Σ(t)|≤U.The objective of this paper is to design a reliable controller for system (2.1) with delayed switching such that the resulting closed-loop system is robust exponentially stable.


## 3. Main Results

To obtain the main results of this paper, we first consider the stability of the following nonlinear delay system(3.1)ẋ(t)=Ax(t)+Adx(t-d(t))+f(x(t),t),(3.2)x(t)=φ(t),t∈[t0-d,t0],
where x(t)∈Rn is the state vector, φ(t) is a continuous vector-valued initial function, A,Ad are real-valued matrices with appropriate dimensions, f(·,·):Rn×R→Rn is unknown nonlinear functions satisfying
(3.3)∥f(x(t),t)∥≤∥U̅x(t)∥,
where U̅ is known real constant matrix.


### 3.1. Stability Analysis

Lemma 3.1.
Consider system (3.1)-(3.2), for given positive constant  α, ρ1, ρ2, ε1, η1, if there exist positive definite symmetric matrices  P,S,Q,R, and any matrices  G,W  with appropriate dimensions, such that
(3.4)Q≤ρ1I,P≤ρ2I,(3.5)[ΣGTWTAdATQ0d(WT+P)ATQ0U̅T*-(1-τ)e-αdR-G-GT-GAdTQ000AdTQ0**-d-1e-αdQ0AdTS0000***-d-1Q00000****-d2S0000*****-S000******-d-1ε1Q00*******-d-1η1Q0********Φ]<0,
holds, then for Lyapunov functional candidate
(3.6)V(x(t))=xT(t)Px(t)+∫t-d(t)tẋT(s)e-α(t-s)Rẋ(s)ds+∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ,
along the trajectory of system (3.1), there holds the following inequality:
(3.7)V(x(t))<e-α(t-t0)V(x(t0)),
where  Σ=(A+Ad)TP+P(A+Ad)+(α+1)P+R,Φ=-[τρ1(1+ε1+η1)+ρ2]-1I.Proof.
LetV1(x(t))=xT(t)Px(t),V2(x(t))=∫t-d(t)tẋT(s)e-α(t-s)Rẋ(s)ds,V3(x(t))=∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ.
Notice thatx(t)-x(t-d(t))=∫t-d(t)tẋ(r)dr, (3.1) can be written as
(3.8)ẋ(t)=(A+Ad)x(t)+f(x(t),t)-Ad∫t-d(t)tẋ(r)dr.
Along the trajectory of system (3.1), the time derivative of V1(x(t)) is given by
(3.9)V̇1(x(t))=2xT(t)P{(A+Ad)x(t)+f(x(t),t)}-∫t-d(t)t2xT(t)PAdẋ(r)dr.
Let a=Adẋ(r),b=Px(t), from Lemma 2.5, we can obtain
(3.10)-2xT(t)PAdẋ(r)≤ẋT(r)AdTXAdẋ(r)+2ẋT(r)AdTXMPx(t)+xT(t)P(MTX+I)X-1(XM+I)Px(t).
Substituting (3.10) into (3.9) leads to
(3.11)V̇1(x(t))≤xT(t){(A+Ad)TP+P(A+Ad)+d(t)P(MTX+I)X-1(MX+I)P}x(t)+xT(t)Pf(x(t),t)+fT(x(t),t)Px(t)+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)AdTXAdẋ(r)dr≤xT(t){(A+Ad)TP+P(A+Ad)+τP(MTX+I)X-1(XM+I)P}x(t)+2xT(t)Pf(x(t),t)+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)AdTXAdẋ(r)dr.
Differentiating V2(x(t)) and V3(x(t)) along the trajectory of system (3.1), we have
(3.12)V̇2(x(t))≤-α∫t-d(t)te-α(t-s)ẋT(s)Rẋ(s)ds+xT(t)Rx(t)-(1-τ)e-αdxT(t-d(t))Rx(t-d(t)),V̇3(x(t))=dẋT(t)Qẋ(t)-∫t-dtẋT(r)e-α(t-r)Qẋ(r)dr-α∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ≤dẋT(t)Qẋ(t)-e-αd∫t-d(t)tẋT(r)Qẋ(r)dr-α∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ=dxT(t)ATQAx(t)+2dxT(t-d(t))AdTQAx(t)+dxT(t-d(t))AdTQAdx(t-d(t))-e-αd∫t-d(t)tẋT(r)Qẋ(r)dr-α∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ+d[2xT(t)ATQf(x(t),t)+2xT(t-d(t))AdTQf(x(t),t)+fT(x(t),t)Qf(x(t),t)].
By Lemma 2.6, we have
(3.13)2xT(t)Pf(x(t))≤xT(t)Px(t)+fT(x(t))Pf(x(t)),2xT(t)ATQf(x(t),t)≤ε1-1xT(t)ATQAx(t)+ε1fT(x(t),t)Qf(x(t),t),2xT(t-d(t))AdTQf(x(t),t)≤η1-1xT(t-d(t))AdTQAdx(t-d(t))+η1fT(x(t),t)Qf(x(t),t).
Therefore
(3.14)V̇(x(t))+αV(x(t))=V̇1(x(t))+V̇2(x(t))+αV(x(t))≤xT(t){(A+Ad)TP+P(A+Ad)+dP(MTX+I)X-1(XM+I)P+(α+1)P+d(1+ε1-1)ATQA}x(t)+2dxT(t-d(t))AdTQAx(t)+d(1+η1-1)xT(t-d(t))AdTQAdx(t-d(t))+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)(AdTXAd-e-αdQ)ẋ(r)dr+d(1+ε1+η1)fT(x(t),t)×Qf(x(t),t)+fT(x(t))Pf(x(t))≤xT(t){(A+Ad)TP+P(A+Ad)+dP(MTX+I)X-1(XM+I)P+(α+1)P+d(1+ε1-1)ATQA+d(1+ε1+η1)U̅TQU̅+U̅TPU̅}x(t)+2dxT(t-d(t))AdTQAx(t)+d(1+η1-1)xT(t-d(t))AdTQAdx(t-d(t))+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)(AdTXAd-e-αdQ)ẋ(r)dr.
Notice that 2xT(t-d(t))G[x(t)-x(t-d(t))-∫t-d(t)tẋ(r)dr]=0, where G is any matrix with appropriate dimension, we have
(3.15)V̇(x(t))+αV(x(t))≤1d(t)∫t-d(t)tξT(t,r)Zξ(t,r)dr,
where ξT(t,r)=[xT(t)xT(t-d)ẋT(r)],
(3.16)Z=[Ψ-U̅TΦ1-1U̅τATQAd+GTd(t)WTAdτAdTQA+GτAdTQAd-(1-τ)e-αdR-G-GT-d(t)Gd(t)AdTW-d(t)GTd(t)(τ-1AdTSAd-e-ατQ)],
where Ψ=(A+Ad)TP+P(A+Ad)+dP(MTX+I)X-1(XM+I)P+(α+1)P+R+d(1+ε1-1)ATQA.
LetW=XMP,S=τX, by Schur complement lemma, (3.5) is equivalent to the following inequality:
(3.17)[ΨdATQAd+GTWTAdU̅TdAdTQA+GdAdTQAd-(1-τ)e-αd-G-GT-G0AdTW-GTd-2AdTSAd-d-1e-ατQ0U̅00Φ]<0.
Using diag{I,I,d(t)I,I} to pre- and post- multiply the left term of (3.17), respectively, we can obtain Z<0.  Therefore, V̇(x(t))+αV(x(t))<0.
The proof is completed.Lemma 3.2.
Consider system (3.1)-(3.2), for given positive constant  β, ρ1, ρ2, ε1,  η1, if there exist positive definite symmetric matrices  P,S,Q, and any matrices  G,W with appropriate dimensions, such that
(3.18)Q≤ρ1I,P≤ρ2I,(3.19)[ΣGTWTAdATQ0d(WT+P)ATQ0U̅T*-G-GT-GAdTQ000AdTQ0**-d-1e-αdQ0AdTS0000***-d-1Q00000****-d2S0000*****-S000******-d-1ε1Q00*******-d-1η1Q0********Φ]<0
holds, then for Lyapunov functional candidate
(3.20)V(x(t))=xT(t)Px(t)+∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ,
along the trajectory of system (3.1), there holds the following inequality
(3.21)V(x(t))<eβ(t-t0)V(x(t0)),
where  Σ=(A+Ad)TP+P(A+Ad)-(β-1)P.Proof.
Similarly to the proof line of Lemma3.1, we can obtain Lemma 3.2.


