Elsevier

Automatica

Volume 40, Issue 8, August 2004, Pages 1435-1439
Automatica

Technical Communique
Delay-dependent criteria for robust stability of time-varying delay systems

https://doi.org/10.1016/j.automatica.2004.03.004Get rights and content

Abstract

This paper deals with the problem of delay-dependent robust stability for systems with time-varying structured uncertainties and time-varying delays. Some new delay-dependent stability criteria are devised by taking the relationship between the terms in the Leibniz–Newton formula into account. Since free weighting matrices are used to express this relationship and since appropriate ones are selected by means of linear matrix inequalities, the new criteria are less conservative than existing ones. Numerical examples suggest that the proposed criteria are effective and are an improvement over previous ones.

Introduction

Stability criteria for time-delay systems have been attracting the attention of many researchers. They can be classified into two categories: delay-dependent and delay-independent criteria. Since delay-dependent criteria make use of information on the length of delays, they are less conservative than delay-independent ones. For delay-dependent criteria (see, for example, Su & Huang, 1992; Li & de Souza, 1997; Gu, Wang, Li, Cheng, & Qian, 1998; Cao, Sun, & Cheng, 1998; de Souza & Li, 1999; Park, 1999; Han & Gu, 2001; Kim, 2001; Moon, Park, Kwon, & Lee, 2001; Han 2002a, Han 2002b; Yue & Won, 2002; Fridman and Shaked 2002, Fridman and Shaked 2003), the main approaches currently consist of four model transformations of the original system (Fridman & Shaked, 2003). The first type is a first-order transformation. Since additional eigenvalues are introduced into the transformed system, it is not equivalent to the original one (Gu & Niculescu, 2000). The second type is a neutral transformation. The system obtained by this method is not equivalent to the original one, either (Gu & Niculescu, 2001); and this method requires an additional assumption to obtain the stability condition for the system. In addition, the inequality used to determine the stability of the system is −2aTb⩽aTXa+bTX−1b,a,b∈Rn,X>0, which is known to be conservative. Park (1999) introduced a free matrix, M, to obtain a less conservative inequality −2aTb⩽(a+Mb)TX(a+Mb)+bTX−1b+2bTMb, and Moon et al. (2001) extended it to a more general form, −2aTb⩽abTXYT−IY−IZab,XYTYZ⩾0. The third type of model transformation employs these inequalities and yields a transformed system that is equivalent to the original one. However, in the derivative of the Lyapunov functional, Park (1999) and Moon et al. (2001) used the Leibniz–Newton formula and just replaced some of the terms x(tτ) with x(t)−t−τtẋ(s)ds in the derivative of the Lyapunov functional in order to make it easy to handle. For example, in Moon et al. (2001), x(tτ) was replaced with x(t)−t−τtẋ(s)ds in the expression 2xT(t)PA1ẋ(t), but not in τẋT(t)Zẋ(t). Since both x(tτ) and x(t)−t−τtẋ(s)ds affect the result, there must be some relationship between them; and there must exist optimal weighting matrices for those terms. However, they did not give a method for determining them, but just selected some fixed weighting matrices. Fridman and Shaked 2002, Fridman and Shaked 2003 combined a descriptor model transformation (Fridman, 2001) with Park and Moon's inequalities to yield the fourth type of transformation. This method produces less conservative criteria. However, since the basic approach in Fridman and Shaked 2002, Fridman and Shaked 2003 is also based on the substitution of x(t)−t−τtẋ(s)ds for x(tτ), it does not entirely overcome the conservatism of the methods given by Park (1999) and Moon et al. (2001).

This paper presents new criteria based on a new method with some interesting features. First, it deals with the system model directly and does not employ any system transformation, thus avoiding the conservatism that results from such a transformation. Second, it does not use the above inequality or the improved inequality to estimate the upper bound of −2aTb. This also reduces the conservatism in the derivation of the stability condition. Third, some free weighting matrices are employed to express the influence of the terms in the Leibniz–Newton formula, in contrast to existing methods, which preselect fixed ones. The matrices are determined by solving linear matrix inequalities (LMIs). This is the main advantage of our method, and is the essential difference between existing methods and ours. Compared with Moon et al. (2001), and Fridman and Shaked (2002), our new criteria overcome some of the main sources of conservatism, and contain the criteria in Moon et al. (2001) as a special case. Furthermore, the new criteria also contain the well-known delay-independent stability condition in Gu, Kharitonov, and Chen (2003) and Hale and Verduyn Lunel (1993). For two examples studied numerically, the new criteria are shown to be effective, offering significant improvements over previously published criteria.

Section snippets

Preliminaries

Consider a nominal system Σ0 with a time-varying delay given byΣ0:ẋ(t)=Ax(t)+Bx(t−d(t)),t>0,x(t)=φ(t),t∈[−τ,0],where x(t)∈Rn is the state vector. The time delay, d(t), is a time-varying continuous function that satisfies0⩽d(t)⩽τ,ḋ(t)⩽μ<1,where τ and μ are constants and the initial condition, φ(t), is a continuous vector-valued initial function of t∈[−τ,0].

When the system contains time-varying structured uncertainties, it can be described byΣ1:ẋ(t)=(A+ΔA(t))x(t)+(B+ΔB(t))x(t−d(t)),t>0,

Main results

First, the nominal system, Σ0, is discussed. The Leibniz–Newton formula is employed to obtain a delay-dependent condition, and the relationship between the terms in the formula is taken into account. Specifically, the terms on the left side of the equation2[xT(t)Y+xT(t−d(t))T]x(t)−t−d(t)tẋ(s)ds−x(t−d(t))=0are added to the derivative of the Lyapunov functional, V̇(xt). In this equation, the free weighting matrices Y and T indicate the relationship between the terms in the Leibniz–Newton

Numerical examples

In this section, some examples are used to demonstrate that the method presented in this paper is effective and is an improvement over existing methods.

Example 5

Consider the uncertain system Σ1 with the following parameters:A=−200−1,B=−10−1−1,Ea=diag{1.6,0.05},Eb=diag{0.1,0.3},D=I.

This example was given in Kim (2001) and Yue and Won (2002). The upper bounds on the time delay for different μ obtained from Theorem 4 are shown in Table 1. For comparison, the table also lists the upper bounds obtained from

Conclusion

This paper presents a new method of determining delay-dependent stability criteria that takes the relationship between x(td(t)) and x(t)−t−d(t)tẋ(s)ds into account. Some free weighting matrices that express the influence of these two terms are determined based on linear matrix inequalities, which makes it easy to choose suitable ones. It was shown that the criteria in Moon et al. (2001) are a special case of this new method, and that the new method is less conservative than existing ones.

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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ricu Middleton under the direction of Editor Paul Van den Hof.

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