# Robust Stability andH∞ Stabilization of Switched Systems with Time-Varying Delays Using Delta Operator Approach

Authors: Chen Qin; Zhengrong Xiang; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper considers the problems of the robust stability and robustH∞ controller design for time-varying delay switched systems using delta operator approach. Based on the average dwell time approach and delta operator theory, a sufficient condition of the robust exponential stability is presented by choosing an appropriate Lyapunov-Krasovskii functional candidate. Then, a state feedback controller is designed such that the resulting closed-loop system is exponentially stable with a guaranteed H∞ performance. The obtained results are formulated in the form of linear matrix inequalities (LMIs). Finally, a numerical example is provided to explicitly illustrate the feasibility and effectiveness of the proposed method.


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## Body

## 1. Introduction

Switched systems are a kind of hybrid systems consisting of a set of discrete event dynamic subsystems or continuous variable dynamic subsystems and a switching rule which defines a particular subsystem working during a certain interval of time. Switched systems have numerous applications in network control systems [1], robot control systems [2], intelligent traffic control systems [3], chemical industry control systems [4], and many other areas [5, 6]. Many important achievements on stability and stabilization of switched systems have been developed [7–10]. It was shown in the literature that the average dwell time (ADT) method is a powerful tool to deal with the stability of switched systems.The delta operator which is a novel approach with good finite word length performance under fast sampling rates has been investigated by many researchers due to their extensive applications [11–13], for instance, optimal filtering [14], signal processing [15], robust control [16], system identification [17], and so forth. As stated in [15], the standard shift operator was mostly adopted in the study of control theories for discrete-time systems. However, the dynamic response of a discrete system does not converge smoothly to its continuous counterpart when the sampling period tends to zero; namely, data are taken at high sampling rates. The delta operator method can solve the above problem. In addition, it was shown in [15] that delta operator requires smaller word length when implemented in fixed-point digital control processors than shift operator does. So far, some useful results on delta operator systems have been formulated in [18–21]. As is well known, time delay phenomena which often cause instability or undesirable performance in control systems are involved in a variety of real systems, such as chaotic systems, and hydraulic pressure systems [22]. In the past years, a mass of results on delta operator systems with time delay have appeared [23–27]. The delta operator is defined by
(1)δx(t)={dx(t)dt,T=0,(x(t+T)-x(t))T,T≠0,
where T is a sampling period. When T→0, the delta operator model will approach the continuous system before discretization and reflect a quasicontinuous performance [28].It should be noted that external disturbances are generally inevitable, and the output will be subsequently affected by disturbances in the system. Some results onH∞ control were developed by many researchers to restrain the external disturbances [29–33]. The H∞ control problem for a class of discrete systems was solved by using delta operator approach [34]. Low order sampled data H∞ control using the delta operator was reported in the literature [35]. Robust H∞ control for a class of uncertain switched systems using delta operator was investigated [36]. However, few results on the issues of robust stability and H∞ controller design for delta operator switched systems with time-varying delay are presented, which motivates the present investigations.In this paper, we concentrate our interest on investigating the stability andH∞ controller design problems for delta operator switched systems with time-varying delay. The main contributions of this paper can be summarized as follows: (1) by constructing a new Lyapunov-Krasovskii functional candidate and using the average dwell time approach, an exponential stability criterion for the considered system is proposed and (2) a state feedback controller design scheme is developed such that the corresponding closed-loop system is exponentially stable with a guaranteed H∞ performance.The remainder of the paper is organized as follows. The formulation of the considered systems and some corresponding definitions and lemmas is given in Section2. In Section 3, the exponential stability analysis and H∞ control for the underlying system are developed. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method in Section 4. Finally, concluding remarks are presented in Section 5.Notations. ∥·∥2 denotes the Euclidean norm. λmin(·) and λmax(·) denote the minimum and maximum eigenvalues of a matrix, respectively; AT means the transpose of matrix A; R denotes the set of all real numbers; Rn represents the n-dimensional real vector space; Rm×n is the set of all (m×n)-dimensional real matrices. The notation A>0(≥0)   means that the matrix A is positive (nonnegative) definite; diag{⋯} refers to the block-diagonal matrix; I is the identity matrix of appropriate dimension. l2[k0,∞) stands for the space of square summable functions on [k0,∞).