### 3.2. Stabilizing Controller Design

In this subsection, we will design a stabilizing controller for system (2.12) with delayed switching.In our design approach we only require the subsystems to be stable during matched period, and the subsystems are allowed to be unstable during mismatched period. Under delayed switching controlleru(t)=Kσ′(t)x(t), the corresponding closed-loop system is given by
(3.22)ẋ(t)=(Aσ(t)+Bσ(t)Kσ′(t))x(t)+Adσ(t)x(t-dσ(t)(t))+fσ(t)(x(t),t),(3.23)x(t)=φ(t),t∈[t0-d,t0].LetT+(t0,t) denote the total mismatched period during [t0,t), T-(t0,t) denote the total matched period during [t0,t), then we have the following result.Theorem 3.3.
Consider system (2.12), for given positive constants  α,  β,  ε1,  η1,  ε2,  η2,  ρ1,  ρ2,  ρ3,  ρ4, if there exist positive definite symmetric matrices  Xi,Ζi,Si,Pij,Qij,Sij, and any matrices  Yi,Gij,Wij with appropriate dimensions, such that, for  i,j∈N̲,i≠j,
(3.24)Zi≥ρ1-1I,Xi≥ρ2-1I,Qij≤ρ3I,Pij≤ρ4I,(3.25)[ΠiXiAdiZiΞi02dSiΞi0XiUiT*-2Zi-ZiZiAdiT000ZiAdiT0**-d-1e-αdZi0ZiAdiT0000***-d-1Zi00000****-d2Si0000*****-Si000******-d-1ε1Zi00*******-d-1η1Zi0********Φ1]<0,(3.26)[ΛijGijTWijTAdjΞij0d(WijT+Pij)Ξij0UjTGij-Gij-GijT-GijAdjTQij000AdjTQij0AdjTWij-GijT-d-1Qij0AdjTSij0000ΞijTQijAdj0-d-1Qij0000000SijAdj0-d2Sij0000τ(Wij+Pij)0000-Sij000ΞijT00000-d-1ε2Qij000QijAdj00000-d-1η2Qij0Uj0000000Φ2]<0
holds, then under the switching controller  u(t)=Kσ′(t)x(t),Ki=YiXi-1, and the following average dwell-time scheme:
(3.27)inft>t0T-(t0,t)T+(t0,t)≥β+λ*α-λ*,τa>τa*=ln(μ1μ2)λ*,
the corresponding closed-loop system is exponentially stable, where  0<λ*<α,μ1,μ2≥1 satisfying  Xi-1<μ1Pij,   Pij<μ2Xi-1,    Zi-1<μ1Qij,   Qij<μ2Zi-1,   Φ1=-[dρ1(1+ε1+η1)+ρ2]-1I,   Ξi=XiAiT+YiTBiT,   Πi=(Ai+Adi)Xi+Xi(Ai+Adi)T+(α+1)Xi+BiYi+YiTBiT+Ri,   Φ2=-[dρ3(1+ε2+η2)+ρ4]-1I,   Ξij=(Aj+BjYiXi-1)TQij,   Λij=(Aj+Adj)TPij+Pij(Aj+Adj)-(β-1)Pij+PijBjYiXi-1+Xi-1YiTBjTPij.Proof.
Suppose that theith subsystem is activated at the switching instant tk-1, the jth subsystem is activated at the switching instant tk, then the corresponding switchings of the controller occur at the switching instant tk-1+Δk-1 and tk+Δk, respectively.
Whent∈[tk-1+Δk-1,tk), system (3.22) can be written as
(3.28)ẋ(t)=AKix(t)+Adix(t-di(t))+fi(x(t),t),
where AKi=Ai+BiKi.
Consider Lyapunov functional candidate as follows:(3.29)Vi(x(t))=xT(t)Pix(t)+∫t-d(t)tẋT(s)e-α(t-s)Riẋ(s)ds+∫-d0∫t+θtẋT(r)e-α(t-r)Qiẋ(r)drdθ.
For given positive constantsα,ρ1,ρ2,ε1,η1, if there exist positive definite symmetric matrices Pi,S̃i,Qi,R̃i, and any matrices Gi,Wi with appropriate dimensions such that the follows matrix inequalities (3.30)-(3.31) hold, then from Lemma 3.1 we have V̇i(x(t))+αVi(x(t))<0:
