Elsevier

Automatica

Volume 75, January 2017, Pages 11-15
Automatica

Technical communique
Relaxed conditions for stability of time-varying delay systems

https://doi.org/10.1016/j.automatica.2016.08.011Get rights and content

Abstract

In this paper, the problem of delay-dependent stability analysis of time-varying delay systems is investigated. Firstly, a new inequality which is the modified version of free-matrix-based integral inequality is derived, and then by aid of this new inequality, two novel lemmas which are relaxed conditions for some matrices in a Lyapunov function are proposed. Based on the lemmas, improved delay-dependent stability criteria which guarantee the asymptotic stability of the system are presented in the form of linear matrix inequality (LMI). Two numerical examples are given to describe the less conservatism of the proposed methods.

Introduction

Time delay is a natural phenomenon in real world. It is well known that the existence of time delay often causes the oscillation, deterioration of system performance, and even instability, so the stability analysis of time-delay systems strongly requires before experimental stage. As these reason, the stability analysis of time-delay system has formed a sturdy research field during the past years (Gu, Kharitonov, & Chen, 2003).

Let us consider the following time-varying delay systems: {ẋ(t)=Ax(t)+Bx(th(t)),x(t)=ϕ(t),t[h,0] where x(t)Rn is the state vector, h(t) is the time-varying delay satisfying 0h(t)h and μḣ(t)μ<1, ϕ(t) is initial function, and A, B are known real constant matrices with appropriate dimensions.

In stability problems of time-delay systems, to derive less conservative criteria guaranteeing the stability of the system (1) is a key purpose. The maximal allowable upper bound (MAUB) of time-delay is one of the important indexes to check conservatism of stability criteria in the system. Therefore, many researchers have tried to develop such conditions which ensure the stability for MAUB of time-delay as large as possible. In line with this, several remarkable approaches have been reported such as free-weighting matrix approach, delay partitioning approach, reciprocally convex approach, augmented Lyapunov method, and reduction approach for Jensen’s inequality (Kim, 2016, Seuret and Gouaisbaut, 2013, Zeng et al., 2015a, Zeng et al., 2015b).

Recently, authors in  Xu, Lam, Zhang, and Zou (2015) gave a new insight for reducing the conservatism of stability criteria. Most existing works on the stability of time-delay systems require the positive definiteness of all matrices in Lyapunov functions to meet their positive definiteness. In Xu et al. (2015), a relaxed condition for a matrix in the Lyapunov function instead of its positive definiteness was proposed, i.e. the matrix does not need to be positive definite. After this work, several works about relaxed conditions were reported (Zhang et al., 2015a, Zhang et al., 2015b).

Motivated by above discussion, this paper focuses on to develop relaxed conditions for time-varying delay systems because above commented works on relaxed conditions can be applied only integral terms with constant time-delay interval, i.e.  thtf(s)ds. To this end, a new inequality is derived based on free-matrix-based integral inequality, and then by utilizing this new inequality two new relaxed conditions are presented.

NotationI denotes the identity matrix with appropriate dimensions. in a matrix represents the elements below the main diagonal of a symmetric matrix. Sym{X} indicates X+XT. X[f(t)]Rm×n means that the elements of the matrix X include the values of f(t). For XRm×n, X denotes a basis for the null-space of X.

Section snippets

Preliminaries

The following lemmas will play a key role to derive main results.

Lemma 1

Zeng et al., 2015a

Let x be a differentiable function: [α,β]Rn. For symmetric matrices RRn×n and Z1,Z3R3n×3n, and any matrices Z2R3n×3n and N1,N2R3n×n satisfying[Z1Z2N1Z3N2R]0,the following inequality holds:αβẋT(s)Rẋ(s)dsϖ1T(α,β)Ψ1ϖ1(α,β),whereϖ1(α,β)=[xT(β),xT(α),1βααβxT(s)ds]T,Ψ1=(βα)(Z1+13Z3)+Sym{N1[I,I,0]+N2[I,I,2I]}.

Lemma 2

Let xRn be a continuous function and admits a continuous derivative differentiable function in [α,β]. For

Main results

In this section, new stability criteria are derived by utilization of Lemma 3. Before deriving the main result, let us define ei(i=1,,8)R8n×n as block entry matrices and a vector as ζT(t)=[xT(t),xT(th(t)),xT(th),ẋT(t),ẋT(th(t)),ẋT(th),1h(t)th(t)txT(s)ds,1hh(t)thth(t)xT(s)ds].

Theorem 1

For given a positive constant h, the system   (1)   is asymptotically stable, if there exist positive definite matrices QiR3n×3n (i=1,2 ), RRn×n, symmetric matrices PR5n×5n, ZiR3n×3n (i=1,,4 ), GiR6n×6n(

Numerical example

In this section, two numerical examples are given to illustrate the validity and superiority of the proposed scheme.

Example 1

Consider the system (1) with the following parameters: A=[2000.9],B=[1011].Table 1 shows comparative results on MAUB, h, of time-varying delay, h(t), for various μ. It should be noticed that, for all cases of μ, the results obtained by our Theorem 1 are better than the other literature, which means that our Theorem 1 enhances feasible region of the stability criterion. In

Conclusions

This paper suggested relaxed conditions for some matrices in a Lyapunov function. To this end, we presented a new inequality which is a modified version of free-matrix-based integral inequality, and then by utilizing derived new inequality, two novel lemmas were proposed which provide relaxed conditions, i.e. some matrices in Lyapunov function do not need to be positive definite. From two numerical examples, the effectiveness and superiority of proposed methods were shown.

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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Yasuaki Oishi under the direction of Editor André L. Tits.

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