## 2. Problem Formulation

Consider the following delta operator switched system with time-varying delay:(2)δx(k)=A^σ(k)x(k)+A^dσ(k)x(k-τ(k))+Dσ(k)w(k),z(k)=Cσ(k)x(k)+Gσ(k)w(k),x(k0+θ)=ϕ(θ),θ=-τ-,-τ-+1,…,0,
where x(k)∈Rn is the state vector, z(k)∈Rl denotes the controlled output, and w(k)∈Rw represents the disturbance input belonging to l2[k0,∞). k means the time t=kT and T>0 is the sampling period; k0 is the initial instant. σ(k):[k0,∞)→N_={1,2,…,N} is the switching signal with N being the number of subsystems. τ(k) is the time-varying delay satisfying 0≤τ_≤τ(k)≤τ- for known constants τ_ and τ-. ϕ(θ) is the discrete vector-valued initial function. Ci, Di, and Gi are constant matrices with proper dimensions. A^i and A^di are uncertain real-valued matrices with appropriate dimensions and have the following form:
(3)[A^iA^di]=[AiAdi]+HiFi(k)[EaiEadi],
where Ai, Adi, Hi, Eai, and Eadi are known real constant matrices of suitable dimensions and Fi(k) is an unknown time-varying matrix which satisfies
(4)FiT(k)Fi(k)≤I.To obtain the main results, we first give some definitions and lemmas which will be essential in our later development.Definition 1 (see [36]).
Consider system (2) with w(k)=0. It is said to be exponentially stable under a switching signal σ(k) if, for the initial condition x(k0+θ)=ϕ(θ), θ=-τ-,-τ-+1,…,0, there exist constants α>0 and β>0 such that the solution x(k) satisfies
(5)∥x(k)∥≤α∥x(k0)∥ce-β(k-k0),∀k≥k0,
where ∥x(k0)∥c=sup-τ-≤k≤0∥x(k0+θ)∥.Definition 2.
For given0<α<1/T and γ>0, system (2) is said to have an H∞ performance level γ if there exists a switching signal σ(k) such that the following conditions are satisfied:(1)
system (2) is exponentially stable when w(k)=0;(2)
under the zero-initial condition, that is,ϕ(θ)=0, θ=-τ-,-τ-+1,…,-1,0, system (2) satisfies
(6)∑k=k0∞(1-Tα)(k-k0)∥z(k)∥2≤γ2∑k=k0∞∥w(k)∥2,∑k=k0∞(1-Tα)(k-k0)∥z(k)∥2∀w(k)∈l2[k0,∞).Definition 3 (see [37]).
For any switching signalσ(k) and any k2>k1≥0, let Nσ(k)(k1,k2) denote the number of switching of σ(k) over the interval [k1,k2). For given τa>0 and N0≥0, if the inequality
(7)Nσ(k)(k1,k2)≤N0+k2-k1τa
holds, then the positive constant τa is called the average dwell time and N0 is called the chattering bound. As commonly used in the literature, we choose N0=0 in this paper.Lemma 4 (see [20]).
For a given matrixS=[S11S12S12TS22], where S11 and S22 are square matrices, the following conditions are equivalent:(i)
S<0;(ii)
S11<0, S22-S12TS11-1S12<0;(iii)
S22<0, S11-S12S22-1S12T<0.Lemma 5 (see [20]).
LetU, V, W, and X be real matrices of appropriate dimensions with X satisfying X=XT; then, for all VTV≤I,
(8)X+UVW+WTVTUT<0
if and only if there exists a scalar ε such that
(9)X+εUUT+ε-1WTW<0.Lemma 6 (see [28]).
For any time functionx(t) and y(t), the following equation holds:
(10)δ(x(t)y(t))=δ(x(t))y(t)+x(t)δ(y(t))+Tδ(x(t))δ(y(t)),
where T is the sampling period.The objectives of the paper are(1) to find a class of switching signal σ(k) such that system (2) is exponentially stable with a guaranteed H∞ performance and (2) to determine a class of switching signal and design a state feedback controller u(k)=Kσ(k)x(k) for the following delta operator switched system with time-varying delay:
(11)δx(k)=A^σ(k)x(k)+A^dσ(k)x(k-τ(k))+B^σ(k)u(k)+Dσ(k)w(k),z(k)=Cσ(k)x(k)+Gσ(k)w(k),x(k0+θ)=ϕ(θ),θ=-τ-,-τ-+1,…,0
such that the corresponding closed-loop system is exponentially stable with a guaranteed H∞ performance.