(3.30)Qi≤ρ1I,Pi≤ρ2I,(3.31)[ΘiGiTWiTAdiAKiTQi0d(WiT+Pi)AKiTQi0UiT*-(1-τ)e-αdR̃i-Gi-GiT-GiAdiTQi000AdiTQi0**-d-1e-αdQi0AdiTS̃i0000***-d-1Qi00000****-d2S̃i0000*****-S̃i000******-d-1ε1Qi00*******-d-1η1Qi0********Φ1]<0,
where
(3.32)Θi=(AKi+Adi)TPi+Pi(AKi+Adi)+(α+1)Pi+R̃iΦ1=-[dρ1(1+ε1+η1)+ρ2]-1I.
Using diag{Pi-1,Qi-1,Qi-1,Qi-1,Si-1,Si-1,Qi-1,Qi-1,I} to pre- and postmultiply the left term of (3.31), and denoting Pi-1=Xi,KiPi-1=Yi,Qi-1=Zi,S̃i-1=Si, Wi=Pi, Gi=Qi, R̃i=0, we have that (3.31) is equivalent to (3.25).
Therefore, we have(3.33)Vi(x(t))<e-α(t-t0i)Vi(x(t0i)),
where t0i represents the initial value of the ith subsystem.
Whent∈[tk,tk+Δk), system (3.22) can be written as
(3.34)ẋ(t)=AKijx(t)+Adjx(t-dj(t))+fj(x(t),t),
where AKij=Aj+BjKi.
Consider Lyapunov functional candidate for system (3.34) as follows:
(3.35)Vij(x(t))=xT(t)Pijx(t)+∫-τ0∫t+θtẋT(r)eβ(t-r)Qijẋ(r)drdθ.
For given positive constants β,ρ3,ρ4,ε2,η2, if there exist positive definite symmetric matrices Pij,Sij,Qij, and any matrices Gij,Wij with appropriate dimensions such that the follows matrix inequalities (3.36)-(3.37) hold, then from Lemma 3.2 we have V̇ij(x(t))-βVij(x(t))<0:
(3.36)Qij≤ρ3I,Pij≤ρ4I,(3.37)[ΘijGijTWijTAdjAKijTQij0τ(WijT+Pij)AKijTQij0UjTGij-Gij-GijT-GijAdjTQij000AdjTQij0AdjTWij-GiT-τ-1Qij0AdjTSij0000QijAKijQijAdj0-τ-1Qij0000000SijAdj0-τ2Sij0000τ(Wij+Pij)0000-Sij000QijAKij00000-τ-1ε2Qij000QijAdj00000-τ-1η2Qij0Uj0000000Φ2]<0,
where Θij=(AKij+Adj)TPij+Pij(AKij+Adj)-(β-1)Pij,Φ2=-[dρ3(1+ε2+η2)+ρ4]-1I.
DenotingKi=YiXi-1, we know that (3.37) is equivalent to (3.26). Thus, we have
(3.38)Vij(x(t))<eβ(t-t0ij)Vij(x(t0ij)),
where t0ij represents the initial value of the jth subsystem.
Lett0,t1,…,tk denote the switching instants in [t0,t), from (3.33) and (3.38), for t≥tk+Δk, we have
(3.39)V(t)<e-α(t-tk-Δk)V(tk+Δk)<(μ1μ2)ke-α[(t-tk-Δk)+(tk-tk-1-Δk-1)+⋯+(t2-t1-Δ1)+(t1-t0-Δ0)]+β(Δk+Δk-1+⋯+Δ1+Δ0)V(t0)=(μ1μ2)ke-αT-(t0,t)+βT+(t0,t)V(t0).
By Definition 2.4, we have
(3.40)k≤N0+t-t0τa.
From (3.27), it follows that
(3.41)-T-(t0,t)λ-+T+(t0,t)λ+≤-λ*(t-t0).
Substituting (3.40) and (3.41) into (3.39), we have
(3.42)V(t)<(μ1μ2)N0+(t-t0)/τae-λ*(t-t0)V(t0)=(μ1μ2)N0e[ln(μ1μ2)/τa-λ*](t-t0)V(t0).
Thus
(3.43)∥x(t)∥<ba·(μ1μ2)N0/2e[ln(μ1μ2)/τa-λ*](t-t0)/2∥x(t0)∥h,
where  a=mini,j∈N̲,i≠j{λmin(Xi-1),λmin(Pij)},b=maxi,j∈N̲,i≠j{λmax(Xi-1)+(τ2/2)λmax(Zi-1),λmax(Pij)+(τ2/2)λmax(Qij)}.
The proof is completed.Remark 3.4.
Whenμ1=μ2=1, we haveτa*=0, which implies that switching signals can be arbitrary.


