Elsevier

Automatica

Volume 46, Issue 2, February 2010, Pages 466-470
Automatica

Technical communique
Improved delay-range-dependent stability criteria for linear systems with time-varying delays

https://doi.org/10.1016/j.automatica.2009.11.002Get rights and content

Abstract

This paper is concerned with the stability analysis of linear systems with time-varying delays in a given range. A new type of augmented Lyapunov functional is proposed which contains some triple-integral terms. In the proposed Lyapunov functional, the information on the lower bound of the delay is fully exploited. Some new stability criteria are derived in terms of linear matrix inequalities without introducing any free-weighting matrices. Numerical examples are given to illustrate the effectiveness of the proposed method.

Introduction

Time delay is encountered in many dynamic systems such as chemical or process control systems and networked control systems and often results in poor performance and can lead to instability (Gu et al., 2003, Niculescu, 2001). The subject of the stability analysis of systems with time-varying delay has attracted considerable attention during the past few years (Kharitonov and Niculescu, 2003, Lin et al., 2006, Xu and Lam, 2005, Yue et al., 2005, Zhang et al., 2009).

Existing stability criteria can be classified into two categories, that is, delay-independent ones and delay-dependent ones. It is well known that delay-independent ones are usually more conservative than the delay-dependent ones, so much attention has been paid in recent years to the study of delay-dependent stability conditions (Gao & Wang, 2003; Han,2004a,2004b;Haoussi and Tissir, 2009, Jiang and Han, 2005, Niculescu, 2001, Park, 1999, Suplin et al., 2006). Most of the existing delay-dependent stability criteria are obtained using Lyapunov–Krasovskii approach or Lyapunov–Razumikhin approach. To address this issue, a descriptor model transformation method is introduced in Fridman and Shaked (2003). This method significantly reduces the conservatism of the results since it is based on an equivalent model of the original system. In order to reduce further the conservatism introduced by model transformation and bounding techniques, a free-weighting matrices method is proposed in He, Wu, She, and Liu (2004). Using the Newton–Leibniz formula, some free-weighting matrices are introduced in the derivation of the stability condition. The free-weighting matrices method was further improved in He, Wang, Lin, and Wu (2005) by constructing an augmented Lyapunov functional. However, only a system with a constant delay was considered in He et al. (2005). By considering some useful terms when estimating the upper bound of the derivative of the Lyapunov functional some less conservative results have been obtained (He, Wang, Xie, & Lin, 2007). Recently, the results in He et al. (2007) have been further improved in Shao (2009) where a new Lyapunov functional is constructed and fewer matrix variables are involved. However, there still exists room for further improvement. It can be seen that the Lyapunov functional in Shao (2009) contains some integral terms such as tτ2txT(s)Q2x(s)ds and td(t)txT(s)Q3x(s)ds. The integral upper limit of these terms are all ‘t’. Therefore, the information on the lower bound of the delay, τ1, is not used adequately when the lower bound is not zero. These terms should be replaced by tτ2tτ1xT(s)Q2x(s)ds and td(t)tτ1xT(s)Q3x(s)ds, respectively. Furthermore, the Lyapunov functional in Shao (2009) does not contain any triple-integral terms. Adding some triple-integral terms in the Lyapunov functional may be helpful for the reduction of the conservatism such as reported in Ariba and Gouaisbaut (2007). In addition, in our recent work (Sun, Liu, & Chen, 2009), a novel augmented Lyapunov functional containing a triple-integral term has been used to derive a less conservative stability condition for neutral time-delay systems. However, Sun et al. (2009) has only considered the constant delay case. Therefore, in this paper the method reported in Sun et al. (2009) is extended to the time-varying delay case. In this paper, a new type of augmented Lyapunov functional which contains some triple-integral terms is introduced. The information on the lower bound of the delay is fully used in the Lyapunov functional. Some less conservative stability criteria are derived without introducing any free-weighting matrices. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.

Section snippets

Problem formulation and main results

Consider the following linear system with time-varying delay: ẋ(t)=Ax(t)+A1x(tτ(t)),t>0x(t)=ϕ(t),t[τ2,0] where x(t)Rn is the state vector; The initial condition ϕ(t) is a continuously differentiable vector-valued function; ARn×n and A1Rn×n are constant system matrices; τ(t) is a time-varying differentiable function and satisfies 0<τ1τ(t)τ2τ̇(t)μ where 0<τ1<τ2, and 0μ are constants.

Let τ12=τ2τ1 and τs=12(τ22τ12). The following lemma is introduced which has an important role in the

Numerical examples

In this section, two numerical examples are given to show that results proposed in this paper are less conservative than the existing ones.

Example 8

Consider the following system (He et al., 2007, Shao, 2009) with A=[2000.9],A1=[1011]. For various μ and unknown μ, the maximum upper bounds on delay (MUBDs), τ2, such that the system is asymptotically stable for given lower bound, τ1, are listed in Table 1, Table 2, respectively. From Table 1, Table 2, it is easy to see that our method can obtained

Conclusions

In this paper, the problem of the stability of time-delay systems has been investigated. Without introducing any free-weighting matrices, new delay-dependent stability criteria have been derived by introducing a new type of Lyapunov functional which contains some triple-integral terms and fully uses the information on the lower bound of the delay. The obtained criteria have been shown to be less conservative than the existing ones. Numerical examples have been given to illustrate the

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    This work is supported by the Beijing Education Committee Cooperation Building Foundation Project XK100070532 and China Postdoctoral Science Foundation 20080440308. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Keqin Gu under the direction of Editor André L. Tits.

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