## 3. Main Results

### 3.1. Robust Stability Analysis

In this section, we will focus on the stability of system (2) with w(k)=0.Theorem 7.
For a given positive constant0<α<1/T, if there exist scalars εi and positive definite symmetric matrices Xi and Qi, i∈N_, with appropriate dimensions, such that
(12)[EiAdiXiTXiAiTεiHiXiEaiTXiAdiT-(1-Tα)(τ-+1)QiTXiAdiT0XiEadiTTAiXiTAdiXi-TXiεiTHi0εiHiT0εiTHiT-εiI0EaiXiEadiXi00-εiI]<0,
where Ei=AiXi+XiAiT+αXi+(1-Tα)(τ--τ_+1)Qi, then system (2) with w(k)=0 is exponentially stable for any switching signal σ(k) with the following average dwell time scheme:
(13)τa>τa*=-lnμln(1-Tα),
where μ≥1 satisfies
(14)Xi≤μXj,Qi≤μQj,∀i,j∈N_.Proof.
Choose the following Lyapunov-Krasovskii functional candidate for theith subsystem
(15)Vi(k)=Vi1(k)+Vi2(k)+Vi3(k),∀i∈N_,
where
(16)Vi1(k)=xT(k)Pix(k),Vi2(k)=T∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s),Vi3(k)=T∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)k-sxT(s)Six(s).
Taking the delta operator manipulations of Lyapunov functional candidateVi(k) along the trajectory of system (2), by Lemma 6, we have
(17)δVi1(k)=δ(xT(k)Pix(k))=δ(xT(k)Pi)x(k)+xT(k)Piδ(x(k))+Tδ(xT(k)Pi)δ(x(k))=(A^ix(k)+A^dix(k-τ(k)))TPix(k)+xT(k)Pi(A^ix(k)+A^dix(k-τ(k)))+T(A^ix(k)+A^dix(k-τ(k)))T×Pi(A^ix(k)+A^dix(k-τ(k)))=xT(k)PiA^ix(k)+xT(k)PiA^dix(k-τ(k))+xT(k)A^iTPix(k)+xT(k-τ(k))A^diTPix(k)+TxT(k)A^iTPiA^ix(k)+TxT(k)A^iTPiA^dix(k-τ(k))+TxT(k-τ(k))A^diTPiA^ix(k)+TxT(k-τ(k))A^diTPiA^dix(k-τ(k))=[x(k)x(k-τ(k))]T×[PiA^i+A^iTPi+TA^iTPiA^iPiA^di+TA^iTPiA^diA^diTPi+TA^diTPiA^iTA^diTPiA^di]×[x(k)x(k-τ(k))],(18)δVi2(k)=1T(Vi2(k+1)-Vi2(k))=1T(T∑s=k+1-τ(k+1)k+1-1(1-Tα)(k+1-s)xT(s)Six(s)=551T-T∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s))≤-Tα∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s)≤55-Tα+(1-Tα)xT(k)Six(k)≤55-Tα-(1-Tα)(τ-+1)xT(k-τ(k))Six(k-τ(k))+∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s),(19)δVi3(k)=1T(Vi3(k+1)-Vi3(k))=1T(T∑l=-τ-+1-τ_∑s=k+1+lk-1+1(1-Tα)(k+1-s)xT(s)Six(s)≤5-Tα-T∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)(k-s)xT(s)Six(s))=-Tα∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)(k-s)xT(s)Six(s)+(1-Tα)(τ¯-τ_)xT(k)Six(k)-∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s).
Combining (17)–(19), we have
(20)δVi(k)+αVi(k)=[x(k)x(k-τ(k))]T×[PiA^i+A^iTPi+TA^iTPiA^iPiA^di+TA^iTPiA^diA^diTPi+TA^diTPiA^iTA^diTPiA^di]×[x(k)x(k-τ(k))]+αPi+(1-Tα)(τ--τ_+1)xT(k)Six(k)-(1-Tα)(τ-+1)xT(k-τ(k))Six(k-τ(k))≤[x(k)x(k-τ(k))]TΩi[x(k)x(k-τ(k))],
where(21)Ωi=[PiA^i+A^iTPi+αPi+(1-Tα)×(τ--τ_+1)Si+TA^iTPiA^iPiA^di+TA^iTPiA^diA^diTPi+TA^diTPiA^iTA^diTPiA^di-(1-Tα)(τ-+1)Si].Applying Lemma 4, we can obtain that Ωi<0 is equivalent to(22)[PiA^i+A^iTPi+αPi+(1-Tα)(τ--τ_+1)SiPiA^diTA^iTA^diTPi-(1-Tα)(τ-+1)SiTA^diTTA^iTA^di-TPi-1]<0.Using diag{Pi-1Pi-1I} to pre- and post-multiply both sides of (22), respectively, we have
(23)[E-iA^diPi-1TPi-1A^iTPi-1A^diT-(1-Tα)(τ-+1)Pi-1SPi-1TPi-1A^diTTA^iPi-1TA^diPi-1-TPi-1]<0,
where E-i=A^iPi-1+Pi-1A^iT+αPi-1+(1-Tα)(τ--τ_+1)Pi-1SiPi-1.
DenoteQi=Pi-1SiPi-1 and Xi=Pi-1; then, substituting (3) into (23) and applying Lemmas 4 and 5, we obtain that (23) is equivalent to (12). Thus, from (12), we can easily obtain
(24)δVi(k)+αVi(k)≤0.
It follows from (24) that
(25)δVi(k)=Vi(k+1)-Vi(k)T≤-αVi(k),Vi(k+1)-Vi(k)≤-αiTVi(k),Vi(k+1)≤(1-αT)Vi(k).
Let k1<⋯<kq denote the switching instants of σ(k) over the interval [k0,k). Consider the following piecewise Lyapunov functional for system (2):
(26)≤5TαV(k)=Vσ(k)(k)=Vσ(kp)(k),∀k∈[kp,kp+1),p=0,1,…,q.
From (14), we obtain
(27)Vσ(kp)(kp)≤μVσ(kp-)(kp-),p=0,1,…,q.
It can be obtained from (24), (27), and Definition 3 that
(28)Vσ(k)(k)≤(1-Tα)(k-kq)Vσ(kq)(kq)≤μ(1-Tα)(k-kq)Vσ(kq-)(kq-)≤μ(1-Tα)(k-kq-1)Vσ(kq-1)(kq-1)≤μ2(1-Tα)(k-kq-1)Vσ(kq-1-)(kq-1-)≤⋯≤μNσ(k)(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)≤μ(k-k0)/τa(1-Tα)(k-k0)Vσ(k0)(k0)=(μ1/τa(1-Tα))(k-k0)Vσ(k0)(k0).
Considering the definition of Vσ(k)(k), it yields that
(29)Vσ(k)(k)≥a∥x(k)∥2,(30)Vσ(k)(k0)≤b∥x(k0)∥c2,
where
(31)a=mini∈N_λmin(Pi),b=maxi∈N_{λmax(Pi)+T(τ-2-τ-τ_+τ-)λmax(Si)},∥x(k0)∥c=sup-τ-≤θ≤0∥x(k0+θ)∥.
Combining (29) and (30), we have
(32)∥x(k)∥2≤ba(μ1/τa(1-Tα))(k-k0)∥x(k0)∥2.
Therefore, system (2) with w(k)=0 is exponentially stable under the average dwell time scheme (13).
The proof is completed.Remark 8.
Whenμ=1 in (14), which leads to Xi=Xj, Qi=Qj, ∀i,j∈N_, and τa*=0 by (13), system (2) has a common Lyapunov-Krasovskii functional and the switching signal can be arbitrary.
Whenτ(k)=0, system (2) with w(k)=0 becomes the following system:
(33)δx(k)=(A^σ(k)+A^dσ(k))x(k),z(k)=Cσ(k)x(k).
Then we have the following corollary.Corollary 9.
For a given positive constant0<α<1/T, if there exist scalars εi and positive definite symmetric matrices Xi, ∀i∈N_, of appropriate dimensions, such that(34)[(Ai+Adi)Xi+Xi(AiT+AdiT)+αXiTXi(AiT+AdiT)εiHiXi(EaiT+EadiT)T(Ai+Adi)Xi-TXiεiTHi0εiHiTεiTHiT-εiI0(Eai+Eadi)Xi00-εiI]<0,then system (33) is exponentially stable for any switching signal σ(k) with average dwell time scheme (13), where μ≥1 satisfies
(35)Xi≤μXj,∀i,j∈N_.