### 3.3. Robust Reliable Controller Design

Now, we are in a position to present sufficient conditions for the existence of robust reliable controller for system (2.1) with delayed switching.Theorem 3.5.
Consider system (2.1), for given positive constants  α,  β,  ε1,  η1,  ρ1,  δ1,  ε2,  η2,  ρ2,  δ2,  ρ3, δ3, ρ4, δ4, if there exist positive definite symmetric matrices  Xi,Ζi,Ri,  Pij,Qij,Sij, and any matrices  Yi,Gij,Wij with appropriate dimensions, such that, for  i,j∈N̲,i≠j,
(3.44)Zi≥ρ1-1I,Xi≥ρ2-1I,Qij≤ρ3I,Pij≤ρ4I(3.45)[ΩiϑiΦi*Ψi0**Γi]<0,(3.46)[ΩijϑijΦij*ΨijΔij**Γij]<0
hold, then under the reliable switching controller  u(t)=Kσ′(t)x(t),  Ki=YiXi-1, and the following average dwell-time scheme:
(3.47)inft>t0T-(t0,t)T+(t0,t)≥β+λ*α-λ*,τa>τa*=ln(μ1μ2)λ*,
the corresponding closed-loop system is exponentially stable,where  0<λ*<α,μ1,μ2≥1 satisfying Xi-1<μ1Pij,Pij<μ2Xi-1, Zi-1<μ1Qij,
(3.48)Qij<μ2Zi-1,Ωi=[Ωi11XiAdiZi*-2Zi-Zi**-d-1e-αdZi],Ωi11=(Ai+Adi)Xi+Xi(Ai+Adi)T+(α+1)Xi+BiYi+YiTBiT+δ1HiHiT+δ2BiJiBiT,ϑi=[XiAiT+YiTBiT02dRiXiAiT+YiTBiT0XiUiTZiAdiT000ZiAdiT00ZiAdiT0000],Φi=[Xi(E1i+E2i)T0XiE1iT+YiTE2iTXiE1iT+YiTE2iT0YiTMi0Ji1/20ZiE2iTZiE2iT000ZiE2iT000ZiE2iT0],Ψi=diag{-d-1Zi+δ1HiHiT+δ2BiJiBiT,-d2Ri+δ1HiHiT,-Ri,-d-1ε1Zi+δ1HiHiT+δ2BiJiBiT,-d-1η1Zi+δ1HiHiT,-[dρ1(1+ε1+η1)+ρ2]-1I},Γi=diag{-δ1I,-δ1I,-δ1I,-δ1I,-δ1I,-δ2I},Ωij=[Ωij1GijTWijTAdj*-Gij-GijT+2δ2E2jTE2j-Gij**-τ-1Qij+δ3E2jTE2j],Ωij1=(Aj+Adj)TPij+Pij(Aj+Adj)-(β-1)Pij+PijBjYiXi-1+Xi-1YiTBjTPij+3δ3E1jTE1j+δ4PijBjJiBjTPij,ϑij=[(Aj+BjYiXi-1)TQij0τ(WijT+Pij)(Aj+BjYiXi-1)TQij0UjTAdjTQij000AdjTQij00AdjTSij0000],Φij=[PijHj000WijTHj0000000000],Δij=[00QijHj000000SijHj00000000QijHj00QijHj00000000],Ψij=diag{-d-1Qij+δ4QijBjJiBjTPij,-d2Sij,-Sij,-d-1ε2Qij+δ4QijBjJiBjTQij,-d-1η2Qij,-[dρ3(1+ε2+η2)+ρ4]-1I},Γij=diag{-δ3I,-δ3I,-δ3I,-δ3I,-δ3I,-δ4I}.Proof.
Consider the following inequalities (3.49) and (3.50):
(3.49)Ti=[Π̂iXiÂdiZiΞ̂i02dRiΞ̂i0XiUiT*-2Zi-ZiZiÂdiT000ZiÂdiT0**-d-1e-αdZi0ZiÂdiT0000***-d-1Zi00000****-d2Ri0000*****-Ri000******-d-1ε1Zi00*******-d-1η1Zi0********Φ1]<0,(3.50)Tij=[Λ̂ijGijTWijTÂdjΞ̂ij0d(WijT+Pij)Ξ̂ij0UjT*-Gij-GijT-GijÂdjTQij000ÂdjTQij0**-d-1Qij0ÂdjTSij0000***-d-1Qij00000****-d2Sij0000*****-Sij000******-d-1ε2Qij00*******-d-1η2Qij0********Φ2]<0,
where Φ1=-[dρ1(1+ε1+η1)]-1I,Ξ̂i=XiÂiT+YiTMiBiT,Π̂i=(Âi+Âdi)Xi+Xi(Âi+Âdi)T+(α+1)Xi+BiMiYi+YiTMiBiT,Φ2=-[dρ2(1+ε2+η2)]-1I,Ξ̂ij=(Âj+BjMiYiXi-1)TQij,Λ̂ij=(Âj+Âdj)TPij+Pij(Âj+Âdj)-(β-1)Pij+PijBjMiYiXi-1+Xi-1YiTMiBjTPij.
Substituting (2.3) and (2.10) into (3.49), we can obtain
(3.51)Ti=T1i+T2i,
where
(3.52)T1i=[ΠiXiAdiZiΞi02dRiΞi0XiUiT*-2Zi-ZiZiAdiT000ZiAdiT0**-d-1e-αdZi0ZiAdiT0000***-d-1Zi00000****-d2Ri0000*****-Ri000******-d-1ε1Zi00*******-d-1η1Zi0********Φ1],(3.53)T2i=[Υi0κiϕiT00ϕiT00*00κiT000κiT0**00κiT0000***000000****00000*****0000******000*******00********0],
where Πi=(Ai+Adi)Xi+Xi(Ai+Adi)T+(α+1)Xi+BiMi0Yi+YiTMi0BiT,Υi=HiFi(E1i+E2i)Xi+Xi(E1i+E2i)TFiTHiT+BiMi0LiYi+YiTLiMi0BiT,κi=HiFiE2iZi,ϕi=HiFiE1iXi+BiMi0LiYi.
By Lemma2.7, Lemma 2.8, and Schur complement lemma, we know that (3.49) is equivalent to (3.45).  Similarly, substituting (2.3) into (3.50), we can obtain that (3.50) is equivalent to (3.46).
From Theorem3.3, we know that Theorem 3.5 holds.
The proof is completed.Remark 3.6.
In actual operation, the condition (3.27) or (3.47) is difficult to check. Let Δmax be a known positive scalar that represents the maximum delayed period, a simple condition of the average dwell time is proposed, that is to say, (3.27) or (3.47) can be reduced to the following condition:
(3.54)τa>τa*=max{ln(μ1μ2)λ*,(β+λ*α-λ*+1)Δmax},0<λ*<α.
Proof of (3.54): According to (3.27), we have
(3.55)T-(t0,t)T+(t0,t)≥β+λ*α-λ*⇒(t-t0)-T+(t0,t)T+(t0,t)≥β+λ*α-λ*⇒t-t0≥β+λ*α-λ*T+(t0,t)+T+(t0,t)⇒t-t0≥(β+λ*α-λ*+1)T+(t0,t).
On the other hand
(3.56)(β+λ*α-λ*+1)T+(t0,t)≤(β+λ*α-λ*+1)Nσ(t)(t0,t)Δmax≤(β+λ*α-λ*+1)t-t0τa*Δmax.
Obviously, if the following inequality (3.57) is satisfied, then we have that (3.55) holds
(3.57)t-t0≥(β+λ*α-λ*+1)t-t0τa*Δmax.
From (3.57), we have τa*≥((β+λ*)/(α-λ*)+1)Δmax.
The proof is completed.


