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Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 929725, 10 pages

http://dx.doi.org/10.1155/2013/929725

## New Delay-Dependent Robust Stability Criterion for LPD Discrete-Time Systems with Interval Time-Varying Delays

^{1}Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand^{2}Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand^{3}Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 18 October 2012; Accepted 13 January 2013

Academic Editor: Xiaohui Liu

Copyright © 2013 Narongsak Yotha and Kanit Mukdasai. 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.

#### Abstract

This paper investigates the problem of robust stability for linear parameter-dependent (LPD) discrete-time systems with interval time-varying delays. Based on the combination of model transformation, utilization of zero equation, and parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent robust stability conditions are obtained and formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.

#### 1. Introduction

Systems with time delay exist in many fields such as electric systems, chemical processes systems, networked control systems, telecommunication systems, and economical systems. Over the past decades, the problem of robust stability analysis for uncertain systems with time delay has been widely investigated by many researchers. Commonly, stability criteria for uncertain systems with time delay are generally divided into two classes: a delay-independent one and a delay-dependent one. The delay-independent stability criteria tends to be more conservative, especially for a small size delay; such criteria do not give any information on the size of delay. On the other hand, delay-dependent stability criteria are concerned with the size of delay and usually provide a maximal delay size.

Discrete-time systems with state delay have strong background in engineering applications, among which network-based control has been well recognized to be a typical example. If the delay is constant in discrete systems, one can transform a delayed system into a delay-free one by using state augmentation techniques. However, when the delay is large, the augmented system will become much complex and thus difficult to analyze and synthesize [1]. In recent years, robust stability analysis of continuous-time and discrete-time systems subject to time-invariant parametric uncertainty has received considerable attention. An important class of linear time-invariant parametric uncertain system is a linear parameter-dependent (LPD) system in which the uncertain state matrices are in the polytope consisting of all convex combination of known matrices. To address this problem, several results have been obtained in terms of sufficient (or necessary and sufficient) conditions, see [1–24] and references cited therein. Most of these conditions have been obtained via the Lyapunov theory approaches in which the parameter-dependent Lyapunov functions have been employed. These conditions are always expressed in terms of LMIs which can be solved numerically by using available tools such as the LMI Toolbox in MATLAB. Recently, delay-dependent robust stability criteria for LPD continuous-time systems with time delay have been taken into consideration. Sufficient conditions for robust stability of time-delay systems have been presented via Lyapunov approaches [8, 16, 21]. However, much attention has been focused on the problem of robust stability analysis for LPD discrete-time systems with time delay [10, 13, 22].

In this paper, we focus on the delay-dependent robust stability criterion for LPD discrete-time systems with interval time-varying delays. Based on the combination of model transformation, utilization of zero equation, and parameter-dependent Lyapunov functional, new delay-dependent robust stability conditions are obtained and formulated in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to illustrate that the resulting criterion outperforms the existing stability condition.

#### 2. Problem Formulation and Preliminaries

We introduce some notations, definitions, and propositions that will be used throughout the paper. denotes the set of nonnegative integer numbers; denotes the -dimensional space with the vector norm ; denotes the Euclidean vector norm of , that is, ; denotes the space of all real matrices of -dimensions; denotes the transpose of the matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; ; ; ; ; matrix is called a semipositive definite () if , for all ; is a positive definite () if for all ; matrix is called a seminegative definite () if , for all ; is a negative definite () if for all ; means ; means ; represents the elements below the main diagonal of a symmetric matrix.

Consider the following uncertain LPD discrete-time system with interval time-varying delays of the form where , is the system state and is an initial value at . , are uncertain matrices belonging to the polytope of the form In addition, we assume that the time-varying delay is upper and lower bounded. It satisfies the following assumption of the form where and are known positive integers.

*Definition 1 (see [19]). * The system (1) is said to be robustly stable if there exists a positive definite function such that
along any trajectory of the solution of the system (1).

*Definition 2 (see [19]). *The system (1) when , , and is said to be asymptotically stable if there exists a positive definite function such that
along any trajectory of the solution of the system (1) when , .

