# Delay-Dependent Stability Analysis for Uncertain Switched Time-Delay Systems Using Average Dwell Time

Authors: Yangming Zhang; Peng Yan
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We are concerned with the stability
problem for linear discrete-time switched systems with time
delays. The problem is solved by using multiple Lyapunov
functions to develop constructive tools for the exponential
stability analysis of the switched time-delay system. Furthermore,
the uncertainties of the switched systems are also taken
into consideration. Sufficient delay-dependent conditions are
derived in terms of the average dwell time for the exponential
stability based on linear matrix inequalities (LMIs). Finally,
numerical examples are provided to illustrate the effectiveness
of the proposed method.


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## Body

## 1. Introduction

Switched systems represent dynamical systems described by a collection of differential equations with both continuous-time dynamics and discrete-time elements [1]. In recent years, hybrid and switched dynamic systems have attracted much attention because of their wide applications in control of mechanical systems, electrical systems [2, 3], networked systems [4–7], and many other fields. One of the important topics in the study of switched systems is stability analysis, and many results have been reported for linear switched system. By exploiting average dwell time, Hespanha and Morse derived some sufficient conditions for the uniform exponential stability of the switched linear systems [8]. A concise and timely survey on analysis and synthesis of switched linear system is presented in [9]. In [10], the stability of switched linear system is analyzed by using multiple Lyapunov functions and Lyapunov-Metzler inequalities. Note that these results can not be extended to switched time-delay systems due to the infinite dimensionality of time-delay systems.Most existing results in switched systems are based on finite dimensional systems free of time delays. However, time-delay phenomena are very common in most practical industrial control systems [11]. As a matter of fact, switched time-delay systems have often appeared in the mathematical models of networked systems, hereditary systems, Lotka-Volterra systems, and so on. More importantly, the controller design of time-delay systems sometimes requires switching controller when one single controller cannot meet the design requirements. Thus, it is of great importance to investigate switched systems with time delays. To investigate the time-delay problem for switched systems, some important research efforts have been conducted. Sufficient conditions for exponential stability and weighted L2-gain were developed for a class of switched systems with time-varying delays [12]. In [13], an average dwell time approach was used to analyze switched linear systems with time-varying delays. Furthermore, the literatures [14, 15] extended the average dwell time approach to switched singular time-delay systems. By using a Lyapunov functional and LMI approach, various delay-independent and delay-dependent stability results were provided for linear switched time-delay system in [16]. In [17, 18], the piecewise Lyapunov-Razumikhin functions were introduced for the stability analysis of the switched time-delay systems. It should be noticed that most of the aforementioned results do not consider the uncertainties of switched linear systems.Due to the existence of model uncertainties in real applications, it is very desirable to consider the impact of uncertainties for the switched systems [19]. To the best of our knowledge, such problems for the switched systems with both uncertainties and time delay have rarely been studied till present. In [20], some sufficient conditions for the robust stabilization of a class of uncertain switched time-delay systems were developed based on average dwell time. Using a common Lyapunov function, several sufficient delay-independent conditions for the robust stability of the uncertain linear hybrid systems with time delay were given in [21]. Nevertheless, it may be hard to construct a common Lyapunov function for all the subsystems in most of the application cases. In addition, the research for the delay-dependent stability analysis is relatively a new topic. Generally speaking, the delay-dependent stability analysis is considered less conservative than the delay-independent case [22].Motivated by the challenges discussed above, this paper considers generalized uncertain time-delay systems in a discrete domain where some sufficient delay-dependent conditions are derived by using multiple Lyapunov functions and average dwell time to guarantee global exponential stability of the closed loop systems. Compared with [21], stronger stability results are provided, that is, the exponential stability rather than the asymptotic stability.The remainder of this paper is organized as follows. In Section2, the mathematical model of the uncertain switched system with delay time is presented and some preliminaries are given. In Section 3, the stability of uncertain switched time-delay systems in the discrete-time domain is analyzed; some sufficient conditions with the dwell time for switching signal are given. In Section 4, some numerical examples are provided to illustrate the effectiveness of the results. Finally, some conclusions are drawn in Section 5.The following notations will be used throughout this paper. LetR=(-∞,+∞) and R+=[0,+∞), let Rn be the n-dimensional Euclidean space, and Z+ denotes the set of all nonnegative integers and Rn×m is the space of n×m real matrices. P∈Rn×n≥0(P∈Rn×n≤0) means that matrix P is symmetric and semipositive (seminegative) definite. P∈Rn×n>0(P∈Rn×n<0) means that matrix P is symmetric and positive (negative) definite. In denotes the n×n real identity matrix. AT denotes the transpose of a square matrix A. λmin⁡(·) and λmax⁡(·) denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively. For x=(x1,x2,…,xn)T∈Rn, the norm of x is x=∑i=1n‍xi21/2. For A∈Rn×m, the norm of A is A=λmax⁡(ATA).


## 2. Problem Definition and Preliminaries

In this section, we introduce an uncertain linear discrete-time switched system with delays of the following general form:(1)xk+1=Aσk+ΔAσkkxk+[Bσ(k)+ΔBσ(k)(k)]x(k-h)
and the initial condition
(2)xk0θ=ϕθ,θ∈-h,0,
where h≥0, k∈Z+, the state x(k)∈Rn. Let S={1,…,N}, the switching signal σ(k)=i∈S, xk0(θ)=x(k0+θ) and the set {kq} denotes switching sequence, which is assumed to be a closed discrete subset of Z+ with k0=0<k1<k2<⋯<kq<⋯, and limq→∞kq=∞. For any σ(k), Aσ(k),Bσ(k)∈Rn×n are given constant matrices, and ΔAσ(k)(k),ΔBσ(k)(k)∈Rn×n are the parameter uncertainties which satisfy the following assumptions:
(3)ΔAik=EiFkFi,ΔBik=HiFkGi,
where Ei, Fi, Hi, and Gi are given constant matrices of appropriate dimensions. The uncertain matrix F(k) is assumed to satisfy the condition FT(k)F(k)≤I.Definition 1.
The discrete-time uncertain linear switched system with time delay (1) is robustly stable if there exist a positive definite scalar function V(x(k)) for all x(k)∈Rn and a switching signal σ(k)∈S such that
(4)ΔVxk=Vxk+1-Vxk<0.Definition 2.
The induced norm of a matrixA is denoted by
(5)A=sup⁡Axx:x∈Rn,x≠0,
where A and x satisfy the inequality
(6)Ax≤Ax.Definition 3.
A switching signalσ is said to have an average dwell time τa if there exist two positive numbers N0 and τa such that
(7)N(k0,k)≤N0+k-k0τa,∀k≥k0≥0,
where N(k0,k) is the number of switches in the interval [k0,k).Definition 4 (see [11, 23]).
The switched system (1)-(2) is said to be exponentially stable if its solutions satisfy
(8)x(k)≤cλ-k-k0ϕL,∀k≥k0,
for any initial conditions (k0,ϕ), where ϕL=sup⁡k0-h≤θ≤k0⁡ϕ(θ), c>0, and λ>0 is the decay rate.Lemma 5 (see [24]).
LetM, P, and Q be given matrices such that Q>0, Q=QT. Then, the linear matrix inequality (LMI) PMMT-Q<0 holds if and only if P+MQ-1MT<0.Lemma 6 (see [25, 26]).
LetP>0, FT(k)F(k)≤I, and M, N are constant matrices. If there exists ε>0 such that εI-MTPM>0, then
(9)A+MFkNTPA+MFkN≤ATR-1A+εNTN,
where R=P-1-(1/ε)MMT.Lemma 7 (see [27]).
For a quadratic positive definiteV(x)=xTPx, there exist λmin⁡(P) and λmax⁡(P) such that
(10)λmin⁡Px2≤xTPx≤λmax⁡Px2.


## 3. Stability Analysis

In this section, we analyze the stability for uncertain switched time-delay systems, and some sufficient conditions are given. Firstly, let us consider switched time-delay systems in discrete-time domain instead.