## 3.1. Stability Analysis

Lemma 3.1.
Consider system (3.1)-(3.2), for given positive constant  α, ρ1, ρ2, ε1, η1, if there exist positive definite symmetric matrices  P,S,Q,R, and any matrices  G,W  with appropriate dimensions, such that
(3.4)Q≤ρ1I,P≤ρ2I,(3.5)[ΣGTWTAdATQ0d(WT+P)ATQ0U̅T*-(1-τ)e-αdR-G-GT-GAdTQ000AdTQ0**-d-1e-αdQ0AdTS0000***-d-1Q00000****-d2S0000*****-S000******-d-1ε1Q00*******-d-1η1Q0********Φ]<0,
holds, then for Lyapunov functional candidate
(3.6)V(x(t))=xT(t)Px(t)+∫t-d(t)tẋT(s)e-α(t-s)Rẋ(s)ds+∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ,
along the trajectory of system (3.1), there holds the following inequality:
(3.7)V(x(t))<e-α(t-t0)V(x(t0)),
where  Σ=(A+Ad)TP+P(A+Ad)+(α+1)P+R,Φ=-[τρ1(1+ε1+η1)+ρ2]-1I.Proof.
LetV1(x(t))=xT(t)Px(t),V2(x(t))=∫t-d(t)tẋT(s)e-α(t-s)Rẋ(s)ds,V3(x(t))=∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ.
Notice thatx(t)-x(t-d(t))=∫t-d(t)tẋ(r)dr, (3.1) can be written as
(3.8)ẋ(t)=(A+Ad)x(t)+f(x(t),t)-Ad∫t-d(t)tẋ(r)dr.
Along the trajectory of system (3.1), the time derivative of V1(x(t)) is given by
(3.9)V̇1(x(t))=2xT(t)P{(A+Ad)x(t)+f(x(t),t)}-∫t-d(t)t2xT(t)PAdẋ(r)dr.
Let a=Adẋ(r),b=Px(t), from Lemma 2.5, we can obtain
(3.10)-2xT(t)PAdẋ(r)≤ẋT(r)AdTXAdẋ(r)+2ẋT(r)AdTXMPx(t)+xT(t)P(MTX+I)X-1(XM+I)Px(t).
Substituting (3.10) into (3.9) leads to
(3.11)V̇1(x(t))≤xT(t){(A+Ad)TP+P(A+Ad)+d(t)P(MTX+I)X-1(MX+I)P}x(t)+xT(t)Pf(x(t),t)+fT(x(t),t)Px(t)+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)AdTXAdẋ(r)dr≤xT(t){(A+Ad)TP+P(A+Ad)+τP(MTX+I)X-1(XM+I)P}x(t)+2xT(t)Pf(x(t),t)+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)AdTXAdẋ(r)dr.
Differentiating V2(x(t)) and V3(x(t)) along the trajectory of system (3.1), we have
(3.12)V̇2(x(t))≤-α∫t-d(t)te-α(t-s)ẋT(s)Rẋ(s)ds+xT(t)Rx(t)-(1-τ)e-αdxT(t-d(t))Rx(t-d(t)),V̇3(x(t))=dẋT(t)Qẋ(t)-∫t-dtẋT(r)e-α(t-r)Qẋ(r)dr-α∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ≤dẋT(t)Qẋ(t)-e-αd∫t-d(t)tẋT(r)Qẋ(r)dr-α∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ=dxT(t)ATQAx(t)+2dxT(t-d(t))AdTQAx(t)+dxT(t-d(t))AdTQAdx(t-d(t))-e-αd∫t-d(t)tẋT(r)Qẋ(r)dr-α∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ+d[2xT(t)ATQf(x(t),t)+2xT(t-d(t))AdTQf(x(t),t)+fT(x(t),t)Qf(x(t),t)].
By Lemma 2.6, we have
(3.13)2xT(t)Pf(x(t))≤xT(t)Px(t)+fT(x(t))Pf(x(t)),2xT(t)ATQf(x(t),t)≤ε1-1xT(t)ATQAx(t)+ε1fT(x(t),t)Qf(x(t),t),2xT(t-d(t))AdTQf(x(t),t)≤η1-1xT(t-d(t))AdTQAdx(t-d(t))+η1fT(x(t),t)Qf(x(t),t).
Therefore
(3.14)V̇(x(t))+αV(x(t))=V̇1(x(t))+V̇2(x(t))+αV(x(t))≤xT(t){(A+Ad)TP+P(A+Ad)+dP(MTX+I)X-1(XM+I)P+(α+1)P+d(1+ε1-1)ATQA}x(t)+2dxT(t-d(t))AdTQAx(t)+d(1+η1-1)xT(t-d(t))AdTQAdx(t-d(t))+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)(AdTXAd-e-αdQ)ẋ(r)dr+d(1+ε1+η1)fT(x(t),t)×Qf(x(t),t)+fT(x(t))Pf(x(t))≤xT(t){(A+Ad)TP+P(A+Ad)+dP(MTX+I)X-1(XM+I)P+(α+1)P+d(1+ε1-1)ATQA+d(1+ε1+η1)U̅TQU̅+U̅TPU̅}x(t)+2dxT(t-d(t))AdTQAx(t)+d(1+η1-1)xT(t-d(t))AdTQAdx(t-d(t))+2xT(t)PMTXAd∫t-d(t)tẋ(r)dr+∫t-d(t)tẋT(r)(AdTXAd-e-αdQ)ẋ(r)dr.
Notice that 2xT(t-d(t))G[x(t)-x(t-d(t))-∫t-d(t)tẋ(r)dr]=0, where G is any matrix with appropriate dimension, we have
(3.15)V̇(x(t))+αV(x(t))≤1d(t)∫t-d(t)tξT(t,r)Zξ(t,r)dr,
where ξT(t,r)=[xT(t)xT(t-d)ẋT(r)],
(3.16)Z=[Ψ-U̅TΦ1-1U̅τATQAd+GTd(t)WTAdτAdTQA+GτAdTQAd-(1-τ)e-αdR-G-GT-d(t)Gd(t)AdTW-d(t)GTd(t)(τ-1AdTSAd-e-ατQ)],
where Ψ=(A+Ad)TP+P(A+Ad)+dP(MTX+I)X-1(XM+I)P+(α+1)P+R+d(1+ε1-1)ATQA.
LetW=XMP,S=τX, by Schur complement lemma, (3.5) is equivalent to the following inequality:
(3.17)[ΨdATQAd+GTWTAdU̅TdAdTQA+GdAdTQAd-(1-τ)e-αd-G-GT-G0AdTW-GTd-2AdTSAd-d-1e-ατQ0U̅00Φ]<0.
Using diag{I,I,d(t)I,I} to pre- and post- multiply the left term of (3.17), respectively, we can obtain Z<0.  Therefore, V̇(x(t))+αV(x(t))<0.
The proof is completed.Lemma 3.2.
Consider system (3.1)-(3.2), for given positive constant  β, ρ1, ρ2, ε1,  η1, if there exist positive definite symmetric matrices  P,S,Q, and any matrices  G,W with appropriate dimensions, such that
(3.18)Q≤ρ1I,P≤ρ2I,(3.19)[ΣGTWTAdATQ0d(WT+P)ATQ0U̅T*-G-GT-GAdTQ000AdTQ0**-d-1e-αdQ0AdTS0000***-d-1Q00000****-d2S0000*****-S000******-d-1ε1Q00*******-d-1η1Q0********Φ]<0
holds, then for Lyapunov functional candidate
(3.20)V(x(t))=xT(t)Px(t)+∫-d0∫t+θtẋT(r)e-α(t-r)Qẋ(r)drdθ,
along the trajectory of system (3.1), there holds the following inequality
(3.21)V(x(t))<eβ(t-t0)V(x(t0)),
where  Σ=(A+Ad)TP+P(A+Ad)-(β-1)P.Proof.
Similarly to the proof line of Lemma3.1, we can obtain Lemma 3.2.