Proposition 3 ([7, the Schur complement lemma]). *Given constant symmetric matrices , and of appropriate dimensions with , then if and only if
*

Proposition 4 (see [9]). * For any constant matrix , two integers and satisfying , and vector function , the following inequality holds:
*

Rewrite the system (1) in the following system:

#### 3. Robust Stability Conditions

In this section, we study the robust stability criteria for the system (1) by using the combination of model transformation, the linear matrix inequality (LMI) technique, and the Lyapunov method. We introduce the following notations for later use: where

Theorem 5. *The system (1)-(2) is robustly stable if there exist positive definite symmetric matrices , and , any appropriate dimensional matrices ,, and , , satisfying the following LMIs:
*

*Proof. *Consider the following parameter-dependent Lyapunov-Krasovskii function for the system (9) of the form
where
Evaluating the forward deference of , it is defined as
Let us define, for ,
Then along the solution of the system (9), we obtain
From Proposition 4, we have
We can show that
It is easy to see that
By (22) and (23), we can obtain
and we conclude that
It is obvious that
The following equations are true for any polytopic matrices with appropriate dimensions:
It follows from (18)–(30) that
where and is defined in (10). Due to the fact that , we obtain the following identities:
By (31)–(33), if the conditions (12)–(15) are true, then
where . This means that the system (1)-(2) is robustly stable. The proof of the theorem is complete.

If and when then the system (1)-(2) reduces to the following system: Take the Lyapunov-Krasovskii functional as (16), where , , , , , , , , , , and when . Moreover, let us set polytopic matrices with appropriate dimensions of the forms , and when . According to Theorem 5, we have Corollary 6 for the delay-dependent asymptotically stability criterion of the system (35)-(36). We introduce the following notations for later use: where

Corollary 6. *The system (35)-(36) is asymptotically stable if there exist positive definite symmetric matrices , and , any appropriate dimensional matrices , and , , satisfying the following LMIs:
*

#### 4. Numerical Examples

*Example 7. *Consider the following LPD discrete-time system with interval time-varying delays (1)-(2) with
with initial condition , . The numerical solutions and of (1)-(2) with (40) are plotted in Figure 1.

*Solution*

By using the LMI Toolbox in MATLAB (with accuracy 0.01) and conditions (12)–(15) of Theorem 5, this system is robustly stable for discrete delay time satisfying , , and
For the given , Table 1 lists the comparison of the upper bounds delay for robust stability of the system (1)-(2) with (40) by the different method. By a conditions in Theorem 5, we can see from Table 1 that our result is superior to those in [22, Theorem 1].

*Example 8. *Consider the system (35) with
For the given , we calculate the allowable maximum value of that guarantees the asymptotic stability of the system (35) with (42). By using different methods, the calculated results are presented in Table 2. From the table, we can see that Corollary 6 in this paper provides the less conservative results.

#### Acknowledgments

This work is supported by National Research Council of Thailand and Khon Kaen University, Thailand (Grant no. kku fmis 121740).

