ISRN Applied Mathematics

Volume 2011, Article ID 624127, 10 pages

http://dx.doi.org/10.5402/2011/624127

## Stability Analysis of Linear Discrete-Time Systems with Interval Delay: A Delay-Partitioning Approach

Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad 211004, India

Received 7 September 2011; Accepted 16 October 2011

Academic Editors: C.-H. Lien, F. Tadeo, and Q. Zhang

Copyright © 2011 Priyanka Kokil et al. 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 considers the problem of asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By using a delay partitioning-based Lyapunov functional, a new criterion for the asymptotic stability of such systems is proposed in terms of linear matrix inequalities (LMIs). The proposed stability condition depends on both the size of delay and partition size. The presented approach is compared with previously reported approaches.

#### 1. Introduction

Mathematical models with time delays are frequently encountered in various physical, industrial, and engineering systems due to measurement and computational delays, transmission and transport lags. During the last two decades, there has emerged a considerable interest in the system theoretic problems of time-delay systems. In many applications, time delays must be taken into account in a realistic system design, for instance, chemical processes, thermal processes, echo cancellation, local loop equalization, multipath propagation in mobile communication, array signal processing, congestion analysis and control in high-speed networks, neural networks and long transmission line in pneumatic systems [1–6]. The presence of time delays may result in instability of the designed systems. An excellent survey on the stability results of time-delay systems has been presented in [7]. According to dependence of delay, the available stability criteria are generally classified into two types: delay-dependent criteria and delay-independent criteria. Delay-dependent stability criteria generally lead to less conservative results than delay-independent ones, especially if the size of time delay is small [8–12].

Many publications relating to the delay-dependent stability analysis of continuous time-delay systems have appeared (see, e.g., [4, 12–21] and the references cited therein). In contrast, little effort has been made for studying the problem of stability of discrete time-delay systems. Utilizing Lyapunov functional method, several delay-dependent stability criteria for discrete-time systems have been reported in the literature [2, 22–27]. By employing novel techniques to estimate the forward difference of Lyapunov functional, stability criteria for linear discrete-time systems with interval-like time-varying delays have been proposed in [26]. The criteria proposed in [26] are less conservative with smaller numerical complexity than [22, 28–31]. The delay partitioning technique has been efficiently utilized in [32–36] to the stability analysis of systems with time-varying delays. Recently, by introducing free-weighting matrices and adopting the concept of delay partitioning, improved stability criteria have been established in [37]. The criteria reported in [37] are not only dependent on the delay but also dependent on the partitioning size. Though the approach in [37] provides less conservative results than [22, 29], it would lead to heavier computational burden and more complicated synthesis procedure.

Motivated by these developments, this paper studies the problem of stability analysis of linear discrete-time system with interval-like time-varying delay in the state. The paper is organized as follows. Section 2 introduces model description and preliminaries. By utilizing the delay partitioning idea of [37], a new linear-matrix-inequality- (LMI-) based criterion for the asymptotic stability of discrete-time state-delayed systems is proposed in Section 3. The proposed criterion depends on the size of delay as well as partition size. An example highlighting the usefulness of the presented criterion is given in Section 4.

#### 2. Model Description and Preliminaries

The following notations are used throughout the paper:: set of real matrices,: set of real vectors,: null matrix or null vector of appropriate dimension; the orders are specified in subscripts as the need arises,: identity matrix,: transpose of the matrix (or vector) , is positive-definite symmetric matrix, is negative-definite symmetric matrix.

In this paper, we consider a linear, autonomous, multivariable discrete-time system with interval-like time-varying delay in the state. Specifically, the system under consideration is represented by the difference equation: where is the system state vector, are constant matrices with appropriate dimensions, and is a positive integer representing interval-like time-varying delay satisfying where and are known positive integers representing the lower and upper delay bounds, respectively, and is an initial value at time . Let the lower bound of the delay be divided into number of partitions such that where is an integer representing partition size.

*Definition 2.1. *The equilibrium state of the system (2.1a) and (2.1b)–(2.3) is asymptotically stable if, for any , there exists such that if , , then , for every and .