## 3.2. Stabilizing Controller Design

In this subsection, we will design a stabilizing controller for system (2.12) with delayed switching.In our design approach we only require the subsystems to be stable during matched period, and the subsystems are allowed to be unstable during mismatched period. Under delayed switching controlleru(t)=Kσ′(t)x(t), the corresponding closed-loop system is given by
(3.22)ẋ(t)=(Aσ(t)+Bσ(t)Kσ′(t))x(t)+Adσ(t)x(t-dσ(t)(t))+fσ(t)(x(t),t),(3.23)x(t)=φ(t),t∈[t0-d,t0].LetT+(t0,t) denote the total mismatched period during [t0,t), T-(t0,t) denote the total matched period during [t0,t), then we have the following result.Theorem 3.3.
Consider system (2.12), for given positive constants  α,  β,  ε1,  η1,  ε2,  η2,  ρ1,  ρ2,  ρ3,  ρ4, if there exist positive definite symmetric matrices  Xi,Ζi,Si,Pij,Qij,Sij, and any matrices  Yi,Gij,Wij with appropriate dimensions, such that, for  i,j∈N̲,i≠j,
(3.24)Zi≥ρ1-1I,Xi≥ρ2-1I,Qij≤ρ3I,Pij≤ρ4I,(3.25)[ΠiXiAdiZiΞi02dSiΞi0XiUiT*-2Zi-ZiZiAdiT000ZiAdiT0**-d-1e-αdZi0ZiAdiT0000***-d-1Zi00000****-d2Si0000*****-Si000******-d-1ε1Zi00*******-d-1η1Zi0********Φ1]<0,(3.26)[ΛijGijTWijTAdjΞij0d(WijT+Pij)Ξij0UjTGij-Gij-GijT-GijAdjTQij000AdjTQij0AdjTWij-GijT-d-1Qij0AdjTSij0000ΞijTQijAdj0-d-1Qij0000000SijAdj0-d2Sij0000τ(Wij+Pij)0000-Sij000ΞijT00000-d-1ε2Qij000QijAdj00000-d-1η2Qij0Uj0000000Φ2]<0
holds, then under the switching controller  u(t)=Kσ′(t)x(t),Ki=YiXi-1, and the following average dwell-time scheme:
(3.27)inft>t0T-(t0,t)T+(t0,t)≥β+λ*α-λ*,τa>τa*=ln(μ1μ2)λ*,
the corresponding closed-loop system is exponentially stable, where  0<λ*<α,μ1,μ2≥1 satisfying  Xi-1<μ1Pij,   Pij<μ2Xi-1,    Zi-1<μ1Qij,   Qij<μ2Zi-1,   Φ1=-[dρ1(1+ε1+η1)+ρ2]-1I,   Ξi=XiAiT+YiTBiT,   Πi=(Ai+Adi)Xi+Xi(Ai+Adi)T+(α+1)Xi+BiYi+YiTBiT+Ri,   Φ2=-[dρ3(1+ε2+η2)+ρ4]-1I,   Ξij=(Aj+BjYiXi-1)TQij,   Λij=(Aj+Adj)TPij+Pij(Aj+Adj)-(β-1)Pij+PijBjYiXi-1+Xi-1YiTBjTPij.Proof.
Suppose that theith subsystem is activated at the switching instant tk-1, the jth subsystem is activated at the switching instant tk, then the corresponding switchings of the controller occur at the switching instant tk-1+Δk-1 and tk+Δk, respectively.
Whent∈[tk-1+Δk-1,tk), system (3.22) can be written as
(3.28)ẋ(t)=AKix(t)+Adix(t-di(t))+fi(x(t),t),
where AKi=Ai+BiKi.
Consider Lyapunov functional candidate as follows:(3.29)Vi(x(t))=xT(t)Pix(t)+∫t-d(t)tẋT(s)e-α(t-s)Riẋ(s)ds+∫-d0∫t+θtẋT(r)e-α(t-r)Qiẋ(r)drdθ.
For given positive constantsα,ρ1,ρ2,ε1,η1, if there exist positive definite symmetric matrices Pi,S̃i,Qi,R̃i, and any matrices Gi,Wi with appropriate dimensions such that the follows matrix inequalities (3.30)-(3.31) hold, then from Lemma 3.1 we have V̇i(x(t))+αVi(x(t))<0:
(3.30)Qi≤ρ1I,Pi≤ρ2I,(3.31)[ΘiGiTWiTAdiAKiTQi0d(WiT+Pi)AKiTQi0UiT*-(1-τ)e-αdR̃i-Gi-GiT-GiAdiTQi000AdiTQi0**-d-1e-αdQi0AdiTS̃i0000***-d-1Qi00000****-d2S̃i0000*****-S̃i000******-d-1ε1Qi00*******-d-1η1Qi0********Φ1]<0,
where
(3.32)Θi=(AKi+Adi)TPi+Pi(AKi+Adi)+(α+1)Pi+R̃iΦ1=-[dρ1(1+ε1+η1)+ρ2]-1I.
Using diag{Pi-1,Qi-1,Qi-1,Qi-1,Si-1,Si-1,Qi-1,Qi-1,I} to pre- and postmultiply the left term of (3.31), and denoting Pi-1=Xi,KiPi-1=Yi,Qi-1=Zi,S̃i-1=Si, Wi=Pi, Gi=Qi, R̃i=0, we have that (3.31) is equivalent to (3.25).
Therefore, we have(3.33)Vi(x(t))<e-α(t-t0i)Vi(x(t0i)),
where t0i represents the initial value of the ith subsystem.
Whent∈[tk,tk+Δk), system (3.22) can be written as
(3.34)ẋ(t)=AKijx(t)+Adjx(t-dj(t))+fj(x(t),t),
where AKij=Aj+BjKi.
Consider Lyapunov functional candidate for system (3.34) as follows:
(3.35)Vij(x(t))=xT(t)Pijx(t)+∫-τ0∫t+θtẋT(r)eβ(t-r)Qijẋ(r)drdθ.
For given positive constants β,ρ3,ρ4,ε2,η2, if there exist positive definite symmetric matrices Pij,Sij,Qij, and any matrices Gij,Wij with appropriate dimensions such that the follows matrix inequalities (3.36)-(3.37) hold, then from Lemma 3.2 we have V̇ij(x(t))-βVij(x(t))<0:
(3.36)Qij≤ρ3I,Pij≤ρ4I,(3.37)[ΘijGijTWijTAdjAKijTQij0τ(WijT+Pij)AKijTQij0UjTGij-Gij-GijT-GijAdjTQij000AdjTQij0AdjTWij-GiT-τ-1Qij0AdjTSij0000QijAKijQijAdj0-τ-1Qij0000000SijAdj0-τ2Sij0000τ(Wij+Pij)0000-Sij000QijAKij00000-τ-1ε2Qij000QijAdj00000-τ-1η2Qij0Uj0000000Φ2]<0,
where Θij=(AKij+Adj)TPij+Pij(AKij+Adj)-(β-1)Pij,Φ2=-[dρ3(1+ε2+η2)+ρ4]-1I.
DenotingKi=YiXi-1, we know that (3.37) is equivalent to (3.26). Thus, we have
(3.38)Vij(x(t))<eβ(t-t0ij)Vij(x(t0ij)),
where t0ij represents the initial value of the jth subsystem.
Lett0,t1,…,tk denote the switching instants in [t0,t), from (3.33) and (3.38), for t≥tk+Δk, we have
(3.39)V(t)<e-α(t-tk-Δk)V(tk+Δk)<(μ1μ2)ke-α[(t-tk-Δk)+(tk-tk-1-Δk-1)+⋯+(t2-t1-Δ1)+(t1-t0-Δ0)]+β(Δk+Δk-1+⋯+Δ1+Δ0)V(t0)=(μ1μ2)ke-αT-(t0,t)+βT+(t0,t)V(t0).
By Definition 2.4, we have
(3.40)k≤N0+t-t0τa.
From (3.27), it follows that
(3.41)-T-(t0,t)λ-+T+(t0,t)λ+≤-λ*(t-t0).
Substituting (3.40) and (3.41) into (3.39), we have
(3.42)V(t)<(μ1μ2)N0+(t-t0)/τae-λ*(t-t0)V(t0)=(μ1μ2)N0e[ln(μ1μ2)/τa-λ*](t-t0)V(t0).
Thus
(3.43)∥x(t)∥<ba·(μ1μ2)N0/2e[ln(μ1μ2)/τa-λ*](t-t0)/2∥x(t0)∥h,
where  a=mini,j∈N̲,i≠j{λmin(Xi-1),λmin(Pij)},b=maxi,j∈N̲,i≠j{λmax(Xi-1)+(τ2/2)λmax(Zi-1),λmax(Pij)+(τ2/2)λmax(Qij)}.
The proof is completed.Remark 3.4.
Whenμ1=μ2=1, we haveτa*=0, which implies that switching signals can be arbitrary.