#### References

- H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,”
*Institute of Electrical and Electronics Engineers*, vol. 52, no. 2, pp. 328–334, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - T. Botmart and P. Niamsup, “Robust exponential stability and stabilizability of linear parameter dependent systems with delays,”
*Applied Mathematics and Computation*, vol. 217, no. 6, pp. 2551–2566, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. T. Cui and M. G. Hua, “Robust passive control for uncertain discrete-time systems with time-varying delays,”
*Chaos, Solitons and Fractals*, vol. 29, no. 2, pp. 331–341, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fang, “Delay-dependent stability analysis for discrete singular systems with time-varying delays,”
*Acta Automatica Sinica*, vol. 36, no. 5, pp. 751–755, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - E. Fridman and U. Shaked, “Parameter dependent stability and stabilization of uncertain time-delay systems,”
*Institute of Electrical and Electronics Engineers*, vol. 48, no. 5, pp. 861–866, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - E. Fridman and U. Shaked, “Stability and guaranteed cost control of uncertain discrete delay systems,”
*International Journal of Control*, vol. 78, no. 4, pp. 235–246, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Gu, V. L. Kharitonov, and J. Chen,
*Stability of Time-Delay Systems*, Birkhauser, Berlin, Germany, 2003. - Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties,”
*Institute of Electrical and Electronics Engineers*, vol. 49, no. 5, pp. 828–832, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - X. Jiang, Q. L. Han, and X. Yu, “Stability criteria for linear discrete-time systems with interval-like time-varying delay,” in
*Proceedings of the American Control Conference (ACC '05)*, pp. 2817–2822, June 2005. View at Scopus - X. G. Liu, R. R. Martin, M. Wu, and M. L. Tang, “Delay-dependent robust stabilisation of discrete-time systems with time-varying delay,”
*IEE Proceedings: Control Theory and Applications*, vol. 153, no. 6, pp. 689–702, 2006. View at Publisher · View at Google Scholar · View at Scopus - W.-J. Mao, “An LMI approach to $D$-stability and $D$-stabilization of linear discrete singular system with state delay,”
*Applied Mathematics and Computation*, vol. 218, no. 5, pp. 1694–1704, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - W.-J. Mao and J. Chu, “$D$-stability and $D$-stabilization of linear discrete time-delay systems with polytopic uncertainties,”
*Automatica*, vol. 45, no. 3, pp. 842–846, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - K. Mukdasai, “Robust exponential stability for LPD discrete-time system with interval time-varying delay,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 237430, 13 pages, 2012. View at Zentralblatt MATH · View at MathSciNet - K. Mukdasai and P. Niamsup, “Robust stability of discrete-time linear parameter dependent system with delay,”
*Thai Journal of Mathematics (Annual Meeting in Mathematics)*, vol. 8, no. 4, Special issue, pp. 11–20, 2010. View at Zentralblatt MATH · View at MathSciNet - P. T. Nam, H. M. Hien, and V. N. Phat, “Asymptotic stability of linear state-delayed neutral systems with polytope type uncertainties,”
*Dynamic Systems and Applications*, vol. 19, no. 1, pp. 63–72, 2010. View at Zentralblatt MATH · View at MathSciNet - P. Niamsup and V. N. Phat, “${H}_{\infty}$ control for nonlinear time-varying delay systems with convex polytopic uncertainties,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 11, pp. 4254–4263, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. N. Phat and P. T. Nam, “Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function,”
*International Journal of Control*, vol. 80, no. 8, pp. 1333–1341, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. De la Sen, “Robust stabilization of a class of polytopic linear time-varying continuous systems under point delays and saturating controls,”
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 73–83, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Udpin and P. Niamsup, “Robust stability of discrete-time LPD neural networks with time-varying delay,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 11, pp. 3914–3924, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. G. Wu, P. Shi, H. Su, and J. Chu, “Delay-dependent exponential stability analysis for discrete-time switched neural networks with time-varying delay,”
*Neurocomputing*, vol. 74, no. 10, pp. 1626–1631, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Xia and Y. Jia, “Robust stability functionals of state delayed systems with polytopic type uncertainties via parameter-dependent Lyapunov functions,”
*International Journal of Control*, vol. 75, no. 16-17, pp. 1427–1434, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Zhang, Q. Y. Xie, X. S. Cai, and Z. Z. Han, “New stability criteria for discrete-time systems with interval time-varying delay and polytopic uncertainty,”
*Latin American Applied Research*, vol. 40, no. 2, pp. 119–124, 2010. View at Scopus - W. Zhang, J. Wang, Y. Liang, and Z. Z. Han, “Improved delay-range-dependent stability criterion for discrete-time systems with interval time-varying delay,”
*Journal of Information and Computational Science*, vol. 14, no. 8, pp. 3321–3328, 2011. - W.-A. Zhang and L. Yu, “Stability analysis for discrete-time switched time-delay systems,”
*Automatica*, vol. 45, no. 10, pp. 2265–2271, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet