Lag Quasi-Synchronization of Coupled Delayed ... - IEEE Xplore

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 6, JUNE 2011

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Lag Quasi-Synchronization of Coupled Delayed Systems With Parameter Mismatch Wangli He, Member, IEEE, Feng Qian, Qing-Long Han, and Jinde Cao, Senior Member, IEEE Abstract—This paper is concerned with lag synchronization of two coupled delayed systems with parameter mismatch. Due to parameter mismatch, complete lag synchronization can not be achieved. Therefore, a new lag quasi-synchronization scheme is proposed to ensure that coupled systems are in a state of lag synchronization with an error level. Several simple criteria are derived and the error level is estimated by applying a generalized Halanary inequality and matrix measure. Three examples are given to illustrate the effectiveness of the proposed lag quasi-synchronization scheme. It is shown that as the coupling strength increases, the estimated error level is close to the simulated one, which well supports theoretical results. Index Terms—Lag quasi-synchronization, parameter mismatch, chaotic systems.

C

I. INTRODUCTION

HAOS synchronization has attracted considerable attention due to its theoretical importance and practical applications in various fields such as physics, secure communication, automatic control, and chemical and biological systems [1]. In the existing literature, there are several master–slave synchronization schemes such as complete synchronization (CS) [2]–[13], phase synchronization (PS) [14], lag synchronization (LS) [15]–[24], anticipating synchronization (AS) [24], [25], and generalized synchronization (GS) [26]–[28]. Lag synchronization is an interesting phenomenon where the slave system follows the master system not on time but with a time delay, which has been observed in electronic circuits [16], lasers [18], and neuronal models [19]. Manuscript received April 06, 2010; revised July 13, 2010; accepted November 03, 2010. Date of publication January 06, 2011; date of current version May 27, 2011. This work was in part jointly supported by the China Postdoctoral Science Foundation 20100480611, the National Science Foundation of China under Grant 60874088, the National Science Fund for Distinguished Young Scholars under Grant 60625302, the National 973 Project under Grant 2009CB320603, the National 863 Project under Grant 2009AA04Z159, Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT0721, the 111 Project under Grant B08021, the Shanghai Leading Academic Discipline Project under Grant B504, and the Australian Research Council Discovery Projects under Grant DP1096780 and Grant DP0986376, and the Research Advancement Awards Scheme Program (January 2010–December 2012) at Central Queensland University, Australia. This paper was recommended by Associate Editor X. Li. W. He and F. Qian are with the Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China; W. He is also with the Centre for Intelligent and Networked Systems, Central Queensland University, Australia (e-mail: [email protected]; [email protected]). Q.-L. Han is with the Centre for Intelligent and Networked Systems, and the School of Information and Communication Technology, Central Queensland University, Rockhampton QLD 4702, Australia (e-mail: [email protected]). J. Cao is with the Department of Mathematics, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSI.2010.2096116

There have been some research efforts to investigate the effects of parameter mismatch on lag synchronization. In [15]–[17], lag synchronization was observed in studies of synchronization of two nonidentical bidirectionally coupled Rössler oscillators. It should be pointed out that lag synchronization in [15]–[17] was induced by the difference in natural frequencies of two coupled oscillators with parameter mismatch [24], and it was impossible to achieve lag synchronization if two oscillators were completely identical. In [18], lag synchronization was achieved between two unidirectionally coupled chaotic external cavity semiconductor lasers under optical feedback control. It was found that parameter mismatch was essential for the existence of lag synchronization due to the additional term in the slave laser representing the injected light from the master laser. However, none of [15]–[18] falls into the category of master–slave synchronization of coupled systems with the same structure, in which lag synchronization has been well studied [19]–[21]. On one hand, the error system was local stable due to the linearization method, and the discussed models were restricted to Ikeda oscillator, Mackey–Glass equation, or other simple nonlinear systems [19], [20]. On the other hand, the two coupled systems were completely identical [19]–[21]. In practical situations, parameter mismatch is inevitable in synchronization implementations, and it has significant effects on the collective behaviors of coupled systems. In some cases, loss of synchronization occurs in the presence of parameter mismatch [29]. In other cases, complete synchronization can be maintained even if parameter mismatch is large [30]. Some studies [15] have shown that synchronization transition from phase to lag synchronization exists in bidirectionally coupled systems with parameter mismatch. In fact, parameter mismatch plays an important role in synchronization performance. However, the effects of parameter mismatch on lag synchronization between two coupled time-delayed systems have not been fully investigated, which motivates the current study. This paper investigates lag quasi-synchronization of coupled time-delayed systems with parameter mismatch in a general model. It is known that complete lag synchronization is destroyed owing to parameter mismatch. Lag synchronization with an error level, referred to as lag quasi-synchronization, can be achieved. The error level is further estimated and the relationship between the error level and the feedback matrix is also established. Based on the obtained results, one can control the synchronization error in a predetermined level. Moreover, it is shown that the matrix measure method is effective in estimating the synchronization error level with a smaller feedback gain, compared with [34] and [35]. Finally, the accuracy of the theoretical estimation is verified by numerical simulations. There has been some literature concerning master–slave synchronization of coupled systems in the presence of parameter