## 3.3. Robust Reliable Controller Design

Now, we are in a position to present sufficient conditions for the existence of robust reliable controller for system (2.1) with delayed switching.Theorem 3.5.
Consider system (2.1), for given positive constants  α,  β,  ε1,  η1,  ρ1,  δ1,  ε2,  η2,  ρ2,  δ2,  ρ3, δ3, ρ4, δ4, if there exist positive definite symmetric matrices  Xi,Ζi,Ri,  Pij,Qij,Sij, and any matrices  Yi,Gij,Wij with appropriate dimensions, such that, for  i,j∈N̲,i≠j,
(3.44)Zi≥ρ1-1I,Xi≥ρ2-1I,Qij≤ρ3I,Pij≤ρ4I(3.45)[ΩiϑiΦi*Ψi0**Γi]<0,(3.46)[ΩijϑijΦij*ΨijΔij**Γij]<0
hold, then under the reliable switching controller  u(t)=Kσ′(t)x(t),  Ki=YiXi-1, and the following average dwell-time scheme:
(3.47)inft>t0T-(t0,t)T+(t0,t)≥β+λ*α-λ*,τa>τa*=ln(μ1μ2)λ*,
the corresponding closed-loop system is exponentially stable,where  0<λ*<α,μ1,μ2≥1 satisfying Xi-1<μ1Pij,Pij<μ2Xi-1, Zi-1<μ1Qij,
(3.48)Qij<μ2Zi-1,Ωi=[Ωi11XiAdiZi*-2Zi-Zi**-d-1e-αdZi],Ωi11=(Ai+Adi)Xi+Xi(Ai+Adi)T+(α+1)Xi+BiYi+YiTBiT+δ1HiHiT+δ2BiJiBiT,ϑi=[XiAiT+YiTBiT02dRiXiAiT+YiTBiT0XiUiTZiAdiT000ZiAdiT00ZiAdiT0000],Φi=[Xi(E1i+E2i)T0XiE1iT+YiTE2iTXiE1iT+YiTE2iT0YiTMi0Ji1/20ZiE2iTZiE2iT000ZiE2iT000ZiE2iT0],Ψi=diag{-d-1Zi+δ1HiHiT+δ2BiJiBiT,-d2Ri+δ1HiHiT,-Ri,-d-1ε1Zi+δ1HiHiT+δ2BiJiBiT,-d-1η1Zi+δ1HiHiT,-[dρ1(1+ε1+η1)+ρ2]-1I},Γi=diag{-δ1I,-δ1I,-δ1I,-δ1I,-δ1I,-δ2I},Ωij=[Ωij1GijTWijTAdj*-Gij-GijT+2δ2E2jTE2j-Gij**-τ-1Qij+δ3E2jTE2j],Ωij1=(Aj+Adj)TPij+Pij(Aj+Adj)-(β-1)Pij+PijBjYiXi-1+Xi-1YiTBjTPij+3δ3E1jTE1j+δ4PijBjJiBjTPij,ϑij=[(Aj+BjYiXi-1)TQij0τ(WijT+Pij)(Aj+BjYiXi-1)TQij0UjTAdjTQij000AdjTQij00AdjTSij0000],Φij=[PijHj000WijTHj0000000000],Δij=[00QijHj000000SijHj00000000QijHj00QijHj00000000],Ψij=diag{-d-1Qij+δ4QijBjJiBjTPij,-d2Sij,-Sij,-d-1ε2Qij+δ4QijBjJiBjTQij,-d-1η2Qij,-[dρ3(1+ε2+η2)+ρ4]-1I},Γij=diag{-δ3I,-δ3I,-δ3I,-δ3I,-δ3I,-δ4I}.Proof.
Consider the following inequalities (3.49) and (3.50):
(3.49)Ti=[Π̂iXiÂdiZiΞ̂i02dRiΞ̂i0XiUiT*-2Zi-ZiZiÂdiT000ZiÂdiT0**-d-1e-αdZi0ZiÂdiT0000***-d-1Zi00000****-d2Ri0000*****-Ri000******-d-1ε1Zi00*******-d-1η1Zi0********Φ1]<0,(3.50)Tij=[Λ̂ijGijTWijTÂdjΞ̂ij0d(WijT+Pij)Ξ̂ij0UjT*-Gij-GijT-GijÂdjTQij000ÂdjTQij0**-d-1Qij0ÂdjTSij0000***-d-1Qij00000****-d2Sij0000*****-Sij000******-d-1ε2Qij00*******-d-1η2Qij0********Φ2]<0,
where Φ1=-[dρ1(1+ε1+η1)]-1I,Ξ̂i=XiÂiT+YiTMiBiT,Π̂i=(Âi+Âdi)Xi+Xi(Âi+Âdi)T+(α+1)Xi+BiMiYi+YiTMiBiT,Φ2=-[dρ2(1+ε2+η2)]-1I,Ξ̂ij=(Âj+BjMiYiXi-1)TQij,Λ̂ij=(Âj+Âdj)TPij+Pij(Âj+Âdj)-(β-1)Pij+PijBjMiYiXi-1+Xi-1YiTMiBjTPij.
Substituting (2.3) and (2.10) into (3.49), we can obtain
(3.51)Ti=T1i+T2i,
where
(3.52)T1i=[ΠiXiAdiZiΞi02dRiΞi0XiUiT*-2Zi-ZiZiAdiT000ZiAdiT0**-d-1e-αdZi0ZiAdiT0000***-d-1Zi00000****-d2Ri0000*****-Ri000******-d-1ε1Zi00*******-d-1η1Zi0********Φ1],(3.53)T2i=[Υi0κiϕiT00ϕiT00*00κiT000κiT0**00κiT0000***000000****00000*****0000******000*******00********0],
where Πi=(Ai+Adi)Xi+Xi(Ai+Adi)T+(α+1)Xi+BiMi0Yi+YiTMi0BiT,Υi=HiFi(E1i+E2i)Xi+Xi(E1i+E2i)TFiTHiT+BiMi0LiYi+YiTLiMi0BiT,κi=HiFiE2iZi,ϕi=HiFiE1iXi+BiMi0LiYi.
By Lemma2.7, Lemma 2.8, and Schur complement lemma, we know that (3.49) is equivalent to (3.45).  Similarly, substituting (2.3) into (3.50), we can obtain that (3.50) is equivalent to (3.46).
From Theorem3.3, we know that Theorem 3.5 holds.
The proof is completed.Remark 3.6.
In actual operation, the condition (3.27) or (3.47) is difficult to check. Let Δmax be a known positive scalar that represents the maximum delayed period, a simple condition of the average dwell time is proposed, that is to say, (3.27) or (3.47) can be reduced to the following condition:
(3.54)τa>τa*=max{ln(μ1μ2)λ*,(β+λ*α-λ*+1)Δmax},0<λ*<α.
Proof of (3.54): According to (3.27), we have
(3.55)T-(t0,t)T+(t0,t)≥β+λ*α-λ*⇒(t-t0)-T+(t0,t)T+(t0,t)≥β+λ*α-λ*⇒t-t0≥β+λ*α-λ*T+(t0,t)+T+(t0,t)⇒t-t0≥(β+λ*α-λ*+1)T+(t0,t).
On the other hand
(3.56)(β+λ*α-λ*+1)T+(t0,t)≤(β+λ*α-λ*+1)Nσ(t)(t0,t)Δmax≤(β+λ*α-λ*+1)t-t0τa*Δmax.
Obviously, if the following inequality (3.57) is satisfied, then we have that (3.55) holds
(3.57)t-t0≥(β+λ*α-λ*+1)t-t0τa*Δmax.
From (3.57), we have τa*≥((β+λ*)/(α-λ*)+1)Δmax.
The proof is completed.


## 4. Conclusion

In this paper, we have investigated the robust reliable stabilization problem for switched nonlinear systems with time-varying delays and delayed switching. The average dwell-time approach is utilized for stability analysis and controller design. Our future work will focus on extending the proposed design method toH∞ performance analysis for switched nonlinear time-varying systems with delayed switching.



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*Source: 471680-2010-07-28.xml*