Uniform Stability of Discrete Delay Systems and ... - Semantic Scholar
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008
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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 approach 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 explicitly consider the case of networks with complex structures, such as networks with chaotic nodes, which have great practical importance in the area of secure communications. Index Terms—Discrete delay dynamical networks, discrete delay system, global synchronization, impulsive synchronization, Razumikhin technique, stability.
I. INTRODUCTION
T
IME delays occur frequently in many physical systems and control schemes. Delays occur for a variety of reasons, 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 published 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 recommended 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: [email protected]). H. J. Marquez is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]). 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 systems. See [1]–[3], [5], [6], and the references therein. References [7]–[11] study right-continuous impulsive delay systems using the Razumikhin technique. Razumikhin-type stability theorems 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 function. Unlike the case of continuous systems and right-continuous 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 stability 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 stability 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 theorem 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 communications, 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 factors that may cause the failure of the synchronization scheme. The main issues are: 1) uncertainties in the network (for example, 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 dynamical networks is studied using adaptive control and impulsive
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control methods, respectively. Time delays occur commonly in synchronization schemes. [21], and [32]–[34] analyze synchronization of continuous-time delay networks but restrict attention to the case of a single time-delay known a priori. These assumptions are restrictive because time delay present in a typical synchronization scheme is usually not known a priori and, moreover, there may by multiple time-varying time delays. In this paper, we study global synchronization of discrete dynamical networks with coupling time delays. Applying the Razumikhin-type global uniform exponential stability theorem for discrete delay systems established in Section III, we derive several criteria under which global uniform exponential synchronization is achieved. When compared to other previously published results, our solution has the advantage that the synchronization speed with respect to a given admissible error bound can be easily estimated. The remainder of this paper is organized as follows: In Section II, we introduce our notation and provide several definitions that will be used in later sections. In Section III, we establish a Razumikhin-type global uniform exponential stability theorem for discrete delay systems. In Section IV, the results of Section III are applied to the global synchronization of discrete delay dynamical networks. Finally, in Section V, we present several examples to illustrate our results.
is defined by for any . , , so that (1) admits the trivial We assume solution, and is an equilibrium point for the system (1). We also assume that system (1) has a unique solution, denoted , for any given initial data: and by . Remark 2.1: In [4], the function needs to satisfy: , for some positive constant and . In this paper, this assumption is not necessary. any Hence, the discrete delay system considered in this paper is more general than that in [4]. of system (1) Definition 2.1: The equilibrium point is said to be global uniform exponential stable (GUES) if, for , , there exist two positive any initial condition numbers , , where both and are independent and , such that of
and
(2) Remark 2.2: Obviously, if (2) holds, then, the Lyapunov ex. ponent (LE) of system (1) is less than of Remark 2.3: For simplicity, if the equilibrium point the system (1) is GUES, the system (1) is also said to be GUES.
II. PRELIMINARIES
III. RAZUMIKHIN-TYPE THEOREM FOR DISCRETE DELAY SYSTEMS
denote the -dimensional Euclidean space. Let , and , and . Let for some positive integer , let denote the maximum (minimum) eigenthe norm of induced value of a symmetric matrix , and . by the Euclidean vector norm, i.e., denote the function class: Let is a strict increasing function satisfying . , let For a given positive real number
In this section, we consider the discrete delay system (1) and derive Razumikhin-type stability theorems for global uniform exponential stability. Theorem 3.1 (Razumikhin-Type Global Exponential Stability , Theorem): The delay system (1) is GUES with and constants if there exist a positive definite function , , , , , such that: and , 1) for any
Let
(3) is continuous. be a Banach space with topological structure: . to , The restriction of where
2) for any then
(4) 3) for some
(1) , .
,
i.e.,
, will be denoted . , we equip the linear Given a positive integer with norm defined by space . Similarly, we deand its restriction fine . Consider the discrete delay system of the form
where
, if
, if , then (5)
. where be a solution of system (1). Proof: Let . Define Without loss of the generality, let (6) We now prove that
, , represents the delay in the system (1),
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(7)
LIU AND MARQUEZ: UNIFORM STABILITY OF DISCRETE DELAY SYSTEMS
For every fixed
, we define
It follows by the definition of defined and
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which implies that
that
(8) is well-
(9)
(19) Thus, by condition 2), we have (20)
, then, for any
If
From (20) (10) which implies that
(21) and from (18) that, for (11)
We claim that (22) (12) By the definition of
Hence
, we have (23) (13) Therefore, (7) holds and it leads to that, for any
which implies that
(24) (14)
Thus, by the definition of
and (24), we have that
From (14) and condition 3) and (25) From (25) and condition 1) (15)
(26)
Thus, by (15), we have
(16)
Remark 3.1: An important feature of Theorem 3.1 is that the conditions for global uniform exponential stability can be easily tested. We also emphasize that, for systems with stringent convergence requirements such as synchronization of network controlled systems [12], exponential stability is more significant than stability or asymptotical stability. We now apply the Razumikhin-type Theorem 3.1 to a class of discrete delay systems. Consider the discrete delay systems of the form
Hence, (12) holds. From (11) and (12) (17) If
, then for any (18)
(27) where , and , , and . for any Corollary 3.1: Assume that condition 1) of Theorem 3.1 holds, while conditions 2)–3) are replaced by the following : condition
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there exists a positive constant .
, such that
Since 3.1 holds.
, the condition
of Corollary
where IV. SYNCHRONIZATION OF DISCRETE DELAY DYNAMICAL NETWORKS
. . Then, the system (27) is GUES with Proof: We only need to prove that the conditions 2)–3) of , then, from Theorem 3.1 still hold. Let , we get that , and . , if Thus, for any
then, by condition
, we have
It follows from the fact 3.1 is satisfied. Let , if
then, by condition
In this section, we consider synchronization of discrete delay dynamical networks. Using the Razumikhin-type stability Theorem 3.1, we derive conditions for global uniform exponential synchronization of these type of networks. Moreover, we also consider a class of dynamical networks with complex structure, such as chaotic nodes, and derive conditions for global uniform synchronization of these networks using the concept of impulsive synchronization and the Razumikhin technique. Consider a discrete network consisting of identical nodes dimensional discrete systems) and corresponding network ( coupling delays
that the condition 2) of Theorem , for some (31) , is a smooth nonwhere are smooth but unlinear vector functions, known network coupling functions, where , and for , is the coupling time delay funceach tion which represents the delay of the signal transmitted from th node to other nodes with for any , . Clearly, each node of the network has the following form
that
(32)
(28) Hence, condition 3) of Theorem 3.1 also holds. Corollary 3.2: Assume that condition 1) of Theorem 3.1 hold, while conditions 2)–3) of Theorem 3.1 are replaced by : the following condition There exist positive constants , , , such that
The solution of (32) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Remark 4.1: When the network achieves synchronization, , as namely, the state , the coupling terms should vanish: i.e., . Defining the synchronization error as , the error dynamics has the form
(33) (29) If
, then the system (27) is GUES with , where . Proof: From (29), we get that for any (30)
where
,
, and . Definition 4.1: The discrete delay network (31) is said to achieve global uniform synchronization if the error system (33) is global uniform asymptotically stable (GUAS). Moreover, the discrete delay network (31) is said to achieve global uniform exponential synchronization with convergence exponent more . than if the error system (33) is GUES with
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LIU AND MARQUEZ: UNIFORM STABILITY OF DISCRETE DELAY SYSTEMS
Assumption 4.1: There exists a positive constant such that
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with
(39) (34)
(35)
, , . By Corollary 3.2, the discrete delay network (31) achieves global uniform exponential synchronization , where with convergence exponent . By (25) and (26) in Theorem 3.1, we have
Then, the discrete delay network (31) achieves global uniform exponential synchronization with convergence exponent , where satisfies
(40)
Theorem 4.1: Suppose that Assumption 4.1 holds and the coupling is linear, i.e., . Assume that there exist positive consuch that stants ,
(36) Moreover, for a given admissible error bound satisfy the following condition such that
,
must for all
where
, If (37) holds, then, it is easy to show that , for a given admissible error bound . Theorem 4.2: Suppose that Assumption 4.1 holds and the coupling is linear, i.e., . Assume that there are positive definite and positive numbers , , , matrices , such that 1)
(37) (41) stands for the minimal integer not less than , , , and is the initial condition with , , . , where Proof: Let , . Then, by (33), for any
where
2) the following LMIs hold: (42)
where ,
, , and
satisfying (43)
(38) which implies that
The discrete delay network (31) achieves global uniform exponential synchronization with convergence ex, where satisfies: ponent . For a given admissible error bound , for all , if satisfies (44) where , is the initial condition with .
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,
, and ,
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Proof: Let , then, we have
, where
(48) where ,
, and .
It follows from (48) and condition 2) that
(45) From condition 1) and Assumption 4.1, we obtain which implies that for any
(49)
(46) and (50) and , . By Corollary 3.2, the discrete delay network (31) achieves global uniform exponential synchronization with . Moreover, similar to convergence exponent Theorem 4.1, for a given admissible error bound , if (44) for all . holds, then Theorem 4.2 assumes that the coupling functions in the network 3.1 are linear. Theorem 4.3, given next, generalizes this result to the case nonlinear couplings. Theorem 4.3: Suppose that Assumption 4.1 and the following conditions hold: such that for 1) there exist nonnegative constants , Let
(47) Substituting (46)–(47) into (45) gives
(51) 2) there are positive constants such that
,
(52)
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Theorem 4.4: Assume that Assumption 4.1 holds and . Suppose that there exists a positive constant that the following conditions are satisfied:
such
(55) (56) (57)
Fig. 1. Impulsive synchronization control for the ith node S .
Under these conditions, the discrete delay network (31) achieves global uniform exponential synchronization with convergence , where satisfies exponent
satisfying . where The discrete delay network (31) achieves global uniform syn. chronization under the impulsive controller , where Proof: Let , then, for any , , we have, for
(53) (58) , if (37) holds, where For a given admissible error bound satisfies (53), then, for all . Proof: The proof follows immediately using an argument analogous to Theorem 4.1 and Corollary 3.1. The details are omitted. Remark 4.2: It should be noted that condition (35) in Theorem 4.1 and condition 2) in Theorem 4.3 are very stringent. Indeed, in many dynamical networks, such as the networks in which the discrete chaotic system is used as the nodes, we have and with . that Hence, the results of Theorems 4.1–4.3 cannot be applied. A natural approach is then to consider the use of linear feedback in the th node so to achieve the syncontrol chronization. Unfortunately, when , the gain of still satisfies even if . In order to solve in this problem, we include an impulsive controller the synchronization scheme, as shown in Fig. 1. Here, the error system (33) can be rewritten as
By (58), it yields that, for any
(59) For any
, if for any , , it then follows from (59) that
(54) . The sequence where satisfies 1) , with 2) for all . The impulsive synchronization scheme subject to network stands for th node, coupling is shown in Fig. 1, where is the isolate node (32) and is the delay network coupling of th node, .
,
(60)
When
, by (55)
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(61)
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which leads to
,
where
, and
,
. (62) Hence, by the Razumikhin-type stability theorem for discrete impulsive hybrid systems with time-delays ([37, Theorem 4.4]), we obtain that the error system (54) is GUAS and, hence the discrete delay network (31) achieves global uniform synchroniza. tion under the impulsive controller V. EXAMPLES
, . The system (64) is uniformly exponentially stable if Let
(65) Moreover,
, where
. If the Lyapunov function is
, then
In this section, we consider several examples that illustrate the results of previous sections. 1) Example 5.1: Consider the nonlinear discrete delay system (63) where
,
,
, or 2,
, and
Let
. , then
,
(66) Hence, by Corollary 3.1, the system (64) is GUES with . Remark 5.1: Example 5.2 is discussed in [4], where the objective is to establish uniformly asymptotic stability. Unlike [4], our focus is on global uniform exponential stability. It is immediate, of course, that global uniform exponential stability guarantees uniformly asymptotic stability. Notice, however, the simplicity of our approach when compared to the rather involved method reported in [4]. 3) Example 5.3: Consider the entire discrete delay network , . (31), where , satisfy the equation The functions , , . shown at the bottom of the next page for for , . ,
Then,
.
which implies that the conditions of Corollary 3.2 hold. Thus, by Corollary 3.3, the discrete delay system (63) is GUES with . The numerical simulation of this example is given in Fig. 2. 2) Example 5.2: Consider the discrete delay system, discussed in [4]
(64)
It is easy to show that and
,
,
. Then
By Corollary 3.1, this discrete delay network achieves global uniform exponential synchronization with convergence expo. nent Let . Figs. 3–5 show that trajectories of the error system for this dynamical network exponentially approach
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LIU AND MARQUEZ: UNIFORM STABILITY OF DISCRETE DELAY SYSTEMS
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Fig. 2. GUES property of system (63).
Fig. 4. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
Fig. 3. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
Fig. 5. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
the origin. Moreover, if the given admissible error bound , then, when
where
, and the coupling functions
,
, satisfy
and synchronization errors satisfy: for all , where . 4) Example 5.4: Consider the discrete delay network (31) and assume that the nodes of the network can be modeled using the chaotic Hénon’s map. A single chaotic Hénon’s map is in form of
. and
Let , that
. It is easy to show , i.e., , and
,
(67) where
,
, and
where
, and when
,
. The network is
Let control law GUAS. If (68)
. By Theorem 4.4, we can design an impulsive such that the error system is ,
, then the
conditions of Theorem 4.4 hold. The impulsive controller
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REFERENCES
Fig. 6. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
Fig. 7. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
achieves global uniform synchronization for this discrete dynamical network as illustrated by the simulations shown in Figs. 6 and 7. VI. CONCLUSION In this paper, we have derived global uniform exponential stability for discrete delay systems based on the Razumikhin technique and Lyapunov theory. Our conditions can be easily tested and should prove useful in practical applications. The stability results obtained in Section III were employed to derive several criteria for global uniform synchronization of discrete dynamical networks with coupling time delays. In particular, we explicitly considered the case of networks with complex structures, such as networks with chaotic nodes, which have great practical importance in the area of secure communications. We also estimated the convergence speed, a useful parameter that can be used to calculate the synchronization time for a given error bound. Several examples were presented illustrating the efficiency of our results. ACKNOWLEDGMENT The authors would like to thank the anonymous referees for helpful comments and suggestions.
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Bin Liu received the M.Sc. degree from the Department of Mathematics, East China Normal University, Shanghai, China, in 1993, and the Ph.D. degree from the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China, in 2003, respectively. He was a Postdoctoral Fellow at the Huazhong University of Science and Technology from July 2003 to July 2005, a Postdoctoral Fellow at the University of Alberta, Edmonton, AB, Canada, from August 2005 to October 2006, and a visiting Research Fellow at the Hong Kong Polytechnic University, Hong Kong, China, in 2004. Since July 1993, he has been with the Department of Information and Computation Science, Hunan University of Technology, Hunan, China, where he became an Associate Professor in 2001, and a Professor in 2004. He is currently a Research Fellow in Department of Information Engineering, The Australian National University, ACT, Australia. He is an editor of The Journal of the Franklin Institute and an associate editor of Dynamics of Continuous, Discrete and Impulsive Systems, Series B. His research interests include stability analysis and applications of nonlinear systems and hybrid systems, optimal control and stability, chaos and network synchronization and control, and Lie algebra. Dr. Liu is the recipient of the prestigious Queen Elizabeth II Fellowship from the Australian Research Council in 2007.
Horacio J. Marquez received the B.Sc. degree from the Instituto Tecnologico de Buenos Aires, Buenos Aires, Argentina, and the M.Sc.E. and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, in 1987, 1990, and 1993, respectively. From 1993 to 1996 he held visiting appointments at the Royal Roads Military College, and the University of Victoria, Victoria, BC, Canada. Since 1996, he has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, where he is currently a Professor and Department Chair. He is currently an Area Editor for the International Journal of Robust and Nonlinear Control. He is the Author of “Nonlinear Control Systems: Analysis and Design” (Wiley, 2003). His current research interests include nonlinear dynamical systems and control, nonlinear observer design, robust control, and applications. Dr. Marquez is the recipient of the 2003-2004 McCalla Research Professorship awarded by the University of Alberta.
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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 approach 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 explicitly consider the case of networks with complex structures, such as networks with chaotic nodes, which have great practical importance in the area of secure communications. Index Terms—Discrete delay dynamical networks, discrete delay system, global synchronization, impulsive synchronization, Razumikhin technique, stability.
I. INTRODUCTION
T
IME delays occur frequently in many physical systems and control schemes. Delays occur for a variety of reasons, 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 published 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 recommended 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: [email protected]). H. J. Marquez is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]). 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 systems. See [1]–[3], [5], [6], and the references therein. References [7]–[11] study right-continuous impulsive delay systems using the Razumikhin technique. Razumikhin-type stability theorems 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 function. Unlike the case of continuous systems and right-continuous 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 stability 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 stability 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 theorem 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 communications, 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 factors that may cause the failure of the synchronization scheme. The main issues are: 1) uncertainties in the network (for example, 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 dynamical networks is studied using adaptive control and impulsive
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control methods, respectively. Time delays occur commonly in synchronization schemes. [21], and [32]–[34] analyze synchronization of continuous-time delay networks but restrict attention to the case of a single time-delay known a priori. These assumptions are restrictive because time delay present in a typical synchronization scheme is usually not known a priori and, moreover, there may by multiple time-varying time delays. In this paper, we study global synchronization of discrete dynamical networks with coupling time delays. Applying the Razumikhin-type global uniform exponential stability theorem for discrete delay systems established in Section III, we derive several criteria under which global uniform exponential synchronization is achieved. When compared to other previously published results, our solution has the advantage that the synchronization speed with respect to a given admissible error bound can be easily estimated. The remainder of this paper is organized as follows: In Section II, we introduce our notation and provide several definitions that will be used in later sections. In Section III, we establish a Razumikhin-type global uniform exponential stability theorem for discrete delay systems. In Section IV, the results of Section III are applied to the global synchronization of discrete delay dynamical networks. Finally, in Section V, we present several examples to illustrate our results.
is defined by for any . , , so that (1) admits the trivial We assume solution, and is an equilibrium point for the system (1). We also assume that system (1) has a unique solution, denoted , for any given initial data: and by . Remark 2.1: In [4], the function needs to satisfy: , for some positive constant and . In this paper, this assumption is not necessary. any Hence, the discrete delay system considered in this paper is more general than that in [4]. of system (1) Definition 2.1: The equilibrium point is said to be global uniform exponential stable (GUES) if, for , , there exist two positive any initial condition numbers , , where both and are independent and , such that of
and
(2) Remark 2.2: Obviously, if (2) holds, then, the Lyapunov ex. ponent (LE) of system (1) is less than of Remark 2.3: For simplicity, if the equilibrium point the system (1) is GUES, the system (1) is also said to be GUES.
II. PRELIMINARIES
III. RAZUMIKHIN-TYPE THEOREM FOR DISCRETE DELAY SYSTEMS
denote the -dimensional Euclidean space. Let , and , and . Let for some positive integer , let denote the maximum (minimum) eigenthe norm of induced value of a symmetric matrix , and . by the Euclidean vector norm, i.e., denote the function class: Let is a strict increasing function satisfying . , let For a given positive real number
In this section, we consider the discrete delay system (1) and derive Razumikhin-type stability theorems for global uniform exponential stability. Theorem 3.1 (Razumikhin-Type Global Exponential Stability , Theorem): The delay system (1) is GUES with and constants if there exist a positive definite function , , , , , such that: and , 1) for any
Let
(3) is continuous. be a Banach space with topological structure: . to , The restriction of where
2) for any then
(4) 3) for some
(1) , .
,
i.e.,
, will be denoted . , we equip the linear Given a positive integer with norm defined by space . Similarly, we deand its restriction fine . Consider the discrete delay system of the form
where
, if
, if , then (5)
. where be a solution of system (1). Proof: Let . Define Without loss of the generality, let (6) We now prove that
, , represents the delay in the system (1),
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(7)
LIU AND MARQUEZ: UNIFORM STABILITY OF DISCRETE DELAY SYSTEMS
For every fixed
, we define
It follows by the definition of defined and
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which implies that
that
(8) is well-
(9)
(19) Thus, by condition 2), we have (20)
, then, for any
If
From (20) (10) which implies that
(21) and from (18) that, for (11)
We claim that (22) (12) By the definition of
Hence
, we have (23) (13) Therefore, (7) holds and it leads to that, for any
which implies that
(24) (14)
Thus, by the definition of
and (24), we have that
From (14) and condition 3) and (25) From (25) and condition 1) (15)
(26)
Thus, by (15), we have
(16)
Remark 3.1: An important feature of Theorem 3.1 is that the conditions for global uniform exponential stability can be easily tested. We also emphasize that, for systems with stringent convergence requirements such as synchronization of network controlled systems [12], exponential stability is more significant than stability or asymptotical stability. We now apply the Razumikhin-type Theorem 3.1 to a class of discrete delay systems. Consider the discrete delay systems of the form
Hence, (12) holds. From (11) and (12) (17) If
, then for any (18)
(27) where , and , , and . for any Corollary 3.1: Assume that condition 1) of Theorem 3.1 holds, while conditions 2)–3) are replaced by the following : condition
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there exists a positive constant .
, such that
Since 3.1 holds.
, the condition
of Corollary
where IV. SYNCHRONIZATION OF DISCRETE DELAY DYNAMICAL NETWORKS
. . Then, the system (27) is GUES with Proof: We only need to prove that the conditions 2)–3) of , then, from Theorem 3.1 still hold. Let , we get that , and . , if Thus, for any
then, by condition
, we have
It follows from the fact 3.1 is satisfied. Let , if
then, by condition
In this section, we consider synchronization of discrete delay dynamical networks. Using the Razumikhin-type stability Theorem 3.1, we derive conditions for global uniform exponential synchronization of these type of networks. Moreover, we also consider a class of dynamical networks with complex structure, such as chaotic nodes, and derive conditions for global uniform synchronization of these networks using the concept of impulsive synchronization and the Razumikhin technique. Consider a discrete network consisting of identical nodes dimensional discrete systems) and corresponding network ( coupling delays
that the condition 2) of Theorem , for some (31) , is a smooth nonwhere are smooth but unlinear vector functions, known network coupling functions, where , and for , is the coupling time delay funceach tion which represents the delay of the signal transmitted from th node to other nodes with for any , . Clearly, each node of the network has the following form
that
(32)
(28) Hence, condition 3) of Theorem 3.1 also holds. Corollary 3.2: Assume that condition 1) of Theorem 3.1 hold, while conditions 2)–3) of Theorem 3.1 are replaced by : the following condition There exist positive constants , , , such that
The solution of (32) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Remark 4.1: When the network achieves synchronization, , as namely, the state , the coupling terms should vanish: i.e., . Defining the synchronization error as , the error dynamics has the form
(33) (29) If
, then the system (27) is GUES with , where . Proof: From (29), we get that for any (30)
where
,
, and . Definition 4.1: The discrete delay network (31) is said to achieve global uniform synchronization if the error system (33) is global uniform asymptotically stable (GUAS). Moreover, the discrete delay network (31) is said to achieve global uniform exponential synchronization with convergence exponent more . than if the error system (33) is GUES with
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LIU AND MARQUEZ: UNIFORM STABILITY OF DISCRETE DELAY SYSTEMS
Assumption 4.1: There exists a positive constant such that
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with
(39) (34)
(35)
, , . By Corollary 3.2, the discrete delay network (31) achieves global uniform exponential synchronization , where with convergence exponent . By (25) and (26) in Theorem 3.1, we have
Then, the discrete delay network (31) achieves global uniform exponential synchronization with convergence exponent , where satisfies
(40)
Theorem 4.1: Suppose that Assumption 4.1 holds and the coupling is linear, i.e., . Assume that there exist positive consuch that stants ,
(36) Moreover, for a given admissible error bound satisfy the following condition such that
,
must for all
where
, If (37) holds, then, it is easy to show that , for a given admissible error bound . Theorem 4.2: Suppose that Assumption 4.1 holds and the coupling is linear, i.e., . Assume that there are positive definite and positive numbers , , , matrices , such that 1)
(37) (41) stands for the minimal integer not less than , , , and is the initial condition with , , . , where Proof: Let , . Then, by (33), for any
where
2) the following LMIs hold: (42)
where ,
, , and
satisfying (43)
(38) which implies that
The discrete delay network (31) achieves global uniform exponential synchronization with convergence ex, where satisfies: ponent . For a given admissible error bound , for all , if satisfies (44) where , is the initial condition with .
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,
, and ,
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Proof: Let , then, we have
, where
(48) where ,
, and .
It follows from (48) and condition 2) that
(45) From condition 1) and Assumption 4.1, we obtain which implies that for any
(49)
(46) and (50) and , . By Corollary 3.2, the discrete delay network (31) achieves global uniform exponential synchronization with . Moreover, similar to convergence exponent Theorem 4.1, for a given admissible error bound , if (44) for all . holds, then Theorem 4.2 assumes that the coupling functions in the network 3.1 are linear. Theorem 4.3, given next, generalizes this result to the case nonlinear couplings. Theorem 4.3: Suppose that Assumption 4.1 and the following conditions hold: such that for 1) there exist nonnegative constants , Let
(47) Substituting (46)–(47) into (45) gives
(51) 2) there are positive constants such that
,
(52)
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Theorem 4.4: Assume that Assumption 4.1 holds and . Suppose that there exists a positive constant that the following conditions are satisfied:
such
(55) (56) (57)
Fig. 1. Impulsive synchronization control for the ith node S .
Under these conditions, the discrete delay network (31) achieves global uniform exponential synchronization with convergence , where satisfies exponent
satisfying . where The discrete delay network (31) achieves global uniform syn. chronization under the impulsive controller , where Proof: Let , then, for any , , we have, for
(53) (58) , if (37) holds, where For a given admissible error bound satisfies (53), then, for all . Proof: The proof follows immediately using an argument analogous to Theorem 4.1 and Corollary 3.1. The details are omitted. Remark 4.2: It should be noted that condition (35) in Theorem 4.1 and condition 2) in Theorem 4.3 are very stringent. Indeed, in many dynamical networks, such as the networks in which the discrete chaotic system is used as the nodes, we have and with . that Hence, the results of Theorems 4.1–4.3 cannot be applied. A natural approach is then to consider the use of linear feedback in the th node so to achieve the syncontrol chronization. Unfortunately, when , the gain of still satisfies even if . In order to solve in this problem, we include an impulsive controller the synchronization scheme, as shown in Fig. 1. Here, the error system (33) can be rewritten as
By (58), it yields that, for any
(59) For any
, if for any , , it then follows from (59) that
(54) . The sequence where satisfies 1) , with 2) for all . The impulsive synchronization scheme subject to network stands for th node, coupling is shown in Fig. 1, where is the isolate node (32) and is the delay network coupling of th node, .
,
(60)
When
, by (55)
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(61)
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which leads to
,
where
, and
,
. (62) Hence, by the Razumikhin-type stability theorem for discrete impulsive hybrid systems with time-delays ([37, Theorem 4.4]), we obtain that the error system (54) is GUAS and, hence the discrete delay network (31) achieves global uniform synchroniza. tion under the impulsive controller V. EXAMPLES
, . The system (64) is uniformly exponentially stable if Let
(65) Moreover,
, where
. If the Lyapunov function is
, then
In this section, we consider several examples that illustrate the results of previous sections. 1) Example 5.1: Consider the nonlinear discrete delay system (63) where
,
,
, or 2,
, and
Let
. , then
,
(66) Hence, by Corollary 3.1, the system (64) is GUES with . Remark 5.1: Example 5.2 is discussed in [4], where the objective is to establish uniformly asymptotic stability. Unlike [4], our focus is on global uniform exponential stability. It is immediate, of course, that global uniform exponential stability guarantees uniformly asymptotic stability. Notice, however, the simplicity of our approach when compared to the rather involved method reported in [4]. 3) Example 5.3: Consider the entire discrete delay network , . (31), where , satisfy the equation The functions , , . shown at the bottom of the next page for for , . ,
Then,
.
which implies that the conditions of Corollary 3.2 hold. Thus, by Corollary 3.3, the discrete delay system (63) is GUES with . The numerical simulation of this example is given in Fig. 2. 2) Example 5.2: Consider the discrete delay system, discussed in [4]
(64)
It is easy to show that and
,
,
. Then
By Corollary 3.1, this discrete delay network achieves global uniform exponential synchronization with convergence expo. nent Let . Figs. 3–5 show that trajectories of the error system for this dynamical network exponentially approach
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LIU AND MARQUEZ: UNIFORM STABILITY OF DISCRETE DELAY SYSTEMS
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Fig. 2. GUES property of system (63).
Fig. 4. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
Fig. 3. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
Fig. 5. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
the origin. Moreover, if the given admissible error bound , then, when
where
, and the coupling functions
,
, satisfy
and synchronization errors satisfy: for all , where . 4) Example 5.4: Consider the discrete delay network (31) and assume that the nodes of the network can be modeled using the chaotic Hénon’s map. A single chaotic Hénon’s map is in form of
. and
Let , that
. It is easy to show , i.e., , and
,
(67) where
,
, and
where
, and when
,
. The network is
Let control law GUAS. If (68)
. By Theorem 4.4, we can design an impulsive such that the error system is ,
, then the
conditions of Theorem 4.4 hold. The impulsive controller
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REFERENCES
Fig. 6. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
Fig. 7. Synchronization errors of e (k ), i = 1; 2; . . . ; 10.
achieves global uniform synchronization for this discrete dynamical network as illustrated by the simulations shown in Figs. 6 and 7. VI. CONCLUSION In this paper, we have derived global uniform exponential stability for discrete delay systems based on the Razumikhin technique and Lyapunov theory. Our conditions can be easily tested and should prove useful in practical applications. The stability results obtained in Section III were employed to derive several criteria for global uniform synchronization of discrete dynamical networks with coupling time delays. In particular, we explicitly considered the case of networks with complex structures, such as networks with chaotic nodes, which have great practical importance in the area of secure communications. We also estimated the convergence speed, a useful parameter that can be used to calculate the synchronization time for a given error bound. Several examples were presented illustrating the efficiency of our results. ACKNOWLEDGMENT The authors would like to thank the anonymous referees for helpful comments and suggestions.
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LIU AND MARQUEZ: UNIFORM STABILITY OF DISCRETE DELAY SYSTEMS
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Bin Liu received the M.Sc. degree from the Department of Mathematics, East China Normal University, Shanghai, China, in 1993, and the Ph.D. degree from the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China, in 2003, respectively. He was a Postdoctoral Fellow at the Huazhong University of Science and Technology from July 2003 to July 2005, a Postdoctoral Fellow at the University of Alberta, Edmonton, AB, Canada, from August 2005 to October 2006, and a visiting Research Fellow at the Hong Kong Polytechnic University, Hong Kong, China, in 2004. Since July 1993, he has been with the Department of Information and Computation Science, Hunan University of Technology, Hunan, China, where he became an Associate Professor in 2001, and a Professor in 2004. He is currently a Research Fellow in Department of Information Engineering, The Australian National University, ACT, Australia. He is an editor of The Journal of the Franklin Institute and an associate editor of Dynamics of Continuous, Discrete and Impulsive Systems, Series B. His research interests include stability analysis and applications of nonlinear systems and hybrid systems, optimal control and stability, chaos and network synchronization and control, and Lie algebra. Dr. Liu is the recipient of the prestigious Queen Elizabeth II Fellowship from the Australian Research Council in 2007.
Horacio J. Marquez received the B.Sc. degree from the Instituto Tecnologico de Buenos Aires, Buenos Aires, Argentina, and the M.Sc.E. and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, in 1987, 1990, and 1993, respectively. From 1993 to 1996 he held visiting appointments at the Royal Roads Military College, and the University of Victoria, Victoria, BC, Canada. Since 1996, he has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, where he is currently a Professor and Department Chair. He is currently an Area Editor for the International Journal of Robust and Nonlinear Control. He is the Author of “Nonlinear Control Systems: Analysis and Design” (Wiley, 2003). His current research interests include nonlinear dynamical systems and control, nonlinear observer design, robust control, and applications. Dr. Marquez is the recipient of the 2003-2004 McCalla Research Professorship awarded by the University of Alberta.
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