Open Access

Stability Criterion for Discrete-Time Systems

Journal of Inequalities and Applications20102010:201459

DOI: 10.1155/2010/201459

Received: 21 November 2009

Accepted: 18 January 2010

Published: 2 February 2010

Abstract

This paper is concerned with the problem of delay-dependent stability analysis for discrete-time systems with interval-like time-varying delays. The problem is solved by applying a novel Lyapunov functional, and an improved delay-dependent stability criterion is obtained in terms of a linear matrix inequality.

1. Introduction

Recently, the problem of delay-dependent stability analysis for time-delay systems has received considerable attention, and lots of significant results have been reported; see, for example, Chen et al. [1], He et al. [2], Lin et al. [3], Park [4], and Xu and Lam [5], and the references therein. Among these references, we note that the delay-dependent stability problem for discrete-time systems with interval-like time-varying delays (i.e., the delay https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq1_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq2_HTML.gif ) has been studied by Fridman and Shaked [6], Gao and Chen [7], Gao et al. [8], and Jiang et al. [9], where some LMI-based stability criteria have been presented by constructing appropriate Lyapunov functionals and introducing free-weighting matrices. It should be pointed out that the Lyapunov functionals considered in these references are more restrictive due to the ignorance of the term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq3_HTML.gif Moreover, the term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq4_HTML.gif is also ignored in Gao and Chen [7] and Gao et al. [8]. The ignorance of these terms may lead to considerable conservativeness.

On the other hand, in the study of stabilization for the discrete-time linear systems, traditional idea of the control schemes is to construct a control signal according to the current system state [10]. However, as pointed out by Xiong and Lam [11], in practice there is often a system that itself is not time-delayed but time-delayed may exist in a channel from system to controller. A typical example for the existence of such delays is the measurement and the network transmission of signals. In this case, a time-delayed controller is naturally taken into account. It is worth noting that the closed-loop system resulting from a delayed controller is actually a time-delay system. Therefore, stability results of time-delay systems could be applied to design time-delayed controller.

The present study, based on a new Lyapunov functional, an improved delay-dependent stability criterion for discrete-time systems with time-varying delays is presented in terms of LMIs. It is shown that the obtained result is less conservative than those by Fridman and Shaked [6], Gao and Chen [7], Gao et al. [8], Jiang et al. [9], and Zhang et al. [12].

2. Preliminaries

Fact 1.

For any positive scalar https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq5_HTML.gif and vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq7_HTML.gif the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ1_HTML.gif
(2.1)

Let us denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq8_HTML.gif

Lemma 2.1 (see [13]).

The zero solution of difference system is asymptotic stability if there exists a positive definite function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq9_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ2_HTML.gif
(2.2)

along the solution of the system. In the case the above condition holds for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq10_HTML.gif , say one that the zero solution is locally asymptotically stable.

Lemma 2.2 (see [13]).

For any constant symmetric matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq11_HTML.gif , scalar https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq12_HTML.gif , vector function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq13_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ3_HTML.gif
(2.3)

3. Improved Stability Criterion

In this section, we give a novel delay-dependent stability condition for discrete-time systems with interval-like time-varying delays. Now, consider the following system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ4_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq14_HTML.gif is the state vector, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq16_HTML.gif are known constant matrices, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq17_HTML.gif is a time-varying delay satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq18_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq20_HTML.gif are positive integers representing the lower and upper bounds of the delay. For (3.1), we have the following result.

Theorem 3.1.

Give integers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq22_HTML.gif . Then, the discrete time-delay system (3.1) is asymptotically stable for any time delay https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq23_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq24_HTML.gif , if there exist symmetric positive definite matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq25_HTML.gif satisfying the following matrix inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ5_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq26_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq27_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq28_HTML.gif .

Proof.

Consider the Lyapunov function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq29_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ6_HTML.gif
(3.3)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq30_HTML.gif being symmetric positive definite solutions of (3.2) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq31_HTML.gif

Then difference of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq32_HTML.gif along trajectory of solution of (3.1) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ7_HTML.gif
(3.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ8_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ9_HTML.gif
(3.6)

where Fact 1 is utilized in (3.6), respectively.

Note that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ10_HTML.gif
(3.7)
and hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ11_HTML.gif
(3.8)
Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ12_HTML.gif
(3.9)
Using Lemma 2.2, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ13_HTML.gif
(3.10)
From the above inequality it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ14_HTML.gif
(3.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq33_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq35_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_Equ15_HTML.gif
(3.12)

By condition (3.2), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq36_HTML.gif is negative definite; namely, there is a number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq37_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F201459/MediaObjects/13660_2009_Article_2082_IEq38_HTML.gif and hence, the asymptotic stability of the system immediately follows from Lemma 2.1. This completes the proof.

Remark 3.2.

Theorem 3.1 gives a sufficient condition for stability criterion for discrete-time systems (3.1). These conditions are described in terms of certain diagonal matrix inequalities, which can be realized by using the linear matrix inequality algorithm proposed in [14]. But Zhang et al. in [12] proved that these conditions are described in terms of certain symmetric matrix inequalities, which can be realized by using the Schur complement lemma and linear matrix inequality algorithm proposed in [14].

4. Conclusions

In this paper, an improved delay-dependent stability condition for discrete-time linear systems with interval-like time-varying delays has been presented in terms of an LMI.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Maejo University
(2)
Department of Optimization and Control, Institute of Mathematics

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Copyright

© K. Ratchagit and V. N. Phat. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.