### 3.2.H∞ Performance Analysis

The following theorem gives sufficient conditions for the existence of anH∞ performance level for system (2).Theorem 10.
For given positive constantsγ and 0<α<1/T, if there exist scalars εi and positive definite symmetric matrices Xi and Qi, i∈N_, of appropriate dimensions, such that
(36)[ΔiAdiXiDiTXiAiTXiCiTεiHiXiEaiTXiAdiT-(1-Tα)τ-+1Qi0TXiAdiT00XiEadiTDiT0-γ2ITDiTGiT00TAiXiTAdiXiTDi-TXi0εiTHi0CiXi0Gi0-I00εiHiT00εiTHiT0-εiI0EaiXiEadiXi0000-εiI]<0,
where Δi=AiXi+XiAiT+αXi+(1-Tα)(τ--τ_+1)Qi, then system (2) is exponentially stable with an H∞ performance level γ for any switching signal σ(k) with average dwell time scheme (13), where μ≥1 satisfies (14).Proof.
Equation (12) in Theorem 7 can be directly derived from (36). Thus, system (2) is exponentially stable. We are now in a position to show the H∞ performance of system (2).
Choosing the Lyapunov-Krasovskii functional candidate (15) and following the proof line of Theorem 7, we get
(37)δVi1(k)=xT(k)Piδx(k)+(δx(k))TPix(k)+T(δx(k))TPiδx(k)=xT(k)Pi(A^ix(k)+A^dix(k-τ(k))+Diw(k))+(A^ix(k)+A^dix(k-τ(k))+Diw(k))TPix(k)+T(A^ix(k)+A^dix(k-τ(k))+Diw(k))T×Pi(A^ix(k)+A^dix(k-τ(k))+Diw(k))=[x(k)x(k-τ(k))w(k)]TΘ~i[x(k)x(k-τ(k))w(k)],
where
(38)Θ~i=[PiA^i+A^iTPi+TA^iTPiA^iPiA^di+TA^iTPiA^diPiDi+TA^iTPiDiA^diTPi+TA^diTPiA^iTA^diTPiA^diTA^diTPiDiDiTPi+TDiTPiA^iTDiTPiA^diTDiTPiDi],δVi2(k)≤-Tα∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s)+(1-Tα)xT(k)Six(k)-(1-αT)(τ-+1)xT(k-τ(k))Six(k-τ(k))+∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s),δVi3(k)=-Tα∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)(k-s)xT(s)Six(s)+(1-Tα)(τ--τ_)xT(k)Six(k)-∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s).
It follows from (37)-(38) that
(39)δVi(k)+αVi(k)+zT(k)z(k)-γ2wT(k)w(k)=δVi(k)+αVi(k)+(Cx(k)+Gw(k))T(Cx(k)+Gw(k))-γ2wT(k)w(k)=[x(k)x(k-τ(k))w(k)]TΘi[x(k)x(k-τ(k))w(k)],
where(40)Θi=[ΠiPiA^di+TA^iTPiA^diPiDi+TA^iTPiDi+CiTGiA^diTPi+TA^diTPiA^iTA^diTPiA^di-(1-Tα)τ¯+1SiTA^diTPiDiDiTPi+TDiTPiA^i+GiTCiTDiTPiA^diTDiTPiDi+GiTGi-γ2I],Πi=PiA^i+A^iTPi+TA^iTPiA^i+αPi+(1-Tα)(τ--τ_+1)Si+CiTCi.
Applying Lemma4, we can obtain that Θi<0 is equivalent to the following inequality:(41)[PiA^i+A^iTPi+αPi+(1-Tα)(τ--τ_+1)SiPiA^diPiDiTA^iTCiTA^diTPi-(1-Tα)τ-+1Si0TA^diT0DiTPi0-γ2ITDiTGiTTA^iTA^diTDi-TPi-10Ci0Gi0-I]<0.
Usingdiag{Pi-1Pi-1III} to pre- and post-multiply both sides of (41), respectively, we have
(42)[MiA^diPi-1DiTPi-1A^iTPi-1CiTPi-1A^diT-(1-Tα)τ-+1Pi-1SiPi-10TPi-1A^diT0DiT0-γ2ITDiTGiTTA^iPi-1TA^diPi-1TDi-TPi-10CiPi-10Gi0-I]<0,
where Mi=A^iPi-1+Pi-1A^iT+αPi-1+(1-Tα)(τ--τ_+1)Pi-1SiPi-1.
SetQi=Pi-1SiPi-1 and Xi=Pi-1; then, substituting (3) into (42) and applying Lemmas 4 and 5, we can obtain that (42) is equivalent to (36).
Therefore, one has, fork∈[kp,kp+1),
(43)V(k)≤(1-Tα)(k-kp)Vσ(kp)(kp)-∑s=kpk-1(1-Tα)(k-1-s)Λ(s),
where Λ(s)=T∥z(s)∥2-γ2T∥w(s)∥2.
Following the proof line of (28), we obtain
(44)Vσ(k)(k)≤μ(1-Tα)(k-kq)Vσ(kq-)(kq-)-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)≤μ(1-Tα)(k-kq-1)Vσ(kq-1)(kq-1)-μ∑s=kq-1kq-1(1-Tα)(k-1-s)Λ(s)-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)=μNσ(k)(kq-1,k)(1-Tα)(k-kq-1)Vσ(kq-1)(kq-1)-μNσ(k)(kq-1,k)∑s=kq-1kq-1(1-Tα)(k-1-s)Λ(s)-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)≤⋯≤μNσ(k)(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-μNσ(k)(k0,k)∑s=k0k1-1(1-Tα)(k-1-s)Λ(s)-μNσ(k)(k1,k)∑s=k1k2-1(1-Tα)(k-1-s)Λ(s)-⋯-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)=μNσ(k)(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-1-s)Λ(s).
Under the zero initial condition, we get(45)0≤-∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-1-s)Λ(s).
Namely,
(46)∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-s)∥z(s)∥2≤γ2∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-s)∥w(s)∥2.
Multiplying both sides of (46) by μ-Nσ(k)(k0,k) leads to
(47)∑s=k0k-1μ-Nσ(k)(k0,s)(1-Tα)(k-s)∥z(s)∥2≤γ2∑s=k0k-1μ-Nσ(k)(k0,s)(1-Tα)(k-s)∥w(s)∥2.
From Definition3 and (13), we have
(48)μ-Nσ(k)(k0,s)≤(1-Tα)s-k0.
Combining (47) and (48) leads to
(49)∑s=k0k-1(1-Tα)(s-k0)(1-Tα)(k-s)∥z(s)∥2≤γ2∑s=k0k-1(1-Tα)(k-s)∥w(s)∥2.
Then, summing both sides of (49) from k0 to ∞ leads to
(50)∑k=k0∞(1-Tα)(k-k0)∥z(k)∥2≤γ2∑k=k0∞∥w(k)∥2.
According to Definition 2, we can conclude that the theorem is true.
The proof is completed.


