Finite-time stabilization for a class of switched time-delay systems under asynchronous switching

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Abstract

Switched control systems can be used to describe many practical processes with switching phenomena, such as power electronics, mechanical systems, etc. In this paper, the problem of finite-time stabilization under asynchronous switching is dealt with for a class of switched time-delay systems with nonlinear disturbances. Firstly, the nonlinear uncertainties are transformed into the linear time-varying forms via the differential mean value theorem under some assumptions. Secondly, by applying the average dwell time method and convexity principle, a finite-time stability condition for the unforced switched time-delay system is established. Finally, an asynchronous switching state feedback controller is designed which renders the considered system finite-time stable. A numerical example is provided to show the effectiveness of the developed results.

Introduction

A switched system is composed of an indexed family of subsystems described by continuous- or discrete-time dynamics and a rule orchestrating the switching between them. Switched systems have strong engineering background in various areas and are often used as a unified modeling tool for a great number of real-world systems such as power electronics, chemical processes, mechanical systems, automotive industry, aircraft and air traffic control and many other fields. Therefore, switched systems have received extensive attention from many researchers and various results are available; see, for example [1], [2], [3], [4], [5], [34], [6], [7], [8], [9], and the references therein.

Time delay is the inherent feature of many physical processes, which may degrade system performance and lead to instability. Switched systems with time delay are referred to as switched time-delay systems. Plenty of important progresses and remarkable achievements have been made concerning the analysis and synthesis problems of switched time-delay systems [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. For example, in [10], the stability analysis was conducted for a class of discrete-time switched time-delay systems. By constructing a Lyapunov functional and using the average dwell time scheme, a delay-dependent sufficient condition was derived such that the considered system is exponentially stable. The problem of robust reliable control for a class of switched nonlinear systems with time delay and actuator failures was considered in [22] under the scheme of asynchronous switching.

Generally, asymptotic stability is enough for practical applications, but there are some cases where large values of the state are not acceptable, for instance in the presence of saturations. In these cases, we need to check that those unacceptable values are not attained by the state; for these purposes the finite-time stability could exert itself. Specifically, a system is said to be finite-time stable, if, once a time interval is fixed, its state does not exceed some bound during this time interval. Dorato [23] introduced the concept of finite-time stability (FTS). In the presence of exogenous disturbances, the concept of finite-time boundedness (FTB) was introduced in [24]. In [25], the problem of finite-time state feedback stabilization for discrete-time linear system was investigated. Sufficient conditions ensuring FTS for nonlinear quadratic systems were derived in [26]. By means of the matrix inequality method, the finite-time stability or stabilization problems were considered for impulsive systems in [27], [28], [29]. For a class of linear continuous-time systems, Meng [30] introduced the definition of finite-time H control. Since then, the problems of FTS and FTB have been widely investigated for many kinds of systems. In [31], the concepts of FTS and FTB were extended to switched linear systems, and some sufficient conditions were provided such that the switched linear systems are finite-time bounded and uniformly finite-time bounded. Recently, the problems of FTB and finite-time weighted L2 gain for switched delay systems with disturbances were dealt with in [32]. Furthermore, the problem of finite-time H control for continuous-time switched linear systems with time-varying delay was considered in [33]. Recently, using the switched Lyapunov function approach proposed in [3], Xiang and Xiao [5], [34] presented solutions to the problem of finite-time H control for a class of discrete-time switched nonlinear systems.

Up to date, most previous results on finite-time stabilization for switched time-delay systems were under the synchronous switching assumption, i.e. the controller of each subsystem can match the practical system switching signal precisely. However, because the system switching signal is usually unknown, it is impractical to realize this point. In fact, it takes some time to identify the system mode and switching instant and then apply the matched controller, thus there inevitably exist asynchronous switchings between the system mode and the controller. The necessities of considering asynchronous switching for efficient controller design have been shown in many mechanical and chemical systems [35], [36]. So far, although there are some results with respect to the asynchronous switching, e.g., [22], [37], [38], [39], the asynchronous finite-time stabilization problem for nonlinear switched time-delay systems have not been investigated yet to the best of our knowledge, which motivates the present study.

