Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 406420, 10 pages
http://dx.doi.org/10.1155/2015/406420

Research Article

New Delay-Range-Dependent Robust Exponential Stability Criteria of Uncertain Impulsive Switched Linear Systems with Mixed Interval Nondifferentiable Time-Varying Delays and Nonlinear Perturbations

Piyapong Niamsup,¹ Narongsak Yotha,² and Kanit Mukdasai³

¹Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
²Department of Applied Mathematics and Statistics, Rajamangala University of Technology Isan, Nakhon Ratchasima 30000, Thailand
³Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received 4 June 2015; Accepted 9 August 2015

Academic Editor: Zizhen Zhang

Copyright © 2015 Piyapong Niamsup 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.

We investigate the problem of robust exponential stability analysis for uncertain impulsive switched linear systems with time-varying delays and nonlinear perturbations. The time delays are continuous functions belonging to the given interval delays, which mean that the lower and upper bounds for the time-varying delays are available, but the delay functions are not necessary to be differentiable. The uncertainties under consideration are nonlinear time-varying parameter uncertainties and norm-bounded uncertainties, respectively. Based on the combination of mixed model transformation, Halanay inequality, utilization of zero equations, decomposition technique of coefficient matrices, and a common Lyapunov functional, new delay-range-dependent robust exponential stability criteria are established for the systems in terms of linear matrix inequalities (LMIs). A numerical example is presented to illustrate the effectiveness of the proposed method.

1. Introduction

The problem of stability analysis for dynamical systems with time delays and uncertainties has been intensively studied since these systems often occur in many industrial systems such as chemical processes, biological systems, population dynamics, neural networks, large-scale systems, and network control systems. The occurrence of the time delays and uncertainties may cause frequently the source of instability or poor performances in various systems. Thus, there has been growing interest in stability analysis and controller design for time-delay systems. However, authors investigated the robust synchronization of coupled fuzzy cellular neural networks with differentiable time-varying delay in [1, 2]. Stability criteria for time-delay systems are generally divided into two classes: delay-independent one and delay-dependent one. 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 are concerned with the size of the delay and usually provide a maximal delay size. Most of the existing delay-dependent stability criteria are presented by using Lyapunov-Krasovskii approach or Lyapunov-Razumikhin approach. In recent years, much attention has been paid to stability analysis of the uncertain linear systems with interval time-varying delay [3–6]. In [5], the authors studied the delay-dependent stability problem for uncertain linear systems with interval time-varying delay. The restriction on the derivative of the interval time-varying delay was removed. Moreover, robust stability analysis of uncertain linear systems with time-varying delays and nonlinear perturbations has received the attention of a lot of theoreticians and engineers in this field over the last few decades [7–14]. Furthermore, authors studied the delay-dependent robust stability criteria for linear systems with discrete interval time-varying delay, discrete constant delay, and nonlinear perturbations in [15]. However, a descriptor model transformation and a corresponding Lyapunov-Krasovskii functional have been introduced for stability analysis of systems with delays in [16]. In [17], the authors studied the problem of stability for linear switching system with time-varying delays.

Over the past decades, the problem of stability analysis for dynamic systems with impulsive effects and switching has arisen in a wide range of disciplines, such as physics, chemical engineering, and biology [18–28]. These systems are usually called impulsive switched systems. In [24], the authors studied the asymptotic stability problem for a class of impulsive switched systems with time-invariant delays based on LMI approach. Stability criteria of uncertain impulsive switched systems with time-invariant delays are introduced in [25]. Most of the existing delay-dependent stability criteria for time-delay systems are obtained as the upper bounds on the derivative time-varying delays by using Lyapunov-Krasovskii functional. However, it appears that few results are available for stability analysis for impulsive switched systems with time-varying delays. In consequence, it is important and interesting to study the problem of robust stability analysis for uncertain impulsive switched systems with interval nondifferentiable time-varying delays and nonlinear perturbations by using a common Lyapunov functional and Halanay lemma.

In this paper, we present the delay-range-dependent robust exponential stability criteria for uncertain impulsive switched linear systems with mixed interval nondifferentiable time-varying delays and nonlinear perturbations. Based on Halanay inequality, mixed model transformation, utilization of zero equations, decomposition technique of coefficient matrices, and a common Lyapunov functional, some new delay-range-dependent robust exponential stability criteria are derived in terms of LMIs for the systems. In order to reduce the complexity of stability criteria for calculation and finding solutions, mixed model transformation [13, 16] and Halanay inequality [29–31] are used. Finally, an illustrative example is given to show the effectiveness and advantages of the developed method.

2. Problem Formulation and Preliminaries

The following notations will be used in this paper: denotes the set of all natural numbers; denotes the set of all real nonnegative numbers; denotes the -dimensional Euclidean space equipped with the Euclidean norm ; denotes the space of all 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 semipositive definite () if , for all ; is positive definite () if for all ; matrix is called seminegative definite () if , for all ; is negative definite () if for all ; means ; means ; ; , .

Consider the following uncertain impulsive switched linear system with time delays:

where denotes the state variable and , . , , , and are given constant matrices of appropriate dimensions. The delays and are interval time-varying bounded continuous functions satisfying

where , , , and are given positive real constants. The uncertainties , , and represent the nonlinear parameter perturbations with respect to the current state , the delayed state , and , respectively. They satisfy that , , , and

where , , and are given positive real constants. The uncertain matrices , , and are norm bounded and can be described as

where , , , and are given constant matrices of appropriate dimensions. The class of parametric uncertainties , which satisfies

is said to be admissible where is a known matrix satisfying

and is uncertain matrix satisfying

Hence, , , and . is the initial function with the norm . We assume that the solution of the impulsive switched system (1) is right continuous; that is, . is an impulsive switching time point and as , and we introduce the quantity

This is called the dwell time of system (1). Under the switching law of system (1), at the time , the system switches to the subsystem from the subsystem.

Definition 1. Given , system (1) is robustly exponentially stable, if there exist switching function and positive real constant such that any solution of the system satisfies

Lemma 2 (see [29] (Halanay lemma)). Let be a positive scalar function and assume that the following condition holds:

where , Then, there exists such that, for all ,

Here, and satisfies

Lemma 3 (see [32] (Schur complement lemma)). Given constant symmetric matrices where , then if and only if

Lemma 4 (see [33]). For given matrices of appropriate dimension, then

for all satisfies , if and only if there exists a positive number , such that

Lemma 5 (see [34]). Suppose that is given by (5)–(7). Let , , and be real matrices of appropriate dimensions with . Then, the inequality

holds if and only if, for any scalar ,

Lemma 6. Let be given matrices as in (1). Let be symmetric positive definite matrix. Then,

if and only if

for are positive real constants, .

Proof. Consider inequality (17); we have

Equivalently,

By using Lemma 3 (Schur complement lemma) in the above inequality, we get

Premultiplying (21) by and postmultiplying by , we obtain the result. The proof of the lemma is complete.

Remark 7. Conditions (6) and (7) guarantee that is invertible. It is easy to show that when , the parametric uncertainty of linear fractional form reduces to a norm-bounded one.

The objectives of this paper are (i) to establish new delay-range-dependent sufficient conditions for exponential stability of nominal system (1) and (ii) to establish new delay-range-dependent sufficient conditions for robust exponential stability of system (1).

3. Main Results

In this section, we first present the exponential stability criteria with delays dependence for nominal system (1) via LMI approach. Rewrite the nominal system (1) in the following descriptor system:

Let us decompose the constant matrices and as

where , , , and are given real constant matrices with appropriate dimensions. By Leibniz-Newton formula, we have

By utilizing the following zero equations, we get

where and are real constant matrices with appropriate dimensions which will be chosen to guarantee the exponential stability of the nominal system (1). By (23)–(25), system (22) can be represented by the form

We now introduce the following notations for later use:

where , ,

Theorem 8. The nominal system (1) is exponentially stable if there exist symmetric positive definite matrix , any appropriate dimensional matrices , , and , , and positive real constants , , , , , , , , , , and with and for all such that the following LMIs hold:

where , , and is the unique positive root of the equation

Proof. Consider a common Lyapunov functional

for and a symmetric positive definite matrix . It is easy to see that

where and The Dini derivative of along the trajectories of system (26) is given by

for any appropriate dimensional matrices Since

for positive real constants , , and , we obtain

where

From (29) and (37), we obtain

By (39) and Lemma 2 with for , we obtain that there exists such that, for all , ,

where Consider the case when In this case, we have

By (30) and (34) and Lemma 6, we get

where For , with , we will show that

where We can prove inequality (43) by mathematical induction. Indeed, when , we have

Since , we have

From (40) and (45), we obtain

Therefore, (43) holds for .
Next, we assume that (43) holds for , . Then, we need to show that (43) holds when . By the above induction assumption, (32), (40), and (43), we have

Hence, it follows from (40) and (47) that

Therefore, (43) holds for all . By (31), we get that , which implies

for . We get

Finally, we conclude that

where , This means that the nominal system (1) is exponentially stable. The proof of the theorem is complete.

Next, we now present the new delay-range-dependent robust exponential stability criteria for system (1). We introduce the following notations for later use:

Theorem 9. System (1) is robustly exponentially stable if there exist symmetric positive definite matrix , any appropriate dimensional matrices , , and , , and positive real constants , , , , , , , , , ,   , and with and for all such that the following LMIs hold:

where , , and is the unique positive root of the equation

Proof. Replacing , , and in (29) with , , and , respectively, we find that

By Lemma 5, we can find that (53) is equivalent to (57) where is positive real constant. The proof of the theorem is complete.

4. Numerical Example

Example 1. Consider the following uncertain impulsive switched linear system with mixed interval time-varying delays and nonlinear perturbations (1) under a given switching law. That is, the switching status alternates as We consider robust exponential stability performance of system (1) by using Theorem 9. System (1) is specified as follows:

Decompose the matrices , , , and , where

It is easy to see that , , , , , , and . By using LMI Toolbox in MATLAB, we use (29)–(32) in Theorem 9. This example shows that the solutions of LMIs are given as follows:

, , , , , and The numerical solutions and of system (1) with , , are plotted in Figure 1. This shows that those solutions converge to zero.

Figure 1: The numerical solutions and are presented for system (1) in Example 1.

5. Conclusions

We have presented the problem of robust exponential stability criteria for uncertain impulsive switched linear systems with mixed interval nondifferentiable time-varying delays and nonlinear perturbations. By using a common Lyapunov functional, mixed model transformation, Halanay inequality, utilization of zero equations, and LMI approach, new delay-range-dependent robust exponential stability criteria for the systems are established in terms of LMIs. Finally, the theoretical result is illustrated well with a simulation example.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by DPST Research (Grant no. 001/2557), Thailand Research Fund (TRF), Khon Kaen University (Grant no. TRG5780069), and Khon Kaen University under Incubation Researcher Project and the authors would like to thank the anonymous referees for their valuable comments.

References

Y. H. Xia, Z. Yang, and M. Han, “Lag synchronization of unknown chaotic delayed Yang-Yang-type fuzzy neural networks with noise perturbation based on adaptive control and parameter identification,” IEEE Transactions on Neural Networks, vol. 20, no. 7, pp. 1165–1180, 2009.

Y. H. Xia, Z. Yang, and M. Han, “Synchronization schemes for coupled identical Yang-Yang type fuzzy cellular neural networks,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 9-10, pp. 3645–3659, 2009.

Y. Ariba, F. Gouaisbaut, and K. H. Johansson, “Stability interval for time-varying delay systems,” in Proceedings of the 49th IEEE Conference on Decision and Control (CDC '10), pp. 1017–1022, IEEE, Atlanta, Ga, USA, December 2010.

Y. He, Q.-G. Wang, C. Lin, and M. Wu, “Delay-range-dependent stability for systems with time-varying delay,” Automatica, vol. 43, no. 2, pp. 371–376, 2007.

X. Jiang and Q.-L. Han, “On H ∞ control for linear systems with interval time-varying delay,” Automatica, vol. 41, no. 12, pp. 2099–2106, 2005.

J. Sun, G. P. Liu, J. Chen, and D. Rees, “Improved delay-range-dependent stability criteria for linear systems with time-varying delays,” Automatica, vol. 46, no. 2, pp. 466–470, 2010.

Y. Chen, A. Xue, R. Lu, and S. Zhou, “On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 68, no. 8, pp. 2464–2470, 2008.

J. Cheng, H. Zhu, S. Zhong, and G. Li, “Novel delay-dependent robust stability criteria for neutral systems with mixed time-varying delays and nonlinear perturbations,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7741–7753, 2013.

O. M. Kwon, J. H. Park, and S. M. Lee, “On robust stability criterion for dynamic systems with time-varying delays and nonlinear perturbations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 937–942, 2008.

P. T. Nam, “Exponential stability criterion for time-delay systems with nonlinear uncertainties,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 374–380, 2009.

P. Niamsup, K. Mukdasai, and V. N. Phat, “Improved exponential stability for time-varying systems with nonlinear delayed perturbations,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 490–495, 2008.

J. H. Park, “Novel robust stability criterion for a class of neutral systems with mixed delays and nonlinear perturbations,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 413–421, 2005.

K. Mukdasai and S. Pinjai, “New robust exponential stability criterion for uncertain neutral systems with discrete and distributed time-varying delays and nonlinear perturbations,” Abstract and Applied Analysis, vol. 2011, Article ID 463603, 16 pages, 2011.

W. Zhang, X.-S. Cai, and Z.-Z. Han, “Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 174–180, 2010.

T. Botmart and P. Niamsupa, “Delay-dependent robust stability criteria for linear systems with interval time-varying delays and nonlinear perturbations,” Advances in Nonlinear Variational Inequalities, vol. 15, no. 1, pp. 13–30, 2012.

J. Tian, L. Xiong, J. Liu, and X. Xie, “Novel delay-dependent robust stability criteria for uncertain neutral systems with time-varying delay,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1858–1866, 2009.

F. Gao, S. Zhong, and X. Gao, “Delay-dependent stability of a type of linear switching systems with discrete and distributed time delays,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 24–39, 2008.

L. Gao, D. Wang, and G. Wang, “Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects,” Applied Mathematics and Computation, vol. 268, pp. 186–200, 2015.

D. J. Hill, Z.-H. Guan, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,” IEEE Transactions on Automatic Control, vol. 50, no. 7, pp. 1058–1062, 2005.

B. Liu, X. Liu, and X. Liao, “Stability and robustness of quasi-linear impulsive hybrid systems,” Journal of Mathematical Analysis and Applications, vol. 283, no. 2, pp. 416–430, 2003.

B. Wang, P. Shi, J. Wang, and Y. Song, “Novel LMI-based stability and stabilization analysis on impulsive switched system with time delays,” Journal of the Franklin Institute, vol. 349, no. 8, pp. 2650–2663, 2012.

Y. Wang, X. Shi, Z. Zuo, M. Z. Q. Chen, and Y. Shao, “On finite-time stability for nonlinear impulsive switched systems,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 807–814, 2013.

H. Xu and K. L. Teo, “Robust stabilization of uncertain impulsive switched systems with delayed control,” Computers and Mathematics with Applications, vol. 56, no. 1, pp. 63–70, 2008.

H. Xu, X. Liu, and K. L. Teo, “A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 1, pp. 38–50, 2008.

H. Xu, X. Liu, and K. L. Teo, “Delay independent stability criteria of impulsive switched systems with time-invariant delays,” Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 372–379, 2008.

C. Yang and W. Zhu, “Stability analysis of impulsive switched systems with time delays,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1188–1194, 2009.

W. Zhu, “Stability analysis of switched impulsive systems with time delays,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 3, pp. 608–617, 2010.

G. Zong, S. Xu, and Y. Wu, “Robust stabilization for uncertain switched impulsive control systems with state delay: an LMI approach,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 4, pp. 1287–1300, 2008.

B. Liu, G. Chen, K. L. Teo, and X. Liu, “Robust global exponential synchronization of general Lur'e chaotic systems subject to impulsive disturbances and time delays,” Chaos, Solitons and Fractals, vol. 23, no. 5, pp. 1629–1641, 2005.

W. Wang, “A generalized Halanay inequality for stability of nonlinear neutral functional differential equations,” Journal of Inequalities and Applications, vol. 2010, Article ID 475019, 2010.

L. Wen, Y. Yu, and W. Wang, “Generalized Halanay inequalities for dissipativity of Volterra functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 347, no. 1, pp. 169–178, 2008.

S. Boyd, L. E. Ghaou, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Control Theory, Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994.

Y. Wang, L. Xie, and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992.

T. Li, L. Guo, and C. Lin, “A new criterion of delay-dependent stability for uncertain time-delay systems,” IET Control Theory & Applications, vol. 1, no. 3, pp. 611–616, 2007.