### 3.3.H∞ Controller Design

In this section, a state feedback controlleru(k)=Kσ(k)x(k) will be designed for system (11) such that the corresponding closed-loop system (51) is exponentially stable and satisfies an H∞ performance. Consider
(51)δx(k)=(A^σ(k)+B^σ(k)Kσ(k))x(k)+A^dσ(k)x(k-τ(k))+Dσ(k)w(k),z(k)=Cσ(k)x(k)+Eσ(k)w(k),x(k0+θ)=φ(θ),θ=-τ-,-τ-+1,…,-1,0,
where Ki, i∈N_, are the controller gains to be determined. B^i are uncertain real-valued matrices with appropriate dimensions and have the following form(52)B^i=Bi+HiFi(k)Ebi.Theorem 11.
Consider system (11). For given positive constants γ and 0<α<1/T, if there exist scalars εi, positive definite symmetric matrices Qi and Xi, and any matrices Wi, i∈N_, of appropriate dimensions, such that(53)[ΥiAdiXiDiT(AiXi+BiWi)TXiCiTεiHi(EaiXi+EbiWi)TXiAdiT-(1-Tα)τ-+1Qi0TXiAdiT00XiEadiTDiT0-γ2ITDiTGiT00T(AiXi+BiWi)TAdiXiTDi-TXi0εiTHi0CiXi0Gi0-I00εiHiT00εiTHiT0-εiI0(EaiXi+EbiWi)EadiXi0000-εiI]<0,where Υi=(AiXi+BiWi)+(AiXi+BiWi)T+αXi+(1-Tα)(τ--τ_+1)Qi, then under the state feedback controller
(54)u(k)=Kσ(k)x(k),Ki=WiXi-1
and the average dwell time scheme (13), the closed-loop system (51) is exponentially stable with a prescribed H∞ performance level γ, where μ≥1 satisfies (14).Proof.
ReplacingA^i in (36) with A^i+B^iKi, we get(55)Θi=[φiAdiXiDiTXi(Ai+BiKi)TXiCiTεiHiXi(Eai+EbiKi)TXiAdiT-(1-Tα)τ-+1Qi0TXiAdiT00XiEadiTDiT0-γ2ITDiTGiT00T(Ai+BiKi)XiTAdiXiTDi-TXi0εiTHi0CiXi0Gi0-I00εiHiT00εiTHiT0-εiI0(Eai+EbiKi)XiEadiXi0000-εiI]<0,where φi=(Ai+BiKi)Xi+Xi(Ai+BiKi)T+αXi+(1-Tα)(τ--τ_+1)Qi.
DenotingWi=KiXi, (53) is directly obtained.
The proof is completed.We are now in a position to give an algorithm for determiningKi and τa*.Algorithm 12.
Step
1. Input the system matrices.Step
2. Choose the parameters 0<α<1/T and γ>0. By solving (53), one can obtain the solutions of εi, Wi, Xi, and Qi.Step
3. By (54), with the obtained Wi and Xi, one can compute the gain matrices Ki.Step
4. Compute μ and τa* by (13)-(14).


