IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 2795 Uniform Stability of Discrete Delay Systems and Synchronization of Discrete Delay Dynamical Networks via Razumikhin Technique Bin Liu and Horacio J. Marquez Abstract—In this paper, we consider discrete delay systems and obtain conditions for global uniform exponential stability. Our ap- proach is based on the use of the Razumikhin technique and the Lyapunov function method. In the second part of the paper, we use our stability results to derive exponential synchronization criteria for discrete dynamical networks with coupling time delays. We ex- plicitly consider the case of networks with complex structures, such as networks with chaotic nodes, which have great practical impor- tance in the area of secure communications. Index Terms—Discrete delay dynamical networks, discrete delay system, global synchronization, impulsive synchronization, Razu- mikhin technique, stability. I. INTRODUCTION T IME delays occur frequently in many physical systems and control schemes. Delays occur for a variety of rea- sons, including finite switching times, hardware speed, and network traffic congestion. It is, therefore, important to analyze systems with time delays. In particular, stability of systems with delay has received considerable attention over the last three decades [1], [2], [13]. Several methods have been proposed to analyze stability of time delay systems, including the Lyapunov functional method, the comparison principle and the Razumikhin technique. The Lyapunov functional method requires a Lyapunov function that decreases in the entire time space. This function is often very difficult to find, making this method applicable to a rather narrow class of systems for which such a function can be found. A similar problem is encountered using the comparison principle. This method requires finding a comparison system with known properties. Stability analysis using this method is based on the fact that, under certain conditions, stability of the comparison system implies stability of the original time-delay Manuscript received October 31, 2005; revised January 14, 2007, August 6, 2007, and January 30, 2008. First published April 18, 2008; current version pub- lished October 29, 2008. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), and the National Science Foundation of China (NSFC) under Grant 60874025. This paper was recom- mended for publication by Associate Editor L. Trajkovic. B. Liu is with the Department of Information and Computation Sciences, Hunan University of Technology, Zhuzhou 412008, China, and also with the Department of Information Engineering, The Australian National University, ACT 0200, Australia (e-mail: bin.liu@anu.edu.au). H. J. Marquez is with the Department of Electrical and Computer Engi- neering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: marquez@ece.ualberta.ca). Digital Object Identifier 10.1109/TCSI.2008.923163 system. Finding a suitable comparison system, however, can be difficult, especially in the case of nonlinear time delay systems. The Razumikhin technique [1], [2] requires also the use of a Lyapunov function but, unlike the Lyapunov functional method, this Lyapunov function is not required to decrease in the entire time space. The Razumikhin technique has been successfully applied to the study of several stability problems for continuous delay sys- tems. See [1]–[3], [5], [6], and the references therein. Refer- ences [7]–[11] study right-continuous impulsive delay systems using the Razumikhin technique. Razumikhin-type stability the- orems for continuous delay systems and right-continuous delay systems are based on the fact that the solution of these type of differential equations is a continuous or right-continuous func- tion. Unlike the case of continuous systems and right-contin- uous systems, however, the solution of a difference equation is not continuous or right-continuous, thus bringing difficulties in the use of the Razumikhin technique when investigating sta- bility of discrete delay systems. [4] considers stability of a class of discrete delay systems and reports a Razumikhin-type uniform asymptotic stability result. The main result in this reference, however, is very difficult to use making it very hard to apply in practical applications. To overcome this problem, in this paper, we proceed inspired by reference [11] and investigate Razumikhin-type exponential sta- bility criteria for general discrete delay systems. Our goal is to obtain stability results that can be easily tested. To the best of our knowledge, no Razumikhin-type exponential stability the- orem for discrete delay systems has been previously reported. In the second part of this paper, we consider an application of our results to the synchronization of chaotic systems with time delay. Given the potential application to secure commu- nications, chaotic synchronization has been an active research area for the past 15 years [14]–[22], [38]–[40], [43], [44]. More recently, synchronization of dynamical networks has also received much attention [23]–[36], [41], [42]. A dynamical network consists of coupled nodes that are usually dynamical systems. It has been reported that when a synchronization scheme is applied to a dynamical network, there are several fac- tors that may cause the failure of the synchronization scheme. The main issues are: 1) uncertainties in the network (for ex- ample, channel noise); and 2) time delays. In order to deal with these undesirable factors, robust synchronization theory has become a promising research area. In [12], [22], [28], [31], the problem of robust synchronization of uncertain dynam- ical networks is studied using adaptive control and impulsive 1549-8328/$25.00 © 2008 IEEE Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on November 24, 2008 at 11:33 from IEEE Xplore. Restrictions apply.