Journal of Applied Mathematics

Volume 2012 (2012), Article ID 237430, 13 pages

doi:10.1155/2012/237430

Research Article

Robust Exponential Stability for LPD Discrete-Time System with Interval Time-Varying Delay

Kanit Mukdasai1,2

¹Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

² Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

Received 5 April 2012; Accepted 11 July 2012

Academic Editor: B. V. Rathish Kumar

Copyright © 2012 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 exponential stability for uncertain linear-parameter dependent (LPD) discrete-time system with delay. The delay is of an interval type, which means that both lower and upper bounds for the time-varying delay are available. The uncertainty under consideration is norm-bounded uncertainty. Based on combination of the linear matrix inequality (LMI) technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions for the robust exponential stability are obtained in terms of LMI. Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.

1. Introduction

Over the past decades, the problem of stability analysis of delay discrete-time systems has been widely investigated by many researchers. Because the existence of time delay is frequent, a source of oscillation instability performances degradation of systems. Stability criteria for discrete-time systems with time delay is generally divided into two classes: delay-independent ones and delay-dependent ones. Delay-independent stability criteria tend to be more conservative, especially for small-size delay; such criteria do not give any information on the size of the delay. On the other hand, delay-dependent stability criteria is concerned with the size of the delay and usually provide a maximal delay size. Moreover, robust stability of linear 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 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 [1–15] and references cited therein. Most of these conditions have been obtained via the Lyapunov theory approaches in which parameter dependent Lyapunov functions have been employed. These conditions are always expressed in terms of linear matrix inequalities (LMIs) which can be solved numerically by using available tools such as LMI toolbox in MATLAB. The results have been obtained for robust stability for LPD systems in which time delay occur in state variable such as [6, 11, 14] present sufficient conditions for robust stability of LPD continuous-time system with delays. However, a few results have been obtained for robust stability for LPD discrete-time systems with delay.

In this paper, we deal with the problem of robust exponential stability for uncertain LPD discrete-time system with interval time-varying delay. Combined with the linear matrix inequality technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions for the robust exponential stability are obtained in terms of LMI. Finally, numerical examples have demonstrated the effectiveness of the criteria.

2. Problem Formulation and Preliminaries

We introduce some notations and definitions that will be used throughout the paper. denotes the set of non negative integer numbers; denotes the -dimensional space with the vector norm ; denotes the Euclidean vector norm of ; that is, ; denotes the space of all matrices of -dimensions; denotes transpose of the Matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; ; Matrix is called semi positive definite () if , for all   is positive definite () if for all ; Matrix is called semi-negative definite () if , for all is 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 delay in the state

where , is the system state and is a initial value at . , are uncertain matrices belonging to the polytope of the form

and are unknown matrices representing time-varying parameter uncertainties, we assumed to be of the form

The class of parametric uncertainties , which satisfies

is said to be admissible where is a known matrix satisfying

and is uncertain matrix satisfying

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 2.1. The uncertain LPD discrete-time-delayed system in (2.1) is said to be robustly exponentially stable if there exist constant scalars and such that

for all admissible uncertainties.

Lemma 2.2 (see [5] (Schur complement lemma)). Given constant matrices of appropriate dimensions with . Then if and only if

Lemma 2.3 (see [2]). Given constant matrices , and of appropriate dimensions with . Then,

where ,  if and only if

for some scalar .

3. Main Results

In this section, we present our main results on the robust exponential stability criteria for uncertain LPD discrete-time system with interval time-varying delays. We introduce the following notation for later use:

Lemma 3.1. For any in (3.1), and given by

are parameter-dependent positive definite Lyapunov matrices such that

if and only if

Proof. Consider

We assume that

Using Lemma 2.2, we obtain

We rewrite the latter inequality as

Using Lemma 2.3, inequality (3.8) holds if and only if there exists such that

If we apply to (3.9), then we obtain

Premultiplying (3.10) by diag and postmultiplying by diag, we get that (3.4) and the lemma is proved.

Lemma 3.2. If there exist positive definite symmetric matrices , , , and positive real numbers such that

then, for any in (3.1), and are parameter-dependent positive definite Lyapunov matrices in Lemma 3.1 such that (3.4) holds.