Lemma 2.2 (see [30]). * For any positive definite matrix , two positive integers and satisfying , and vector function , one has
*

#### 3. Main Result

In this section, an LMI-based criterion for the asymptotic stability of system (2.1a) and (2.1b)–(2.3) is established. The main result may be stated as follows.

Theorem 3.1. *For given positive integers , , and , the system described by (2.1a) and (2.1b)–(2.3) is asymptotically stable if there exist real matrices , , , , and such that
*

*where*

*Proof. *Choose a Lyapunov functional candidate as
where
Taking the forward difference of (3.3) along the solution of (2.1a) and (2.1b), we have
Using Lemma 2.2, we obtain
Note that
Now, we have the following relations:
where and are constant matrices of appropriate dimensions and
It follows from (2.1a) and (3.5) that
Employing (3.6)–(3.11), we have the following inequality:
where is given by (3.2). Equation (3.12) can also be rearranged as
Using [17, Lemma 4.1], it can be shown that there exists a matrix of appropriate dimensions satisfying (3.1) if and only if
which together with (3.13) implies for all nonzero . This completes the proof.

*Remark 3.2. *A comparison of the number of the decision variables involved in several recent stability results is summarized in Table 1. It may be observed that the size of complexity in [22, 26, 31] is only related to state dimension , whereas the complexity of [37] and Theorem 3.1 depends on both and .

The total number of scalar decision variables of Theorem 3.1 is , and the total row size of the LMIs is . The numerical complexity of Theorem 3.1 is proportional to [16]. In [37], the total number of scalar decision variables of Theorem 2 is , the total row size of the LMIs is , and the numerical complexity is proportional to . Therefore, Theorem 3.1 has much smaller numerical complexity than Theorem 2 of [37].

*Remark 3.3. *For a given , the allowable maximum value of for guaranteeing the asymptotic stability of system (2.1a) and (2.1b)–(2.3) can be obtained by iteratively solving the LMI (3.1).

*Remark 3.4. *With , , and , Theorem 3.1 reduces to an equivalent form of [26, Proposition 1]. Thus, as compared to [26, Proposition 1], Theorem 3.1 provides additional degrees of freedom in the selection of , **,** and which would result in an enhanced stability region in the parameter space.

*Remark 3.5. *Using similar steps as in the proof of [37, Proposition 8], it is easy to establish that the conservatism of the stability result obtained via Theorem 3.1 is nonincreasing as the number of partitions increases.

It may be noted that, in the derivation of Theorem 3.1, the lower bound of the delay is assumed to be . For the situation where , we have the following result.

Theorem 3.6. *The system (2.1a) and (2.1b) with is asymptotically stable if there exist real matrices , , , , , and such that
**
where
*

*Proof. *Choosing the Lyapunov functional as
and employing the steps of the proof of Theorem 3.1 discussed earlier, one can easily arrive at the above theorem. The details of the proof of Theorem 3.6 are, therefore, omitted.

*Remark 3.7. *With rows and scalar decision variables, LMI (3.15) in Theorem 3.6 has the numerical complexity proportional to [16], while the condition in [37, Proposition 10] has the numerical complexity proportional to , where and . So, Theorem 3.6 has smaller numerical complexity than [37, Proposition 10].

#### 4. A Numerical Example

To demonstrate the applicability of the presented results and compare them with previous results, we now consider a specific example of system (2.1a), (2.1b) with and . This example has been considered in [22, 29, 37]. Using Matlab LMI toolbox [38, 39], it is found from Theorem 3.6 that the present system is stable for . In this case, the number of decision variables involved in Theorem 3.6 is 40. On the other hand, to obtain the upper bound , [37, Proposition 10] requires 57 decision variables. This demonstrates the numerical efficiency of the proposed method.

