# A Novel Stability Analysis of Uncertain Switched Systems with

Authors: Ganji Huang; Shixian Luo; Linna Wei; Wuhua Chen
Journal: Journal of Control Science and Engineering
 (2015)

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## Abstract

This paper deals with the stability of
switched systems with time-varying delay. The time-varying system parameters
are assumed to be norm-bounded. Based on a novel switched time-varying Lyapunov
functional method, some new LMI-based sufficient conditions
have been obtained to ensure the exponential stability for
the uncertain switched delays systems. Finally, the proposed method is applied to a numerical example
and the simulative results are also given.


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## Body

## 1. Introduction

Switched systems are essentially a kind of hybrid systems, which are made up of several subsystems and a switching rule. There are many reality systems that can be viewed as switched systems, for instance, robotic manufacture systems [1], power systems [2], transportation systems [3], and network control [4]. In addition, switched control has shown its advantages in stabilization of many nonlinear systems compared to using static feedback control strategy. Thus the problem of stability and control design of switched systems is of theoretical and practical importance.As the foremost issue in the research of switched systems, stability analysis has gained a lot of attention and many important results have also been obtained during the several decades; see [5–9] and the references therein. Lyapunov function method plays a key role in stability analysis for switched systems. In general, there are two main methods: one is called common Lyapunov function (CLF) approach. The aim of this method is to construct a common Lyapunov function, which can ensure that the systems are stable under all possible switching signals [10–13]. Another more general technique is the so-called multiple Lyapunov function (MLF) method [14]. Different from the CLF approach, MLF method is to find one or more Lyapunov functions for each subsystem and combine them together to constitute a novel Lyapunov function. The research results have shown that MLF method can obtain less conservative results than by using CLF method [15–18].As is well known, many reality systems’ instability and poor behaviors are due in large to time delays. Switched delay systems are referred to as a class of switched systems with time delays in each subsystem, which has extensive application in engineering, such as networked control systems [19, 20] and power systems [21]. Because of their importance and widespread occurrence, stability analysis for switched systems with time delays has been a very active field in the last two decades [22–25]. It is worth noticing that there are few stability results reporting the uncertain switched system with time-varying delays. In [26], delay-dependent stability conditions and L2-gain are obtained for switched systems by using average dwell time method combining LMI techniques. In [27], by using Lyapunov-Razumikhin functions method, stability conditions involving minimum dwell time are obtained for a class of switched time delays systems. Different from [26, 27], some delay-independent LMI-based sufficient conditions, which guarantee the stability of uncertain switched systems with state delays, are derived by constructing two novel Lyapunov functions/functionals in [28]. Exponential stability of a class of switched delays systems with state jumps is investigated in [29]; two stability results are obtained based on dwell time method. Recently, by constructing a new Lyapunov-Krasovskii functional and combining with average dwell time approach, [30] investigates the stability of a class of switched nonlinear systems with disturbance and delays; some conditions are proposed to ensure the input-to-state stability for the system.In addition, note that stability of switched systems relates to the switching sequence besides subsystem [5]. This leads us to believe that an appropriate Lyapunov function/functional, which relates to the switching sequence, will be able to obtain less conservative results. However, CLF method or time-invariant MLF method is mainly used in the aforesaid results, and both methods often cannot make use of the information from the switching time. Thus, the convergence conditions based on CLF or time-invariant MLF approach may be conservative [31].Inspired by the previous discussion, we will address the problem of stability for uncertain switched delays systems in this paper. We first develop a new type of piecewise time-varying Lyapunov functional and then use the novel Lyapunov functional combining Lyapunov-Krasovskii techniques to derive the exponential stability conditions for the switched systems. It will be shown that the new approach can get better results than the exiting results, and these stability conditions can be expressed in the form of linear matrix inequalities. Finally, an example will be given to show the effectiveness of the proposed approach.


## 2. Problem Formulation

In what follows, for matrixA, we use the notation A>0 (≥,<,≤) to denote a symmetric positive-definite (positive-semidefinite, negative, and negative-semidefinite) matrix, and if not otherwise specified, all matrices are assumed to have suitable dimensions. The symbol · denotes the Euclidean norm for a vector. The symbol I denotes an identity matrix, and N is a set of positive integers.Consider the following switched system with uncertain and time delays:(1)x˙t=Aσttxt+Bσttxt-τσtt,xt0θ=φθ,θ∈-τ,0,where x(t)∈Rn is the system state vector, σ(t):[0,∞)→F≜{1,2,…,N} is the switched signal function, and we define σ(t)=kj∈F for t∈tj,tj+1, where {tj} is the jth switching time instant. The time-varying matrices Ai(t), Bi(t) express parameter uncertainties and satisfy(2)AitBit=AiBi+DiFitEiE¯i,i∈F,where Ai, Bi, Di, Ei, and E¯i are known constant matrices which characterize the structure of the ith subsystem and unknown time-varying matrices Fi(t) satisfy |Fi(t)|≤1. xt∈PC([-τ,0],R) is defined by xt(θ)=x(t+θ) for θ∈[-τ,0] and φ∈PC([-τ,0],R) is the initial function. τi(t), i∈F, denote the continuous time-varying delays of the ith subsystem. In this paper, the time-varying delays τi(t) are assumed to satisfy the following condition:(3)0≤τit=τt≤τ,τ˙t≤d<1,i∈F.As mentioned before, stability performance of the switched system often depends on the switched signal. In this paper, we will consider the following two types of switching sequence:(4)Sδ1,δ2=tj;δ1≤tj-tj-1≤δ2,j∈N,Sδ1,∞=tj;δ1≤tj-tj-1,j∈N,where constants δ1 and δ2 satisfy 0<δ1≤δ2.Definition 1.
If, for arbitrary switching signalσ(t)∈Sδ1,δ2  (orS(δ1,∞)) and any uncertainties satisfying (2), there exist constants c,λ>0 such that(5)xt≤cφe-λt-t0,∀t≥t0,then system (1) is said to be robustly uniformly exponentially stable (RUES) over S(δ1,δ2) (or S(δ1,∞)).Lemma 2 (see [32]).
For any vectorsx,y∈Rn, matrices A,P∈Rn×n, D∈Rn×nf, E,N∈Rnf×n, and F∈Rnf×nf, with P>0, F≤1, and scalar ɛ>0, the following inequalities hold:(i)
DFN+NTFTDT≤ɛ-1DDT+ɛNTN.(ii)
2xTy≤xTP-1x+yTPy.(iii)
IfP-ɛDDT>0, then (6)A+DFtETP-1A+DFtE≤ATP-ɛDDT-1A+ɛ-1ETE.The main purpose of this paper is to develop new Lyapunov functional methods to analyze the stability of switched system (1) and to derive some conditions which can ensure switched system (1) is RUES over S(δ1,δ2) (or S(δ1,∞)).


## 3. Main Results

In this section, we first introduce several piecewise differentiable switched time-related functions, followed by a definition of a novel Lyapunov functional which will be applied to establish stability conditions for switched system (1). For given switching sequence {tj}∈S(δ1,δ2), we defined two time-varying functions ρ,ρ1:[t0,∞)→R+ as follows:(7)ρt=t-tjtj+1-tj,ρ1t=1tj+1-tj,j∈N,for t∈[tj,tj+1). It is easy to know(8)ρt∈0,1,ρtj=0,ρtj-=1.And there exists function ρ2(t)∈[0,1] satisfying(9)ρ1t=1-ρ2tδ1+ρ2tδ2.Furthermore, based on ρ(t), we introduce a piecewise time-varying function as follows: (10)ρ3t=ρt-τt,t-τσtt≥t0,1,t-τσtt∈t0-τ,t0,ϕt=μσtρt-1,where scalars μi≥1, i∈F; then(11)ρ3t∈0,1,ϕt∈μσt-1,1,t≥t0.Now we can introduce the following time-varying Lyapunov functional associated with ρ(t) and ϕ(t):(12)Vt,xt=ϕtxTtPσttxt+∫t-τtte-2γt-sxTsQsxsds,where Pσ(t)(t)=ρ(t)Pσ(t),1+ρ~(t)Pσ(t),2, Q(t)=ρ(t)Q1+ρ~(t)Q2, ρ~(t)=1-ρ(t), Pi,l>0, Ql>0, i∈F, l=1,2.By applying time-varying Lyapunov functional (12) to system (1), we can obtain the following stability result.Theorem 3.
Consider switched system (1) with τ(t) satisfying condition (3) and {tj}∈S(δ1,δ2). If, for given scalars μi≥1, γ>0, i∈F, there exist matrices Pi,j>0, Qj>0, j=1,2, and scalars ɛij>0, j=1,2, such that the following matrix inequalities hold,(13)1μi1Pi1,2≤Pi2,1,∀i1,i2∈F,(14)Θijlq=Ω¯ijlPijBiPijDiPijDi∗Φ¯iq00∗0-ɛi1I0∗00-ɛi2I<0,for i∈F, j,l,q=1,2, where (15)Ω¯i,j,l=ln⁡μiδ1+2γPi,j+Pi,jAi+AiTPi,j+1δlPi,1-Pi,2+ɛi1EiTEi+μiQj,Φ¯i,q=ɛi2E¯iTE¯i-1-dexp⁡-2γτQq,then switched system (1) is RUES over S(δ1,δ2).Proof.
Letλ1=minλminPi,j;i∈F,j=1,2, λ2=maxλmaxPi,j;i∈F,j=1,2, λ3=maxλmaxQj;j=1,2, and μ=maxμi;i∈F. Thus, by the definition of V(t,x(t)), (8), and (11), we have(16)1μλ1xt2≤Vt,xt≤λ2+λ3xt2.Set V(t)=V(t,x(t)), for t∈[tj,tj+1), and, applying Lemma 2, the upper right-hand derivative of V(t) along the trajectories of system (1) is given by (17)D+Vt≤ϕtxTtln⁡μkjδ1Pkjt+ρ1tPkj,1-Pkj,2+2PkjtAkj+ɛkj1-1+ɛkj2-1PkjtDkjDkjTPkjt+ɛkj1EkjTEkj·xt+2xTtPkjtBkjxt-τkjt+2xTtPkjtBkjxt-τkjt+ɛkj2xTt-τkjtE¯kjTE¯kjxt-τkjt+xTtQtxt-2γ∫t-τkjtte-2γt-sxTsQtxsds-1-d·e-2γτxTt-τkjtQt-τkjtxt-τkjt≤-2γV2t+ϕtxTt·ln⁡μkjδ1+2γPkjt+ρ1tPkj,1-Pkj,2+2PkjtAkj+ɛkj1-1+ɛkj2-1PkjtDkjDkjTPkjt+ɛkj1EkjTEkj+μkjQtxt+2xTtPkjt·Bkjxt-τkjt+ɛkj2xTt-τkjt·E¯kjTE¯kjxt-τkjt-1-de-2γt-sxTt-τkjtQt-τkjtxt-τkjt=-2γVt+ϕtηTtΘ¯kjηt,where ηT(t)=xT(t)xT(t-τkj(t)) and (18)Θ¯kj=Ω¯kjPkjtBkj∗Φ¯kj,Ω¯kj=ln⁡μkjδ1+2γPkjt+ρ1tPkj1-Pkj2+2PkjtAkj+ɛkj1-1+ɛkj2-1PkjtDkjDkjTPkjt+ɛkj1EkjTEkj+μkjQt,Φ¯kj=ɛkj2E¯kjTE¯kj-1-dexp⁡-2γτQt-τkjt.Set ρ~2(t)=1-ρ2(t), ρ~3(t)=1-ρ3(t), and, using (9) and (14), it is easy to see that (19)ρtρ2tρ3tΘk0121+ρ~3tΘk0122+ρ~2tρ3Θk0111+ρ~3tΘk0112+ρ~t·ρ2tρ3tΘk0221+ρ~3tΘk0222+ρ~2tρ3Θk0211+ρ~3tΘk0212<0.The above inequality can be rewritten as the following form:(20)Ω~kjPkjtBkjPkjtDkjPkjtDkj∗Φ~kj00∗0-ɛkj1I0∗00-ɛkj2I<0,where (21)Ω~kj=ln⁡μiδ1+2γPkjt+PkjtAkj+AkjTPkjt+ρ1tPkj,1-Pkj,2+ɛkj1EkjTEkj+μkjQt,Φ~kj=ɛkj2E¯kjTE¯kj-1-dexp⁡-2γτQt-τkjt.Using Schur complement, we can obtain Θ¯kj<0. It follows from (17) that, for t∈[tj,tj+1), we have D+V(t)≤-2γV(t), which implies(22)Vt≤Vtj+exp⁡-2γt-tj.On the other hand, at switching time tj, applying (8), (11), and (13), we obtain(23)Vtj=ϕtjxTtjρtjPkj,1+ρ~tjPkj,2xtj+∫tj-τtjtje-2γt-sxTsQsxsds=1μkjxTtj-Pkj,2xtj-+∫tj-τtjtje-2γt-sxTsQsxsds≤xTtj-Pkj-1,1xtj-+∫tj-τtjtje-2γt-sxTsQsxsds=Vtj-.For any given t≥t0, there exists j0∈N, such that t∈[tj0,tj0+1). Combining (22) and (23) together yields(24)Vt≤Vtj0exp⁡-2γt-tj0≤Vtj0-exp⁡-2γt-tj0≤Vtj0-1exp⁡-2γt-tj0-1≤⋯≤Vt0exp⁡-2γt-t0.So by this inequality and (16), we obtain (25)xt≤μλ2+λ3λ1φexp⁡-γt-t0,t>t0,which implies that switched system (1) with τ(t) satisfying condition (3) is RUES over S(δ1,δ2).Remark 4.
The stability analysis of system (1) is performed by applying the Lyapunov functional V(t) given in (12). In V(t), the time-varying matrices Pσt(t) and Q(t) are time-varying convex combinations of two constant positive matrices, in which the combination are related to the switched time sequences. However, the traditional Lyapunov functional is usually chosen the case of V(t) when Pσ(t),1=Pσ(t),2 (i.e., Pi1=Pi2, i∈F) and Q1=Q2. Therefore, compared with the traditional Lyapunov functional, the time-varying Lyapunov functional can capture the switched characteristic of the considered system and bring less conservative results.In the case of{tj}∈S(δ1,∞), based on Theorem 3, we have the following corollary.Corollary 5.
Consider switched system (1) with τ(t) satisfying condition (3) and {tj}∈S(δ1,∞). If, for given scalars μi≥1, γ>0, i∈F, there exist matrices Pi>0, Qj>0, j=1,2, and scalars ɛij>0, j=1,2, such that the following matrix inequalities hold,(26)1μi1Pi1≤Pi2,∀i1,i2∈F,Θijq=Ω¯ijPiBiPiDiPiDi∗Φ¯iq00∗0-ɛi1I0∗00-ɛi2I<0,for i∈F, j,q=1,2, where (27)Ω¯i,j=ln⁡μiδ1+2γPi+PiAi+AiTPi+ɛi1EiTEi+μiQj,Φ¯i,q=ɛi2E¯iTE¯i-1-dexp⁡-2γτQq,then switched system (1) with τ(t) satisfying condition (3) is RUES over S(δ1,∞).Proof.
Suppose that LMIs (26) are feasible. Choosing Pi,j=Pi, i∈F, j=1,2, then one can verify that matrix inequalities (13) and (14) are also feasible.Remark 6.
In the earlier discussion, we have assumed that there are no impulsive effects at the switching points. However, similar approach can also be used in the stability analysis for impulsive switched systems.Remark 7.
We can also make a further study in the problem of stabilization for system (1) by using the method proposed in this paper. In the stabilization setting, different subsystem has different controller gain and Lyapunov matrix, so a key question is how to convert the stabilization issue to solve a set of LMIs; at this point, one can refer to the recent paper [31], where the problem of observers design for switched linear systems has been tackled.


## 4. Numerical Examples

In this section, we will give an example to show the effectiveness of our new approach.Example 1.
Consider switched system (1) with two subsystems, and the parameters are given as follows:(28)A1=01-10-1,A2=01-0.1-0.5,B1=θ10.10-0.010.05,B2=θ20.020-0.010.02,σt∈1,2.In order to compare our results with the existing ones, we calculate the lower bound δ1 on the minimum dwell time from three cases as followings.Case 1 (θi=0, Ei=E¯i=Di=0, i=1,2).
By applying Theorem3 with choice of μ1=21.6, μ2=1.01, the lower bound of the minimum dwell time δ1 that can be found is 3.13. In the results of [27, 28], it was found that the corresponding lower bound of the minimum dwell time is 3.42 and 34.8579, respectively.Case 2 (θi=1, Ei=E¯i=Di=0, i=1,2).
By applying Theorem3 and choosing μ1=4.5, μ2=1, the obtained lower bound of the minimum dwell time is δ1=12.18. By Theorem  1 in [26], the average dwell time is 14.79. It was reported in [28] that the lower bound of the minimum dwell time is δ1=14.19.Case 3 (θi=1, Di=0.1I, i=1,2, E1=E¯1=0.02I, E2=E¯2=0.08I).
We note that, for this case, the condition of [28] cannot be satisfied. However, by using Theorem 3 in this paper and choosing μ1=2.6 and μ2=1.025, the obtained lower bound of the minimum dwell time is δ1=22.
Based on the analysis above, our result is much less conservative than the results of [26–28]. For simulation studies, we consider Case 1. Choose dwell time δ1=3.14. With the initial functions x(0)=[8-5]T, the time response curves of the system are plotted in Figure 1 and the switching signal is shown in Figure 2.Figure 1
State responses of the system.Figure 2
Switching signalσ(t) with minimum dwell time δ1=3.13.


## 5. Conclusion

By introducing a new type of Lyapunov functional method, some delay-dependent sufficient conditions of stability for a class of switched systems with uncertain and state delays have been presented, and these conditions can be expressed in the form of LMI. The numerical analysis results have shown that the proposed approach can obtain better results than the earlier results.



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