Stability and stabilization of switched linear dynamic systems with time delay and uncertainties

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Abstract

This paper considers the problem of exponential stability and stabilization of switched linear time-delay systems. The system parameter uncertainties are time-varying and unknown but norm-bounded. The delay in the system states is also time-varying. By using an improved Lyapunov–Krasovskii functional, a switching rule for the exponential stability and stabilization is designed in terms of the solution of Riccati-type equations. The approach allows for computation of the bounds that characterize the exponential stability rate of the solution. Numerical examples are given to illustrate the results.

Introduction

Switching systems belong to an important class of hybrid systems, which are described by a family of differential equations together with specified rules to switch between them. A switching system can be represented by a differential equation of the formx˙(t)=fσ(t,x),t0,where {fσ(·,·):σI} is a family of functions parameterized by some index set I, which is typically a finite set, and σ(·), which depends on the system state at each time, is the switching rule/signal determining a switching sequence for the given system.

Switching systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models, such as manufacturing, communication networks, automotive engineering control, chemical processes (e.g. see [5], [7], [14] and the references therein). In the last two decades, there has been increasing interest in stability analysis and control design for switched systems (e.g. [5], [8], [13], [14], [20]). Also, during the last decades, the stability problem of uncertain linear time-delay systems and applications to control theory has attracted a lot of attention [2], [3], [4], [11], [12]. The main approach for stability analysis relies on the use of Lyapunov–Krasovskii functionals and linear matrix inequality (LMI) for constructing suitable Lyapunov–Krasovskii functionals.

Although some important results have been obtained for linear switched systems, there are few results concerning the stability of switched linear systems with time delay and uncertainties. In [15], the problem of stabilization via state feedback and/or state-based switching for switched linear systems with multiple time-varying delays without uncertainties was considered. It was proved in [15] that the switched linear delay system will be stabilizable via state feedback and/or switching if the corresponding system with zero delays has a Hurwitz stable convex combination and the delays less than an appropriate upper bound that satisfies a set of LMIs. In [16], [18], delay-dependent asymptotic stability conditions are extended to discrete-time linear switching systems with time delay. Considering switching systems composed of a finite number of linear point time-delay differential equations, it has been shown recently in [6], that the asymptotic stability may be achieved by using a common Lyapunov function method switching rule. There are some other results concerning asymptotic stability for switching linear systems with time delay, but most of them provide conditions for the asymptotic stability or stabilization of switched systems without focusing on exponential stability. The exponential stability problem was considered in [21] for switching linear systems with impulsive effects by using the matrix measure concept, and in [19] for nonholonomic chained systems with strongly nonlinear input/state driven disturbances and drifts. On the other hand, it is worth noting that the existing stability conditions for time-delay systems must be solved upon a grid of the parameter space, which results in testing a nonlinear Riccati-type equation or a finite number of LMIs. In this case, the results using finite gridding points are unreliable and the numerical complexity of the tests grows rapidly. Therefore, finding new conditions for the robust exponential stability of uncertain linear switching time-delay systems is of interest.

In this paper, we study the problem of robust exponential stability for a class of uncertain linear hybrid time-delay systems. Different from [6], [15], [19], [21], the system considered in this paper is subject to time-varying uncertainties and time-varying delay. Our objective is to derive delay-dependent conditions for the exponential stability by using an improved Lyapunov–Krasovskii functional. The conditions will be presented in terms of the solution of Riccati-type equations. Comparing with the previous results, a simple geometric design is employed to find the switching rule and our approach allows to compute simultaneously the two bounds that characterize the exponential stability rate of the solution. The result is applied to obtain new sufficient conditions for stabilization of linear uncertain control switching systems. The paper can be considered as an extension of existing results for linear switching time-delay systems.

The paper is organized as follows. Section 2 presents notations, definitions and a technical lemma required for the proof of the main results. Sufficient conditions for the exponential stability and application to stabilization together with illustrative examples are presented in Section 3. The paper ends with a conclusion followed by cited references.

Section snippets

Preliminaries

The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product ·,· and the vector norm ·; Rn×r denotes the space of all matrices of (n×r)-dimensions. AT denotes the transpose of A; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax(A)=max{Reλ:λλ(A)}; λmin(A)=min{Reλ:λλ(A)}; A matrix A is semi-positive definite (A0) if Ax,x0, for all xRn;A is

Main results

In the sequel, for the sake of brevity, we will denote σ for the switching signal σ(·).

For given numbers β,h,μ and symmetric positive definite matrix P we setτ=(1-μ)-1,η=τe2βh+2β;Si=E0iE0iT+e2βhE1iE1iT,Q=i=1NDiTPDi,R=i=1NH1iTH1i,Li(P)=AiTP+PAi+H0iTH0i+PSiP+Q+τR+ηP,α1=λmin(P),α2=λmax(P)+hi=1Nλmax(DiTPDi)+τi=1Nλmax(H1iTH1i).

Theorem 3.1

The system (2.1) is β-exponentially stable if there exists a symmetric positive definite matrix P such that the system of matrices {Li(P)},i=1,2,,N is strictly complete.

Conclusion

This paper has proposed a switching design for the exponential stability and stabilization of uncertain linear switching time-delay systems. The stability conditions are derived in terms of the solution of Riccati-type equations. The approach allows for the use of efficient techniques for computation of the two bounds that characterize the exponential stability rate of the solution, as well as the feedback control.

Acknowledgements

The authors would like to thank Dr. Melvin Scott and the anonymous reviewers for their constructive comments. This work was supported by the National Foundation for Science and Technology Development, Viet Nam and by the Australian Research Council, Australia.

References (21)

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