Next, consider the system described by (2.1a), (2.1b)–(2.3), and (4.1). For different values of , the admissible upper bound is listed in Table 2. From Table 2, it is clear that Theorem 3.1 can provide a larger upper bound than the previously reported stability results [22, 23, 26, 29]. It may also be observed that, for the example under consideration, Theorem 3.1 leads to upper bound which is identical to that arrived at via Theorem 2 in [37]. However, as discussed in Remark 3.2, Theorem 3.1 has much smaller numerical complexity than Theorem 2 in [37].

#### 5. Conclusion

In this paper, the problem concerning the asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state has been considered. Using the concept of delay partitioning, an LMI-based criterion for the asymptotic stability of such systems has been established. The criterion depends on the size of delay as well as partition size. The presented approach may imply asymptotic stability for a broader class of time-varying state-delayed systems, as compared to previous approaches [22, 23, 26, 29]. The presented criterion is numerically less complex than [37]. The stability results discussed in this paper can easily be extended to delayed discrete-time systems with parameter uncertainties.

#### References

- J. K. Hale,
*Functional Differential Equations*, Springer, New York, NY, USA, 1971. View at Zentralblatt MATH - H. Huang and G. Feng, “Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay,”
*IET Control Theory & Applications*, vol. 4, no. 10, pp. 2152–2159, 2010. View at Publisher · View at Google Scholar - J.-P. Richard, “Time-delay systems: an overview of some recent advances and open problems,”
*Automatica*, vol. 39, no. 10, pp. 1667–1694, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Shao, “Delay-dependent approaches to globally exponential stability for recurrent neural networks,”
*IEEE Transactions on Circuits and Systems II*, vol. 55, no. 6, pp. 591–595, 2008. View at Publisher · View at Google Scholar · View at Scopus - H. Shao, “Delay-dependent stability for recurrent neural networks with time-varying delays,”
*IEEE Transactions on Neural Networks*, vol. 19, no. 9, pp. 1647–1651, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus - V. J. S. Leite and M. F. Miranda, “Robust stabilization of discrete-time systems with time-varying delay: an LMI approach,”
*Mathematical Problems in Engineering*, vol. 2008, Article ID 875609, 15 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Xu and J. Lam, “A survey of linear matrix inequality techniques in stability analysis of delay systems,”
*International Journal of Systems Science*, vol. 39, no. 12, pp. 1095–1113, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. F. Chen, “Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities,”
*Chaos, Solitons and Fractals*, vol. 42, no. 2, pp. 1251–1257, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. S. Mahmoud, “Robust stability and stabilization of a class of uncertain nonlinear systems with delays,”
*Mathematical Problems in Engineering*, vol. 4, no. 2, pp. 165–185, 1998. View at Google Scholar · View at Scopus - M. S. Mahmoud, F. M. Al-Sunni, and Y. Shi, “Switched discrete-time delay systems: delay-dependent analysis and synthesis,”
*Circuits, Systems, and Signal Processing*, vol. 28, no. 5, pp. 735–761, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. S. Mahmoud and Y. Xia, “Robust stability and stabilization of a class of nonlinear switched discrete-time systems with time-varying delays,”
*Journal of Optimization Theory and Applications*, vol. 143, no. 2, pp. 329–355, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Xu and J. Lam, “On equivalence and efficiency of certain stability criteria for time-delay systems,”
*IEEE Transactions on Automatic Control*, vol. 52, no. 1, pp. 95–101, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - E. Fridman and U. Shaked, “A descriptor system approach to ${H}_{\infty}$ control of linear time-delay systems,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 2, pp. 253–270, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - Q.-L. Han, “On stability of linear neutral systems with mixed time delays: a discretized Lyapunov functional approach,”
*Automatica*, vol. 41, no. 7, pp. 1209–1218, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-L. Han, “A discrete delay decomposition approach to stability of linear retarded and neutral systems,”
*Automatica*, vol. 45, no. 2, pp. 517–524, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q.-L. Han, “Improved stability criteria and controller design for linear neutral systems,”
*Automatica*, vol. 45, no. 8, pp. 1948–1952, 2009. View at Publisher · View at Google Scholar · View at Scopus - Q.-L. Han and K. Gu, “Stability of linear systems with time-varying delay: a generalized discretized lyapunov functional approach,”
*Asian Journal of Control*, vol. 3, no. 3, pp. 170–180, 2001. View at Google Scholar · View at Scopus - X. Jiang and Q.-L. Han, “On ${H}_{\infty}$ control for linear systems with interval time-varying delay,”
*Automatica*, vol. 41, no. 12, pp. 2099–2106, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wu, Y. He, J.-H. She, and G.-P. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems,”
*Automatica*, vol. 40, no. 8, pp. 1435–1439, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. R. Karimi, M. Zapateiro, and N. Luo, “New delay-dependent stability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations,”
*Mathematical Problems in Engineering*, vol. 2009, Article ID 759248, 22 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Du and X. Zhang, “Delay-dependent stability analysis and synthesis for uncertain impulsive switched system with mixed delays,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 381571, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,”
*IEEE Transactions on Automatic Control*, vol. 52, no. 2, pp. 328–334, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - Y. He, M. Wu, G.-P. Liu, and J.-H. She, “Output feedback stabilization for a discrete-time system with a time-varying delay,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 10, pp. 2372–2377, 2008. View at Publisher · View at Google Scholar - V. K. R. Kandanvli and H. Kar, “Delay-dependent LMI condition for global asymptotic stability of discrete-time uncertain state-delayed systems using quantization/overflow nonlinearities,”
*International Journal of Robust and Nonlinear Control*, vol. 21, no. 14, pp. 1611–1622, 2011. View at Publisher · View at Google Scholar - H. Shao, “New delay-dependent stability criteria for systems with interval delay,”
*Automatica*, vol. 45, no. 3, pp. 744–749, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Shao and Q.-L. Han, “New stability criteria for linear discrete-time systems with interval-like time-varying delays,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 3, pp. 619–625, 2011. View at Publisher · View at Google Scholar - K. Ratchagit and V. N. Phat, “Stability criterion for discrete-time systems,”
*journal of Inequalities and Applications*, vol. 2010, Article ID 201459, 6 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - H. Gao, J. Lam, C. Wang, and Y. Wang, “Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,”
*IET Control Theory and Applications*, vol. 151, no. 6, pp. 691–698, 2004. View at Publisher · View at Google Scholar - 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*, pp. 2817–2822, Portland, Ore, USA, June 2005. View at Scopus - B. Zhang, S. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discrete-time systems,”
*Automatica*, vol. 44, no. 11, pp. 2963–2967, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Huang and G. Feng, “State estimation of recurrent neural networks with time-varying delay: a novel delay partition approach,”
*Neurocomputing*, vol. 74, no. 5, pp. 792–796, 2011. View at Publisher · View at Google Scholar · View at Scopus - B. Du, J. Lam, Z. Shu, and Z. Wang, “A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components,”
*IET Control Theory and Applications*, vol. 3, no. 4, pp. 383–390, 2009. View at Publisher · View at Google Scholar - Y. Zhao, H. Gao, J. Lam, and B. Du, “Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach,”
*IEEE Transactions on Fuzzy Systems*, vol. 17, no. 4, pp. 750–762, 2009. View at Publisher · View at Google Scholar · View at Scopus - J. Liu, B. Yao, and Z. Gu, “Delay-dependent ${H}_{\infty}$ filtering for Markovian jump time-delay systems: a piecewise analysis method,”
*Circuits Systems and Signal Processing*, vol. 30, no. 6, pp. 1253–1273, 2011. View at Google Scholar - L. Wu, J. Lam, X. Yao, and J. Xiong, “Robust guaranteed cost control of discrete-time networked control systems,”
*Optimal Control Applications and Methods*, vol. 32, no. 1, pp. 95–112, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Meng, J. Lam, B. Du, and H. Gao, “A delay-partitioning approach to the stability analysis of discrete-time systems,”
*Automatica*, vol. 46, no. 3, pp. 610–614, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*SIAM Studies in Applied Mathematics*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994. - P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali,
*LMI Control Toolbox for Use With Matlab*, The MATH Works, Natick, Mass, USA, 1995.