## 3.1. Robust Stability Analysis

In this section, we will focus on the stability of system (2) with w(k)=0.Theorem 7.
For a given positive constant0<α<1/T, if there exist scalars εi and positive definite symmetric matrices Xi and Qi, i∈N_, with appropriate dimensions, such that
(12)[EiAdiXiTXiAiTεiHiXiEaiTXiAdiT-(1-Tα)(τ-+1)QiTXiAdiT0XiEadiTTAiXiTAdiXi-TXiεiTHi0εiHiT0εiTHiT-εiI0EaiXiEadiXi00-εiI]<0,
where Ei=AiXi+XiAiT+αXi+(1-Tα)(τ--τ_+1)Qi, then system (2) with w(k)=0 is exponentially stable for any switching signal σ(k) with the following average dwell time scheme:
(13)τa>τa*=-lnμln(1-Tα),
where μ≥1 satisfies
(14)Xi≤μXj,Qi≤μQj,∀i,j∈N_.Proof.
Choose the following Lyapunov-Krasovskii functional candidate for theith subsystem
(15)Vi(k)=Vi1(k)+Vi2(k)+Vi3(k),∀i∈N_,
where
(16)Vi1(k)=xT(k)Pix(k),Vi2(k)=T∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s),Vi3(k)=T∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)k-sxT(s)Six(s).
Taking the delta operator manipulations of Lyapunov functional candidateVi(k) along the trajectory of system (2), by Lemma 6, we have
(17)δVi1(k)=δ(xT(k)Pix(k))=δ(xT(k)Pi)x(k)+xT(k)Piδ(x(k))+Tδ(xT(k)Pi)δ(x(k))=(A^ix(k)+A^dix(k-τ(k)))TPix(k)+xT(k)Pi(A^ix(k)+A^dix(k-τ(k)))+T(A^ix(k)+A^dix(k-τ(k)))T×Pi(A^ix(k)+A^dix(k-τ(k)))=xT(k)PiA^ix(k)+xT(k)PiA^dix(k-τ(k))+xT(k)A^iTPix(k)+xT(k-τ(k))A^diTPix(k)+TxT(k)A^iTPiA^ix(k)+TxT(k)A^iTPiA^dix(k-τ(k))+TxT(k-τ(k))A^diTPiA^ix(k)+TxT(k-τ(k))A^diTPiA^dix(k-τ(k))=[x(k)x(k-τ(k))]T×[PiA^i+A^iTPi+TA^iTPiA^iPiA^di+TA^iTPiA^diA^diTPi+TA^diTPiA^iTA^diTPiA^di]×[x(k)x(k-τ(k))],(18)δVi2(k)=1T(Vi2(k+1)-Vi2(k))=1T(T∑s=k+1-τ(k+1)k+1-1(1-Tα)(k+1-s)xT(s)Six(s)=551T-T∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s))≤-Tα∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s)≤55-Tα+(1-Tα)xT(k)Six(k)≤55-Tα-(1-Tα)(τ-+1)xT(k-τ(k))Six(k-τ(k))+∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s),(19)δVi3(k)=1T(Vi3(k+1)-Vi3(k))=1T(T∑l=-τ-+1-τ_∑s=k+1+lk-1+1(1-Tα)(k+1-s)xT(s)Six(s)≤5-Tα-T∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)(k-s)xT(s)Six(s))=-Tα∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)(k-s)xT(s)Six(s)+(1-Tα)(τ¯-τ_)xT(k)Six(k)-∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s).
Combining (17)–(19), we have
(20)δVi(k)+αVi(k)=[x(k)x(k-τ(k))]T×[PiA^i+A^iTPi+TA^iTPiA^iPiA^di+TA^iTPiA^diA^diTPi+TA^diTPiA^iTA^diTPiA^di]×[x(k)x(k-τ(k))]+αPi+(1-Tα)(τ--τ_+1)xT(k)Six(k)-(1-Tα)(τ-+1)xT(k-τ(k))Six(k-τ(k))≤[x(k)x(k-τ(k))]TΩi[x(k)x(k-τ(k))],
where(21)Ωi=[PiA^i+A^iTPi+αPi+(1-Tα)×(τ--τ_+1)Si+TA^iTPiA^iPiA^di+TA^iTPiA^diA^diTPi+TA^diTPiA^iTA^diTPiA^di-(1-Tα)(τ-+1)Si].Applying Lemma 4, we can obtain that Ωi<0 is equivalent to(22)[PiA^i+A^iTPi+αPi+(1-Tα)(τ--τ_+1)SiPiA^diTA^iTA^diTPi-(1-Tα)(τ-+1)SiTA^diTTA^iTA^di-TPi-1]<0.Using diag{Pi-1Pi-1I} to pre- and post-multiply both sides of (22), respectively, we have
(23)[E-iA^diPi-1TPi-1A^iTPi-1A^diT-(1-Tα)(τ-+1)Pi-1SPi-1TPi-1A^diTTA^iPi-1TA^diPi-1-TPi-1]<0,
where E-i=A^iPi-1+Pi-1A^iT+αPi-1+(1-Tα)(τ--τ_+1)Pi-1SiPi-1.
DenoteQi=Pi-1SiPi-1 and Xi=Pi-1; then, substituting (3) into (23) and applying Lemmas 4 and 5, we obtain that (23) is equivalent to (12). Thus, from (12), we can easily obtain
(24)δVi(k)+αVi(k)≤0.
It follows from (24) that
(25)δVi(k)=Vi(k+1)-Vi(k)T≤-αVi(k),Vi(k+1)-Vi(k)≤-αiTVi(k),Vi(k+1)≤(1-αT)Vi(k).
Let k1<⋯<kq denote the switching instants of σ(k) over the interval [k0,k). Consider the following piecewise Lyapunov functional for system (2):
(26)≤5TαV(k)=Vσ(k)(k)=Vσ(kp)(k),∀k∈[kp,kp+1),p=0,1,…,q.
From (14), we obtain
(27)Vσ(kp)(kp)≤μVσ(kp-)(kp-),p=0,1,…,q.
It can be obtained from (24), (27), and Definition 3 that
(28)Vσ(k)(k)≤(1-Tα)(k-kq)Vσ(kq)(kq)≤μ(1-Tα)(k-kq)Vσ(kq-)(kq-)≤μ(1-Tα)(k-kq-1)Vσ(kq-1)(kq-1)≤μ2(1-Tα)(k-kq-1)Vσ(kq-1-)(kq-1-)≤⋯≤μNσ(k)(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)≤μ(k-k0)/τa(1-Tα)(k-k0)Vσ(k0)(k0)=(μ1/τa(1-Tα))(k-k0)Vσ(k0)(k0).
Considering the definition of Vσ(k)(k), it yields that
(29)Vσ(k)(k)≥a∥x(k)∥2,(30)Vσ(k)(k0)≤b∥x(k0)∥c2,
where
(31)a=mini∈N_λmin(Pi),b=maxi∈N_{λmax(Pi)+T(τ-2-τ-τ_+τ-)λmax(Si)},∥x(k0)∥c=sup-τ-≤θ≤0∥x(k0+θ)∥.
Combining (29) and (30), we have
(32)∥x(k)∥2≤ba(μ1/τa(1-Tα))(k-k0)∥x(k0)∥2.
Therefore, system (2) with w(k)=0 is exponentially stable under the average dwell time scheme (13).
The proof is completed.Remark 8.
Whenμ=1 in (14), which leads to Xi=Xj, Qi=Qj, ∀i,j∈N_, and τa*=0 by (13), system (2) has a common Lyapunov-Krasovskii functional and the switching signal can be arbitrary.
Whenτ(k)=0, system (2) with w(k)=0 becomes the following system:
(33)δx(k)=(A^σ(k)+A^dσ(k))x(k),z(k)=Cσ(k)x(k).
Then we have the following corollary.Corollary 9.
For a given positive constant0<α<1/T, if there exist scalars εi and positive definite symmetric matrices Xi, ∀i∈N_, of appropriate dimensions, such that(34)[(Ai+Adi)Xi+Xi(AiT+AdiT)+αXiTXi(AiT+AdiT)εiHiXi(EaiT+EadiT)T(Ai+Adi)Xi-TXiεiTHi0εiHiTεiTHiT-εiI0(Eai+Eadi)Xi00-εiI]<0,then system (33) is exponentially stable for any switching signal σ(k) with average dwell time scheme (13), where μ≥1 satisfies
(35)Xi≤μXj,∀i,j∈N_.


## 3.2.H∞ Performance Analysis

The following theorem gives sufficient conditions for the existence of anH∞ performance level for system (2).Theorem 10.
For given positive constantsγ and 0<α<1/T, if there exist scalars εi and positive definite symmetric matrices Xi and Qi, i∈N_, of appropriate dimensions, such that
(36)[ΔiAdiXiDiTXiAiTXiCiTεiHiXiEaiTXiAdiT-(1-Tα)τ-+1Qi0TXiAdiT00XiEadiTDiT0-γ2ITDiTGiT00TAiXiTAdiXiTDi-TXi0εiTHi0CiXi0Gi0-I00εiHiT00εiTHiT0-εiI0EaiXiEadiXi0000-εiI]<0,
where Δi=AiXi+XiAiT+αXi+(1-Tα)(τ--τ_+1)Qi, then system (2) is exponentially stable with an H∞ performance level γ for any switching signal σ(k) with average dwell time scheme (13), where μ≥1 satisfies (14).Proof.
Equation (12) in Theorem 7 can be directly derived from (36). Thus, system (2) is exponentially stable. We are now in a position to show the H∞ performance of system (2).
Choosing the Lyapunov-Krasovskii functional candidate (15) and following the proof line of Theorem 7, we get
(37)δVi1(k)=xT(k)Piδx(k)+(δx(k))TPix(k)+T(δx(k))TPiδx(k)=xT(k)Pi(A^ix(k)+A^dix(k-τ(k))+Diw(k))+(A^ix(k)+A^dix(k-τ(k))+Diw(k))TPix(k)+T(A^ix(k)+A^dix(k-τ(k))+Diw(k))T×Pi(A^ix(k)+A^dix(k-τ(k))+Diw(k))=[x(k)x(k-τ(k))w(k)]TΘ~i[x(k)x(k-τ(k))w(k)],
where
(38)Θ~i=[PiA^i+A^iTPi+TA^iTPiA^iPiA^di+TA^iTPiA^diPiDi+TA^iTPiDiA^diTPi+TA^diTPiA^iTA^diTPiA^diTA^diTPiDiDiTPi+TDiTPiA^iTDiTPiA^diTDiTPiDi],δVi2(k)≤-Tα∑s=k-τ(k)k-1(1-Tα)(k-s)xT(s)Six(s)+(1-Tα)xT(k)Six(k)-(1-αT)(τ-+1)xT(k-τ(k))Six(k-τ(k))+∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s),δVi3(k)=-Tα∑l=-τ-+1-τ_∑s=k+lk-1(1-Tα)(k-s)xT(s)Six(s)+(1-Tα)(τ--τ_)xT(k)Six(k)-∑s=k+1-τ-k-τ_(1-Tα)(k+1-s)xT(s)Six(s).
It follows from (37)-(38) that
(39)δVi(k)+αVi(k)+zT(k)z(k)-γ2wT(k)w(k)=δVi(k)+αVi(k)+(Cx(k)+Gw(k))T(Cx(k)+Gw(k))-γ2wT(k)w(k)=[x(k)x(k-τ(k))w(k)]TΘi[x(k)x(k-τ(k))w(k)],
where(40)Θi=[ΠiPiA^di+TA^iTPiA^diPiDi+TA^iTPiDi+CiTGiA^diTPi+TA^diTPiA^iTA^diTPiA^di-(1-Tα)τ¯+1SiTA^diTPiDiDiTPi+TDiTPiA^i+GiTCiTDiTPiA^diTDiTPiDi+GiTGi-γ2I],Πi=PiA^i+A^iTPi+TA^iTPiA^i+αPi+(1-Tα)(τ--τ_+1)Si+CiTCi.
Applying Lemma4, we can obtain that Θi<0 is equivalent to the following inequality:(41)[PiA^i+A^iTPi+αPi+(1-Tα)(τ--τ_+1)SiPiA^diPiDiTA^iTCiTA^diTPi-(1-Tα)τ-+1Si0TA^diT0DiTPi0-γ2ITDiTGiTTA^iTA^diTDi-TPi-10Ci0Gi0-I]<0.
Usingdiag{Pi-1Pi-1III} to pre- and post-multiply both sides of (41), respectively, we have
(42)[MiA^diPi-1DiTPi-1A^iTPi-1CiTPi-1A^diT-(1-Tα)τ-+1Pi-1SiPi-10TPi-1A^diT0DiT0-γ2ITDiTGiTTA^iPi-1TA^diPi-1TDi-TPi-10CiPi-10Gi0-I]<0,
where Mi=A^iPi-1+Pi-1A^iT+αPi-1+(1-Tα)(τ--τ_+1)Pi-1SiPi-1.
SetQi=Pi-1SiPi-1 and Xi=Pi-1; then, substituting (3) into (42) and applying Lemmas 4 and 5, we can obtain that (42) is equivalent to (36).
Therefore, one has, fork∈[kp,kp+1),
(43)V(k)≤(1-Tα)(k-kp)Vσ(kp)(kp)-∑s=kpk-1(1-Tα)(k-1-s)Λ(s),
where Λ(s)=T∥z(s)∥2-γ2T∥w(s)∥2.
Following the proof line of (28), we obtain
(44)Vσ(k)(k)≤μ(1-Tα)(k-kq)Vσ(kq-)(kq-)-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)≤μ(1-Tα)(k-kq-1)Vσ(kq-1)(kq-1)-μ∑s=kq-1kq-1(1-Tα)(k-1-s)Λ(s)-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)=μNσ(k)(kq-1,k)(1-Tα)(k-kq-1)Vσ(kq-1)(kq-1)-μNσ(k)(kq-1,k)∑s=kq-1kq-1(1-Tα)(k-1-s)Λ(s)-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)≤⋯≤μNσ(k)(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-μNσ(k)(k0,k)∑s=k0k1-1(1-Tα)(k-1-s)Λ(s)-μNσ(k)(k1,k)∑s=k1k2-1(1-Tα)(k-1-s)Λ(s)-⋯-∑s=kqk-1(1-Tα)(k-1-s)Λ(s)=μNσ(k)(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-1-s)Λ(s).
Under the zero initial condition, we get(45)0≤-∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-1-s)Λ(s).
Namely,
(46)∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-s)∥z(s)∥2≤γ2∑s=k0k-1μNσ(k)(s,k)(1-Tα)(k-s)∥w(s)∥2.
Multiplying both sides of (46) by μ-Nσ(k)(k0,k) leads to
(47)∑s=k0k-1μ-Nσ(k)(k0,s)(1-Tα)(k-s)∥z(s)∥2≤γ2∑s=k0k-1μ-Nσ(k)(k0,s)(1-Tα)(k-s)∥w(s)∥2.
From Definition3 and (13), we have
(48)μ-Nσ(k)(k0,s)≤(1-Tα)s-k0.
Combining (47) and (48) leads to
(49)∑s=k0k-1(1-Tα)(s-k0)(1-Tα)(k-s)∥z(s)∥2≤γ2∑s=k0k-1(1-Tα)(k-s)∥w(s)∥2.
Then, summing both sides of (49) from k0 to ∞ leads to
(50)∑k=k0∞(1-Tα)(k-k0)∥z(k)∥2≤γ2∑k=k0∞∥w(k)∥2.
According to Definition 2, we can conclude that the theorem is true.
The proof is completed.


## 3.3.H∞ Controller Design

In this section, a state feedback controlleru(k)=Kσ(k)x(k) will be designed for system (11) such that the corresponding closed-loop system (51) is exponentially stable and satisfies an H∞ performance. Consider
(51)δx(k)=(A^σ(k)+B^σ(k)Kσ(k))x(k)+A^dσ(k)x(k-τ(k))+Dσ(k)w(k),z(k)=Cσ(k)x(k)+Eσ(k)w(k),x(k0+θ)=φ(θ),θ=-τ-,-τ-+1,…,-1,0,
where Ki, i∈N_, are the controller gains to be determined. B^i are uncertain real-valued matrices with appropriate dimensions and have the following form(52)B^i=Bi+HiFi(k)Ebi.Theorem 11.
Consider system (11). For given positive constants γ and 0<α<1/T, if there exist scalars εi, positive definite symmetric matrices Qi and Xi, and any matrices Wi, i∈N_, of appropriate dimensions, such that(53)[ΥiAdiXiDiT(AiXi+BiWi)TXiCiTεiHi(EaiXi+EbiWi)TXiAdiT-(1-Tα)τ-+1Qi0TXiAdiT00XiEadiTDiT0-γ2ITDiTGiT00T(AiXi+BiWi)TAdiXiTDi-TXi0εiTHi0CiXi0Gi0-I00εiHiT00εiTHiT0-εiI0(EaiXi+EbiWi)EadiXi0000-εiI]<0,where Υi=(AiXi+BiWi)+(AiXi+BiWi)T+αXi+(1-Tα)(τ--τ_+1)Qi, then under the state feedback controller
(54)u(k)=Kσ(k)x(k),Ki=WiXi-1
and the average dwell time scheme (13), the closed-loop system (51) is exponentially stable with a prescribed H∞ performance level γ, where μ≥1 satisfies (14).Proof.
ReplacingA^i in (36) with A^i+B^iKi, we get(55)Θi=[φiAdiXiDiTXi(Ai+BiKi)TXiCiTεiHiXi(Eai+EbiKi)TXiAdiT-(1-Tα)τ-+1Qi0TXiAdiT00XiEadiTDiT0-γ2ITDiTGiT00T(Ai+BiKi)XiTAdiXiTDi-TXi0εiTHi0CiXi0Gi0-I00εiHiT00εiTHiT0-εiI0(Eai+EbiKi)XiEadiXi0000-εiI]<0,where φi=(Ai+BiKi)Xi+Xi(Ai+BiKi)T+αXi+(1-Tα)(τ--τ_+1)Qi.
DenotingWi=KiXi, (53) is directly obtained.
The proof is completed.We are now in a position to give an algorithm for determiningKi and τa*.Algorithm 12.
Step
1. Input the system matrices.Step
2. Choose the parameters 0<α<1/T and γ>0. By solving (53), one can obtain the solutions of εi, Wi, Xi, and Qi.Step
3. By (54), with the obtained Wi and Xi, one can compute the gain matrices Ki.Step
4. Compute μ and τa* by (13)-(14).


## 4. Numerical Example

Consider system (11) with parameters as follows:
(56)A1=[0.5-0.700.4],Ad1=[0.200-0.4],B1=[0.1200.4-1],C1=[0.2-0.18],D1=[0.010],G1=0.01,H1=[0.05-0.05],Ea1=[-0.110.03]T,Ead1=[0.03-0.1]T,Eb1=[0.02-0.01]T,A2=[1.2-1.31.2-0.8],Ad2=[00.30.10],B2=[-0.1000.3],C2=[0.1-0.18],D2=[0.02-0.01],G2=0.05,H2=[0.07-0.1],Ea2=[0.06-0.13]T,Ead2=[0.01-0.03]T,Eb2=[-0.240.01]T,F1(k)=F2(k)=sin(k).
Choosing τ-=1, τ_=0, α=0.9, γ=2, and T=0.25 and solving (53) in Theorem 11, we obtain
(57)X1=[48.993628.906328.906357.2489],Q1=[28.348216.594016.594039.3879],X2=[7.764612.210312.210335.4768],Q2=[4.35239.91879.918735.9927],
and the state feedback gain matrices are as follows:
(58)K1=[-25.260614.3786-11.31177.9375],K2=[24.9665-8.260616.0614-12.3235].According to (14), we have μ=6.5134. Then, from (13), we get τa>τa*=7.3516. Choosing τa=7.5, the simulation results are shown in Figures 1 and 2, where the initial conditions are x(0)=[-11]T, x(k)=[00]T, and  k∈[-1,0) and the exogenous disturbance input is w(k)=0.05e-0.5k. The switching signal with average dwell time τa=7.5 is shown in Figure 1 and the state responses of the corresponding closed-loop system are given in Figure 2.Figure 1
Switching signal.Figure 2
State responses of the closed-loop system.From Figures1 and 2, it is easy to see that the designed controller can guarantee that the resulting closed-loop system is exponentially stable. This demonstrates the effectiveness of the proposed method.


## 5. Conclusions

In this paper, the robust stability andH∞ controller design problems for time-varying delay switched system using delta operator approach have been investigated. By using the average dwell time approach and constructing a Lyapunov-Krasovskii functional candidate, sufficient conditions for the existence of a state feedback H∞ controller are presented. Finally, a numerical example is given to illustrate the feasibility of the proposed approach. In our future work, we will study the problem of robust H∞ filtering for delta operator switched systems with uncertainties and time-varying delays.



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*Source: 230576-2013-12-26.xml*