### 3.1. Switched Systems with Time Delay

Consider the discrete-time switched system with state delays given by(11)x(k+1)=Aσ(k)x(k)+Bσ(k)x(k-h)
and the initial condition
(12)xk0θ=ϕθ,θ∈-h,0.
System (11)-(12) can be obtained from (1)-(2) as long as ΔAσ(k)(k)=0 and ΔBσ(k)(k)=0. σ(k) is the switching signal, for each k∈Z+; we get the following sets:
(13)Aσ(k)∈{A1,…,AN},Bσ(k)∈{B1,…,BN}.Obviously, the switched linear system (11)-(12) has N different subsystems, that is,
(14)xk+1=Aixk+Bixk-h,
and the initial condition
(15)xk0θ=ϕθ,θ∈-h,0,
where h>0, i∈S={1,…,N}, k∈[kq-1,kq), and q∈Z+. Clearly, there exists j∈S such that Ai and Bi are constrained to jump to Aj and Bj among their own sets, respectively, where j≠i.Considering discrete-time switched system (11)-(12), we choose the following Lyapunov function given by
(16)Vixk=xTkPixk+∑s=k-hk-1λ2(s-k)xT(s)Qix(s),
where, ∀Pi, Qi>0 and λ>0 is a given constant.Proposition 8.
For a given scalarλ>1 and the delay h>0, there exist matrices Pi>0, Qi>0, and i∈S such that the following matrix inequality
(17)Γ=λ2AiTPiAi-Pi+Qiλh+2AiTPiBi*λ2h+2BiTPiBi-Qi<0
holds; then the function Vi(x(k)) in (16) along any trajectory of system (14)-(15) guarantees the following growth estimation:
(18)Vixk≤λ-2k-k0Vixk0,k≥k0.Proof.
Applying the transformationx(k)=λ-(k-k0)ξ(k), we obtain the following system from (14)-(15):
(19)ξk+1=λAiξk+λh+1Biξk-h
and the initial condition
(20)ξk0θ=ξk0+θ=λθxk0θ,θ∈-h,0.
Choose the following Lyapunov function for system (19)-(20):
(21)Wiξk=ξTkPiξk+∑s=k-hk-1‍ξTsQiξs.
The forward difference of the Lyapunov function Wi(ξ(k)) along the trajectory of system (19)-(20) is given by
(22)ΔWiξk=Wiξk+1-Wiξk=λAiξk+λh+1Biξk-hT·PiλAiξk+λh+1Biξk-h-ξTkPiξk+ξTkQiξk-ξTk-hQiξk-h=yTΓy,
where Γ is defined in (17) and y=[ξT(k),ξT(k-h)]T. Using (17), we arrive at
(23)ΔWiξk<0
which implies
(24)Wiξk≤Wiξk0.
From (16), we have
(25)Vixk=λ-2(k-k0)ξTkPiξk+∑s=k-hk-1‍λ2(s-k)λ-2(s-k0)xTkQixk=λ-2k-k0Wiξk
and the fact that
(26)Wiξk0=Vixk0.
It follows that
(27)Vixk≤λ-2(k-k0)Vixk0,k≥k0.
The proof is completed.Theorem 9.
For given scalarsλ>1 and μ≥1 and the delay h>0, assume that there exist Pi>0 and Qi>0 such that equality (17) holds. Then, switched delay-time system (11)-(12) is exponentially stable if the following conditions hold:(A1)
the positive definite matricesPi and Qi satisfy
(28)Pj≤μPi,Qj≤μQi∀i,j∈S;(A2)
there exists1<v≤λ such that the average dwell time τa satisfies
(29)τa≥ln⁡μ2ln⁡v.Proof.
Choose the following Lyapunov function for system (11)-(12):
(30)Vσkxk=xTkPσkxk+∑s=k-hk-1‍λ2s-kxTsQσkxs.
For i∈S, using Proposition 8, if equality (17) holds, we obtain
(31)Vixk=λ-2(k-kq-1)Vixkq-1.
Using condition (A1), then we have
(32)Vσkqxkq=xTkqPσkqxkq+∑s=kq-hkq-1‍λ2(s-kq)xTsQσkqxs≤xTkqμPσk-1xkq+∑s=kq-hkq-1λ2(s-kq)xT(s)μQσ(kq-1)x(s)=μVσkq-1xkq.
By virtue of (31) and (32), it follows that
(33)Vσkxk≤λ-2(k-kq)Vσkqxkq≤λ-2(k-kq)μVσkq-1xkq≤μλ-2(k-kq)λ-2kq-kq-1·Vσkq-1xkq-1.
Iterating kq from q-1 to 0, we have
(34)Vσ(k)xk≤μNλ-2k-k0Vσk0xk0.
Applying (29), we have
(35)VNxk≤μN0e-2(ln⁡λ-ln⁡v)(k-k0)Vσk0xk0.
Thus, there exists ln⁡ρ1=ln⁡λ-ln⁡v>0 and K1=μN0 such that
(36)VNxk≤μN0e-2(k-k0)ln⁡ρ1Vσk0xk0=K1ρ1-2(k-k0)Vσk0xk0.
It follows that
(37)β1xk2≤Vσ(k)(x(k))≤K1ρ1-2(k-k0)β2ϕL2
which yields x(k)≤K1β2/β1ρ1-k-k0ϕL, where
(38)β1=min⁡i∈Sλmin⁡(Pi),β2=max⁡i∈Sλmax⁡(Pi)+hmax⁡i∈Sλmax⁡(Qi).
Hence, it is concluded from Definition 4 that switched delay-time system (11)-(12) is exponentially stable. The proof is completed.


### 3.2. Uncertain Discrete-Time System with Time Delay

Consider the following subsystem of switched system (1)-(2):
(39)xk+1=Ai+ΔAikxk+[Bi+ΔBi(k)]x(k-h)
and the initial condition
(40)xk0θ=ϕθ,θ∈-h,0,∀i∈S. Choose the following Lyapunov function for (39)-(40):
(41)Vixk=xTkPixk+∑s=k-hk-1‍λ2s-kxTsQixs,
where ∀Pi, Qi>0 and λ>1 is a given constant.To derive the exponential stability of switched time-delay system (1)-(2), we give the decay estimation of the Lyapunov function Vi(x(k)) along the trajectory (39)-(40) in the following proposition firstly.Proposition 10.
For a given scalarλ>1, ε>0, and any delay h>0, if there exist matrices Pi>0, Qi>0, and εI-EiTPiEiEiTPiHiHiTPiEiHiTPiHi>0 such that the following inequalities,
(42)Xi+YiZi-1Ωi<0,Zi>0,Ri>0,
hold, then function Vi(x(k)) in (41) along any trajectory of switched system (39)-(40) guarantees the decay estimation as follows:
(43)Vixk≤λ-2k-k0Vixk0,k≥k0,
where
(44)Ri=Pi-1-1εEiTEi+HiTHi,Xi=λ2AiTRi-1Ai+εFiTFi-Pi+Qi,Yi=λh+2AiTRi-1Bi,Ωi=λh+2BiTRi-1Ai,Zi=Qi-λ2h+1BiTRi-1Bi+εGiTGi.Proof.
∀
λ
>
1, by applying the transformation x(k)=λ-(k-k0)ξ(k), we obtain the following system from (39)-(40):
(45)xk+1=λAi+ΔAikξk+λh+1[Bi+ΔBi(k)]ξ(k-h)
and the initial condition
(46)ξk0θ=λθxk0θ,θ∈-h,0.
Choose the following Lyapunov function for switched system (45)-(46):
(47)Wiξk=ξTkPσkξk+∑s=k-hk-1‍ξTkQσkξk.
Let A¯i=λ[Ai+ΔAi(k)] and B¯i=λh+1[Bi+ΔBi(k)]. The forward difference of Lyapunov function Wi(k) along any trajectory of system (45)-(46) is given by
(48)ΔWiξk=Wi(ξ(k+1))-Wi(ξ(k))=A¯iξk+B¯iξk-hT·PiA¯iξk+B¯iξk-h-ξT(k)Piξ(k)+ξT(k)Qiξ(k)-ξT(k-h)Qiξ(k-h)=yTM1y,
where
(49)y=ξTk,ξTk-hT,M1=A¯iTPiA¯i-Pi+QiA¯iTPiB¯iB¯iTPiA¯iB¯iTPiB¯i-Qi.
Note that
(50)M1=A¯iTPiA¯i-Pi+QiA¯iTPiB¯iB¯iTPiA¯iB¯iTPiB¯i-Qi=-Pi+Qi00-Qi+A¯iTB¯iTPiA¯iB¯i.
On the other hand, by virtue of Lemma 6, we have
(51)A¯iTB¯iTPiA¯iB¯i=λAiTλh+1BiT+λFiT00λh+1GiTλAiTλh+1BiT+λFiT00λh+1GiT×FTk00FTkEiTHiTPi×λAiλh+1Bi+EiHiF(k)00F(k)×λFi00λh+1Gi≤λAiTλh+1BiTRi-1λAi,λh+1Bi+ελFiT00λh+1GiTλFi00λh+1Gi=(1,1)(1,2)(2,1)(2,2),
where
(52)Ri=Pi-1-1εEiTEi+HiTHi,1,1=λ2AiTRi-1Ai+εFiTFi,1,2=λh+2AiTRi-1Bi,2,1=λh+2BiTRi-1Ai,2,2=λ2(h+1)BiTRi-1Bi+εGiTGi.
Then,
(53)M1=A¯iTPiA¯i-Pi+QiA¯iTPiB¯iB¯iTPiA¯iB¯iTPiB¯i-Qi≤XiYiΩi-Zi.
Now taking into account (42) and using Lemma 5, we have
(54)ΔWiξk<0
which implies that
(55)Wiξk≤Wiξk0.
From (41), we have
(56)Vixk=λ-2k-k0ξTkPiξk+∑k-hk-1‍λ2s-kλ-2s-k0ξTkQiξk=λ-2(k-k0)Wi(ξ(k))
and the fact that
(57)Wiξk0=Vixk0.
It follows that
(58)Vixk≤λ-2k-k0Vixk0.
The proof is completed.Theorem 11.
For given scalarsλ>1 and μ≥1 and any delay h>0, if there exist matrices Pi>0 and Qi>0, such that inequalities (42) and the conditions
(59)Pj≤μPi,Qj≤μQi,∀i,j∈S,(60)τa>ln⁡μ2ln⁡λ*
hold, then the uncertain switched time-delay system (1)-(2) is exponentially stable and guarantees a decay rate ρ2=λ/λ*, where 1<λ*<λ and τa is the average dwell time.Proof.
Considering system (1)-(2), choose the following Lyapunov function given by
(61)Vσkxk=xTkPσkxk+∑s=k-hk-1‍λ2s-kxTsQσkxs.
By virtue of (59), we have
(62)Vσkqxkq≤μVσkq-1xkq.
On the other hand, function Vi(k) ensures the decay estimation (43) under condition (42). Hence, we have
(63)Vσkxk≤λ-2k-kqVσkqxkq≤λ-2(k-kq)μVσ(kq-1)(x(kq))≤μλ-2(k-kq)λ-2(kq-kq-1)Vσ(kq-1)(x(kq-1))≤μN(k,k0)λ-2(k-k0)Vσ(k0)(x(k0))≤μN0e((k-k0)/τa)ln⁡μλ-2(k-k0)Vσ(k0)(x(k0)).
Let K2=μN0; by virtue of (60), we have
(64)Vσkxk≤K2λ*2(k-k0)λ-2(k-k0)Vσk0(x(k0))≤K2ρ2-2(k-k0)Vσk0xk0,
where ρ2=λ/λ*>1. It follows that
(65)α1xk2≤Vσkxk≤K2ρ2-2(k-k0)α2ϕL2
which yields x(k)≤K2α2/α1ρ2-k-k0ϕL, where
(66)α1=min⁡i∈Sλmin⁡Pi,α2=max⁡i∈Sλmax⁡Pi+hmax⁡i∈Sλmax⁡Qi.
Hence, by virtue of Definition 4, the switched system (1)-(2) is exponentially stable. The proof is completed.Remark 12.
In literature [21], the common Lyapunov function was employed to derive the delay-independent conditions. From condition (59), it can be seen that the common Lyapunov function approach can be treated as a special case of Theorem 11 if and only if μ satisfies μ=1. In this sense, we get away from the common Lyapunov conditions as μ increases from 1, which indicates the conservativeness of the common Lyapunov function approach. In contrast, this paper presents the delay-dependent exponential stability conditions by constructing multiple Lyapunov functions.Remark 13.
It should be noted that [11] only considers the switched time-delay systems without the parameter uncertainties. The present paper extends the results in [11] to the uncertain switched time-delay system in discrete-time domain by constructing different Lyapunov functions and employing the concept of the average dwell time.


## 3.1. Switched Systems with Time Delay

Consider the discrete-time switched system with state delays given by(11)x(k+1)=Aσ(k)x(k)+Bσ(k)x(k-h)
and the initial condition
(12)xk0θ=ϕθ,θ∈-h,0.
System (11)-(12) can be obtained from (1)-(2) as long as ΔAσ(k)(k)=0 and ΔBσ(k)(k)=0. σ(k) is the switching signal, for each k∈Z+; we get the following sets:
(13)Aσ(k)∈{A1,…,AN},Bσ(k)∈{B1,…,BN}.Obviously, the switched linear system (11)-(12) has N different subsystems, that is,
(14)xk+1=Aixk+Bixk-h,
and the initial condition
(15)xk0θ=ϕθ,θ∈-h,0,
where h>0, i∈S={1,…,N}, k∈[kq-1,kq), and q∈Z+. Clearly, there exists j∈S such that Ai and Bi are constrained to jump to Aj and Bj among their own sets, respectively, where j≠i.Considering discrete-time switched system (11)-(12), we choose the following Lyapunov function given by
(16)Vixk=xTkPixk+∑s=k-hk-1λ2(s-k)xT(s)Qix(s),
where, ∀Pi, Qi>0 and λ>0 is a given constant.Proposition 8.
For a given scalarλ>1 and the delay h>0, there exist matrices Pi>0, Qi>0, and i∈S such that the following matrix inequality
(17)Γ=λ2AiTPiAi-Pi+Qiλh+2AiTPiBi*λ2h+2BiTPiBi-Qi<0
holds; then the function Vi(x(k)) in (16) along any trajectory of system (14)-(15) guarantees the following growth estimation:
(18)Vixk≤λ-2k-k0Vixk0,k≥k0.Proof.
Applying the transformationx(k)=λ-(k-k0)ξ(k), we obtain the following system from (14)-(15):
(19)ξk+1=λAiξk+λh+1Biξk-h
and the initial condition
(20)ξk0θ=ξk0+θ=λθxk0θ,θ∈-h,0.
Choose the following Lyapunov function for system (19)-(20):
(21)Wiξk=ξTkPiξk+∑s=k-hk-1‍ξTsQiξs.
The forward difference of the Lyapunov function Wi(ξ(k)) along the trajectory of system (19)-(20) is given by
(22)ΔWiξk=Wiξk+1-Wiξk=λAiξk+λh+1Biξk-hT·PiλAiξk+λh+1Biξk-h-ξTkPiξk+ξTkQiξk-ξTk-hQiξk-h=yTΓy,
where Γ is defined in (17) and y=[ξT(k),ξT(k-h)]T. Using (17), we arrive at
(23)ΔWiξk<0
which implies
(24)Wiξk≤Wiξk0.
From (16), we have
(25)Vixk=λ-2(k-k0)ξTkPiξk+∑s=k-hk-1‍λ2(s-k)λ-2(s-k0)xTkQixk=λ-2k-k0Wiξk
and the fact that
(26)Wiξk0=Vixk0.
It follows that
(27)Vixk≤λ-2(k-k0)Vixk0,k≥k0.
The proof is completed.Theorem 9.
For given scalarsλ>1 and μ≥1 and the delay h>0, assume that there exist Pi>0 and Qi>0 such that equality (17) holds. Then, switched delay-time system (11)-(12) is exponentially stable if the following conditions hold:(A1)
the positive definite matricesPi and Qi satisfy
(28)Pj≤μPi,Qj≤μQi∀i,j∈S;(A2)
there exists1<v≤λ such that the average dwell time τa satisfies
(29)τa≥ln⁡μ2ln⁡v.Proof.
Choose the following Lyapunov function for system (11)-(12):
(30)Vσkxk=xTkPσkxk+∑s=k-hk-1‍λ2s-kxTsQσkxs.
For i∈S, using Proposition 8, if equality (17) holds, we obtain
(31)Vixk=λ-2(k-kq-1)Vixkq-1.
Using condition (A1), then we have
(32)Vσkqxkq=xTkqPσkqxkq+∑s=kq-hkq-1‍λ2(s-kq)xTsQσkqxs≤xTkqμPσk-1xkq+∑s=kq-hkq-1λ2(s-kq)xT(s)μQσ(kq-1)x(s)=μVσkq-1xkq.
By virtue of (31) and (32), it follows that
(33)Vσkxk≤λ-2(k-kq)Vσkqxkq≤λ-2(k-kq)μVσkq-1xkq≤μλ-2(k-kq)λ-2kq-kq-1·Vσkq-1xkq-1.
Iterating kq from q-1 to 0, we have
(34)Vσ(k)xk≤μNλ-2k-k0Vσk0xk0.
Applying (29), we have
(35)VNxk≤μN0e-2(ln⁡λ-ln⁡v)(k-k0)Vσk0xk0.
Thus, there exists ln⁡ρ1=ln⁡λ-ln⁡v>0 and K1=μN0 such that
(36)VNxk≤μN0e-2(k-k0)ln⁡ρ1Vσk0xk0=K1ρ1-2(k-k0)Vσk0xk0.
It follows that
(37)β1xk2≤Vσ(k)(x(k))≤K1ρ1-2(k-k0)β2ϕL2
which yields x(k)≤K1β2/β1ρ1-k-k0ϕL, where
(38)β1=min⁡i∈Sλmin⁡(Pi),β2=max⁡i∈Sλmax⁡(Pi)+hmax⁡i∈Sλmax⁡(Qi).
Hence, it is concluded from Definition 4 that switched delay-time system (11)-(12) is exponentially stable. The proof is completed.


## 3.2. Uncertain Discrete-Time System with Time Delay

Consider the following subsystem of switched system (1)-(2):
(39)xk+1=Ai+ΔAikxk+[Bi+ΔBi(k)]x(k-h)
and the initial condition
(40)xk0θ=ϕθ,θ∈-h,0,∀i∈S. Choose the following Lyapunov function for (39)-(40):
(41)Vixk=xTkPixk+∑s=k-hk-1‍λ2s-kxTsQixs,
where ∀Pi, Qi>0 and λ>1 is a given constant.To derive the exponential stability of switched time-delay system (1)-(2), we give the decay estimation of the Lyapunov function Vi(x(k)) along the trajectory (39)-(40) in the following proposition firstly.Proposition 10.
For a given scalarλ>1, ε>0, and any delay h>0, if there exist matrices Pi>0, Qi>0, and εI-EiTPiEiEiTPiHiHiTPiEiHiTPiHi>0 such that the following inequalities,
(42)Xi+YiZi-1Ωi<0,Zi>0,Ri>0,
hold, then function Vi(x(k)) in (41) along any trajectory of switched system (39)-(40) guarantees the decay estimation as follows:
(43)Vixk≤λ-2k-k0Vixk0,k≥k0,
where
(44)Ri=Pi-1-1εEiTEi+HiTHi,Xi=λ2AiTRi-1Ai+εFiTFi-Pi+Qi,Yi=λh+2AiTRi-1Bi,Ωi=λh+2BiTRi-1Ai,Zi=Qi-λ2h+1BiTRi-1Bi+εGiTGi.Proof.
∀
λ
>
1, by applying the transformation x(k)=λ-(k-k0)ξ(k), we obtain the following system from (39)-(40):
(45)xk+1=λAi+ΔAikξk+λh+1[Bi+ΔBi(k)]ξ(k-h)
and the initial condition
(46)ξk0θ=λθxk0θ,θ∈-h,0.
Choose the following Lyapunov function for switched system (45)-(46):
(47)Wiξk=ξTkPσkξk+∑s=k-hk-1‍ξTkQσkξk.
Let A¯i=λ[Ai+ΔAi(k)] and B¯i=λh+1[Bi+ΔBi(k)]. The forward difference of Lyapunov function Wi(k) along any trajectory of system (45)-(46) is given by
(48)ΔWiξk=Wi(ξ(k+1))-Wi(ξ(k))=A¯iξk+B¯iξk-hT·PiA¯iξk+B¯iξk-h-ξT(k)Piξ(k)+ξT(k)Qiξ(k)-ξT(k-h)Qiξ(k-h)=yTM1y,
where
(49)y=ξTk,ξTk-hT,M1=A¯iTPiA¯i-Pi+QiA¯iTPiB¯iB¯iTPiA¯iB¯iTPiB¯i-Qi.
Note that
(50)M1=A¯iTPiA¯i-Pi+QiA¯iTPiB¯iB¯iTPiA¯iB¯iTPiB¯i-Qi=-Pi+Qi00-Qi+A¯iTB¯iTPiA¯iB¯i.
On the other hand, by virtue of Lemma 6, we have
(51)A¯iTB¯iTPiA¯iB¯i=λAiTλh+1BiT+λFiT00λh+1GiTλAiTλh+1BiT+λFiT00λh+1GiT×FTk00FTkEiTHiTPi×λAiλh+1Bi+EiHiF(k)00F(k)×λFi00λh+1Gi≤λAiTλh+1BiTRi-1λAi,λh+1Bi+ελFiT00λh+1GiTλFi00λh+1Gi=(1,1)(1,2)(2,1)(2,2),
where
(52)Ri=Pi-1-1εEiTEi+HiTHi,1,1=λ2AiTRi-1Ai+εFiTFi,1,2=λh+2AiTRi-1Bi,2,1=λh+2BiTRi-1Ai,2,2=λ2(h+1)BiTRi-1Bi+εGiTGi.
Then,
(53)M1=A¯iTPiA¯i-Pi+QiA¯iTPiB¯iB¯iTPiA¯iB¯iTPiB¯i-Qi≤XiYiΩi-Zi.
Now taking into account (42) and using Lemma 5, we have
(54)ΔWiξk<0
which implies that
(55)Wiξk≤Wiξk0.
From (41), we have
(56)Vixk=λ-2k-k0ξTkPiξk+∑k-hk-1‍λ2s-kλ-2s-k0ξTkQiξk=λ-2(k-k0)Wi(ξ(k))
and the fact that
(57)Wiξk0=Vixk0.
It follows that
(58)Vixk≤λ-2k-k0Vixk0.
The proof is completed.Theorem 11.
For given scalarsλ>1 and μ≥1 and any delay h>0, if there exist matrices Pi>0 and Qi>0, such that inequalities (42) and the conditions
(59)Pj≤μPi,Qj≤μQi,∀i,j∈S,(60)τa>ln⁡μ2ln⁡λ*
hold, then the uncertain switched time-delay system (1)-(2) is exponentially stable and guarantees a decay rate ρ2=λ/λ*, where 1<λ*<λ and τa is the average dwell time.Proof.
Considering system (1)-(2), choose the following Lyapunov function given by
(61)Vσkxk=xTkPσkxk+∑s=k-hk-1‍λ2s-kxTsQσkxs.
By virtue of (59), we have
(62)Vσkqxkq≤μVσkq-1xkq.
On the other hand, function Vi(k) ensures the decay estimation (43) under condition (42). Hence, we have
(63)Vσkxk≤λ-2k-kqVσkqxkq≤λ-2(k-kq)μVσ(kq-1)(x(kq))≤μλ-2(k-kq)λ-2(kq-kq-1)Vσ(kq-1)(x(kq-1))≤μN(k,k0)λ-2(k-k0)Vσ(k0)(x(k0))≤μN0e((k-k0)/τa)ln⁡μλ-2(k-k0)Vσ(k0)(x(k0)).
Let K2=μN0; by virtue of (60), we have
(64)Vσkxk≤K2λ*2(k-k0)λ-2(k-k0)Vσk0(x(k0))≤K2ρ2-2(k-k0)Vσk0xk0,
where ρ2=λ/λ*>1. It follows that
(65)α1xk2≤Vσkxk≤K2ρ2-2(k-k0)α2ϕL2
which yields x(k)≤K2α2/α1ρ2-k-k0ϕL, where
(66)α1=min⁡i∈Sλmin⁡Pi,α2=max⁡i∈Sλmax⁡Pi+hmax⁡i∈Sλmax⁡Qi.
Hence, by virtue of Definition 4, the switched system (1)-(2) is exponentially stable. The proof is completed.Remark 12.
In literature [21], the common Lyapunov function was employed to derive the delay-independent conditions. From condition (59), it can be seen that the common Lyapunov function approach can be treated as a special case of Theorem 11 if and only if μ satisfies μ=1. In this sense, we get away from the common Lyapunov conditions as μ increases from 1, which indicates the conservativeness of the common Lyapunov function approach. In contrast, this paper presents the delay-dependent exponential stability conditions by constructing multiple Lyapunov functions.Remark 13.
It should be noted that [11] only considers the switched time-delay systems without the parameter uncertainties. The present paper extends the results in [11] to the uncertain switched time-delay system in discrete-time domain by constructing different Lyapunov functions and employing the concept of the average dwell time.


## 4. Numerical Example

In this section, we use an example to illustrate the results in Section3.Example 1.
Consider the delay-time switched system (11)-(12) given by
(67)A1=00.3-0.20.1,B1=-0.100.10,A2=0.2-0.500.8,B2=00.100.3,
where S={1,2}. Let h=1. Assuming that the average dwell time is τa=2 and μ=1.5, then we have v≥1.1067. If we take v=1.2, then we can choose λ=1.8. From equalities (17) and (28), by using LMIs toolbox, we can obtain the feasible solutions for P1, Q1 and P2, Q2 given by
(68)P1=1.77330.13390.13391.5144,Q1=1.31740.10590.10590.5855,P2=0.09290.02470.02470.0137,Q2=0.53270.05490.05490.2748.
From Theorem 9, it is concluded that the delay-time switched system (11)-(12) is exponentially stable.Example 2.
Consider the delay-time switched system (1)-(2), where i∈{1,2}, and
(69)A1=-0.6000.1,B1=-0.1000.1,A2=0.200-0.8,B2=-0.1000.3,Ei=ei100ei2,Hi=hi100hi2,Fi=fi100fi2,Gi=gi100gi2,
where
(70)e11=0.02,e12=0.04,e21=0.01,e22=0.03,h11=0.01,h12=0.03,h21=0.02,h22=0.04,f11=0.01,f12=0.02,f21=0.01,f22=0.03,g11=0.04,g12=0.05,g21=0.04,g22=0.09.
Let h=1. Assuming that the average dwell time is τa=1 and μ=1.4, then we have λ*≥1.1832. If we take λ*=1.2, then we can choose λ=1.32. By virtue of Theorem 11, the decay rate is ρ2=1.1. If we choose ε=0.5 and the matrices
(71)P1=0.8000.8,P2=0.2000.2,
we can verify the conditions
(72)εI-E1TP1E1E1TP1H1H1TP1E1H1TP1H1>0,εI-E2TP2E2E2TP2H2H2TP2E2H2TP2H2>0,Z1=0.1590000.5874>0,Z2=0.1493000.2016>0.
From equalities (42) and (59), by using LMIs toolbox, we can obtain the feasible solutions for Q1 and Q2 given by
(73)Q1=0.2056000.6365,Q2=0.1609000.3183,
where
(74)R1=1.2490001.2450,R2=4.9990004.9950.
The initial states are given by x0=(2,-3)T. According to Theorem 11, the state curves can be obtained by simulation (as shown in Figure 1).Figure 1
State trajectories of switched systems.


## 5. Conclusions

In this paper, we have investigated the stability for switched time-delay systems in discrete-time domain. Several sufficient conditions have been proposed by utilizing multiple Lyapunov functions with the average dwell time. On one hand, the exponential stability conditions have been derived for the switched systems in the presence of time delays. On the other hand, the exponential stability conditions have been developed for the uncertain switched linear system with time delays based on LMIs conditions. Finally, the illustrative examples have been given to verify the main theoretical results.We are currently working on the switched systems with interval time delays as well as controller synthesis methods for such systems, in the hope that switching controller design can in the long run offer a new look of synthesis of systems with large and/or time-varying delays when a single controller cannot suffice the design requirements.



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*Source: 628693-2015-03-02.xml*