Proof. Consider

Using the fact that , we obtain the following identities:

Then, it follows from (3.11), (3.12), and (3.13) that (3.4) holds. The proof of the lemma is complete.

Theorem 3.3. The system (2.1) is robustly exponentially stable if the LMI conditions (3.11) are feasible.

Proof. Consider the following Lyapunov-Krasovskii function for system (2.1) of the form

where

A Lyapunov-Krasovskii difference for the system (2.1) is defined as

Taking the difference of and , the increments of and are

Form , the two last terms of the right-hand side of the latter equality yield

Thus, we obtain

The increment of is easily computed as

It is easy to see that

for simplicity, we let . Since, , we obtain that

Therefore, we conclude that

It follows form (3.24) that

where and . By (3.11), (3.25), and Lemma 3.1, and 3.2, we obtain

where . By (3.14), it is easy to see that

where

It can be shown that there always exists a scalar satisfying

For any scalar , it follows from (3.26) and (3.27) that

where

Therefore, for any integer , summing up both sides of (3.30) from to gives

For ,

From (3.32) and (3.33), we obtain

Observe

Then, it follows from (3.29), (3.33), and (3.35) that

By Definition 2.1, this means that the system (2.1) is robustly exponentially stable. The proof of the theorem is complete.

4. Numerical Example

Example 4.1. Consider the following uncertain LPD discrete-time system with time-varying delays (2.1) where , that is, and

and . By using LMI Toolbox in MATLAB, we use condition (3.11) in Theorem 3.3 for this example. The solutions of LMI verify as follows of the form , , and (see Figure 1).

Figure 1: The simulation solution of the states and in Example 4.1 for uncertain LPD discrete-time delayed system with initial conditions and , , and by using the method of Runge-Kutta order 4() with Matlab.

Example 4.2. Consider the following the LPD discrete-time system with time-varying delays (2.1) where, with

Table 1 lists the comparison of the upper-bound delay for asymptotic stability of system (2.1) where by different method. We apply Theorem 3.3 and see from Table 1 that our result is superior to those in [7, Theorem 3.2].

Table 1: Comparison of the maximum allowed time delay .

Methods
()
()
()
()

Liu et al. [7] 2006
2
4
5
7

Our results
4
6
7
9

Acknowledgments

This work was supported by the Thailand Research Fund (TRF), the Office of the Higher Education Commission (OHEC), and the Khon Kaen University (Grant number MRG5580006), and the author would like to thank his advisor Associate Professor Dr. Piyapong Niamsup from the Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand.

References

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.

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.

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.

E. Fridman and U. Shaked, “Parameter dependent stability and stabilization of uncertain time-delay systems,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 861–866, 2003.

K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhäuser, 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,” IEEE Transactions on Automatic Control, vol. 49, no. 5, pp. 828–832, 2004.

X. Liu, R. Martin, M. Wu, and M. 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.

W. J. Mao, “An LMI approach to -stability and -stabilization of linear discrete singular system with state delay,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1694–1704, 2011.

W. J. Mao and J. Chu, “-stability and -stabilization of linear discrete time-delay systems with polytopic uncertainties,” Automatica, vol. 45, no. 3, pp. 842–846, 2009.

K. Mukdasai and P. Niamsup, “Robust stability of discrete-time linear parameter dependent system with delay,” Thai Journal of Mathematics, vol. 8, no. 4, pp. 11–20, 2010.

P. Niamsup and V. N. Phat, “ control for nonlinear time-varying delay systems with convex polytopic uncertainties,” Nonlinear Analysis, vol. 72, no. 11, pp. 4254–4263, 2010.

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.

Z. G. Wu, P. Shi, H. Su, and J. Chu, “Delay-dependent exponential stability analysis for discretetime switched neural networks with time-varying delay,” Neurocomputing, vol. 74, no. 10, pp. 1626–1631, 2011.

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.

W. A. Zhang and L. Yu, “Stability analysis for discrete-time switched time-delay systems,” Automatica, vol. 45, no. 10, pp. 2265–2271, 2009.