The main contribution of this paper lies in that the finite-time stability results for the switched time-delay systems with nonlinear disturbances are first proposed. Then the asynchronous switching control problem is studied in the sense of finite-time stability for the considered system. The remainder of the paper is organized as follows. In Section 2, preliminaries and problem formulation are given. Section 3 is devoted to the main results of the paper. By applying the average dwell time method, a sufficient condition ensuring the finite-time stability of the switched time-delay system is first derived. Then a state feedback asynchronous switching controller is designed. In Section 4, an interesting example is provided to show the potential and the validity of the obtained results. In the end, concluding remarks are given in Section 5.

Notations. The notations in this paper are fairly standard. We use A>B (AB) to mean that A-B is positive definite (respectively, positive semi-definite). AT denotes the transpose of a matrix A, and λmax(A) (respectively, λmin(A)) represents the maximum (respectively, minimum) eigenvalue of A. Let R+ denote the set of nonnegative real numbers, Rn represent the n-dimensional Euclidean space, and Z+ stand for the set of nonnegative integers. As is commonly used in other literature, denotes the elements below the main diagonal of a symmetric matrix; max and min, respectively, stand for the maximum and minimum; sup and inf represent the supremum and infimum. Es={es(i)|es(i)=[0,,0,1,0,,0]1×sT,1 is the i-th component of es(i),i=1,,s} denotes the canonical basis of the vector space Rs for s1. The set Co(x,y)={λx+(1-λ)y:0λ1} is the convex hull of (x,y), and the symbol i,j=1q,n means i=1qj=1n. In addition, I refers to an identity matrix with an appropriate dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

Section snippets

Problem formulation and preliminaries

Consider the following switched time-delay system given byẋ(t)=Aσ(t)x(t)+Bσ(t)x(t-d)+Cσ(t)fσ(t)(x(t),x(t-d))+Dσ(t)u(t),x(t)=φ(t),t[-d,0],where σ(t):R+N̲={1,2,,N} is a right continuous piecewise constant mapping called the switching signal, x(t)Rn is the state vector, and u(t)Rm is the control input. d>0 is the time delay. Ai,Bi,Ci,Di are constant matrices of appropriate dimensions for iN̲. φ(t) is the continuous initial function. fi(·,·) are nonlinear functions.

In this paper, the

Finite-time stability analysis

Consider the switched delay system without the control input as followsẋ(t)=Aσ(t)x(t)+Bσ(t)x(t-d)+Cσ(t)fσ(t)(x(t),x(t-d)),x(t)=φ(t),t[-d,0].

In this subsection, a sufficient condition under which system (3) is finite-time stable will first be developed.

Let X(t)=x(t)x(t-d). Taking fi(0,0)=0 into account, then from Lemma 1 there exist vectorsZj(t)Co(X(t),0),forj=1,,qi,such thatfi(x(t),x(t-d))=j,k=1qi,neqi(j)enT(k)fijxk(t)(Zj(t))x(t)+j,k=1qi,neqi(j)enT(k)fijxk(t-d)(Zj(t))x(t-d).

Noticing

Numerical example

Consider system (1) with the parameters as followsA1=157.8529241.2414-217.2737-126.4228,A2=353.1349575.3646-676.7944-280.4123,B1=30.20.21.3,B2=2.80.10.41,C1=0.20.6,C2=0.10.4,D1=1111,D2=1111.The values of δ,ε,T,d, and matrix R are given byδ=1,ε=6.55,T=1,d=0.02,R=I.The nonlinearities are described asf1=sin(x1(t))-1.12sin(x2(t))+0.5sin(x1(t-d))-0.423x2(t-d),f2=0.01sin(x1(t))-0.0094x1(t-d)+0.009cos(x2(t-d))-0.009.From (49), (50), we can getv11=-1-1.12,v12=1-1.12,v13=-11.12,v14=11.12,vd11=-0.5-0.423,

Conclusions

In this paper, we have investigated the finite-time stabilization problem for a class of switched delay systems under asynchronous switching. By employing the differential mean value theorem and the average dwell time approach, a new sufficient condition ensuring the unforced system finite-time stable has been obtained. An asynchronous switching state feedback controller has also been designed. Finally, the usefulness of the obtained results has been demonstrated by a numerical example.

Acknowledgement

This work was supported in part by a research grant from the Australian Research Council, in part by the National Natural Science Foundation of China under grants Nos. 61273123, 61273152, 60904022, 60904030, and 61104136, in part by Shandong Provincial Scientific Research Reward Foundation for Excellent Young and Middle-aged Scientists of China under grant BS2010DX011, and in part by Specialized Research Fund for the Doctoral Programme of Higher Education of China under grant 20113705120003.

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