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mismatch [31]–[37]. In [31], [32], master–slave synchronization of nonidentical Lur’e systems was considered, and several criteria were given in terms of linear matrix inequalities. In [33], the effect of parameter mismatch on impulsive synchronization of coupled chaotic systems was also discussed. However, those studies only examined coupled systems without time delays [31]–[33]. In order to handle the case of coupled systems with time delays and parameter mismatch, Huang et al. [34], [35] proposed a general delayed model, and further considered its master–slave synchronization. Several criteria in term of matrix inequalities were derived. However, multiple matrix inequalities might make the results conservative. Another study [36] employed impulsive control to synchronize nonidentical delayed neural networks. In addition, synchronization of nonidentical delayed systems with distributed delays has also been studied in [37], and other studies have focused on synchronization of complex networks with nonidentical nodes [38]–[40]. Although master–slave synchronization of delayed systems with parameter mismatch has been partially investigated, there are still challenges including estimating the synchronization error level accurately, controlling the error level in a predetermined level with less cost, and obtaining the exact relationship between the error level and the feedback matrix. Inspired by the above-mentioned works, this paper examines the effects of parameter mismatch on lag synchronization between two coupled delayed systems. Several simple and easily applicable criteria are derived. Moreover, the error level is estimated accurately, which is confirmed by simulation results, and the relationship between the error level and the feedback matrix is also established. It is shown that the feedback gain can be greatly reduced by using the matrix measure method. Therefore, the results in this paper improves the existing ones in [34] and [35]. The remainder of this paper is organized as follows: in Section II, the problem statement is presented for lag quasi-synchronization of two coupled systems. In Section III, several simple criteria are given. Also, some corollaries and remarks are listed to show the advantage of this paper. In Section IV, three examples are given to show the effectiveness of the proposed synchronization scheme. In Section V, conclusions are drawn. Notation: denotes the -dimensional Euclidean space, is the set of real matrices. is the correis an induced matrix norm on and sponding matrix measure. The definition of matrix measure and the explicit expressions are given in Appendix A. is the idendenotes tity matrix with appropriate dimensions. the Banach space of continuous vector-valued functions mapping the interval into . II. PROBLEM STATEMENT Consider a general master–slave synchronization scheme with the dynamics of the master system described by

(1)

is the state vector; , where and are constant matrices; are nonlinear functions; is the transmittal delay and satisfies ( is a positive constant); and is the initial condition. The corresponding slave system under consideration is given by (2) where and

is the state vector; , are constant matrices; and is the initial condition. Due to the finite speed of transmission and spreading, the signal which travels from the master system to the slave system is often associated with a time delay, which means that a time delay occurs during the signal transferring from the master system to the slave system. Observing this fact, for the master system (1), the slave system (2), we choose the following coupling (controller) (3)

is the time-varying coupling matrix, and is where the transmittal delay. In many practical situations, is usually selected as a time-invariant coupling strength or feedback gain. Defining the synchronization error as , we have the following error dynamical system:

(4) where (5) and (6) and

(7) , and . with In order to define the initial condition of system (4), we supplement the state on as

Introduce a new notation

as

HE et al.: LAG QUASI-SYNCHRONIZATION OF COUPLED DELAYED SYSTEMS WITH PARAMETER MISMATCH

Then initial condition of system (4) is defined by

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III. MAIN RESULTS A. Synchronization Criteria

Notice that from (4) the origin is not an equilibrium point of the error system. Therefore, complete lag synchronization can not be achieved due to parameter mismatch. However, approximate lag synchronization can be observed numerically with a slight mismatch. That is, there exists a synchronization error, which does not decay to zero but a relative small value. In this paper we refer such synchronization as lag quasi-synchronization, which is defined in the following. Definition 1: The master system (1) and the slave system (2) are said to be lag quasi-synchronized with an error level if there exists a compact set , such that for any , the error signal converges to as goes to infinity. For the nonlinear functions and , we need the following assumption. Assumption 1: For any , there exist constants such that

In this subsection, based on matrix measure theory and generalized Halanay inequality [41], we will derive some sufficient conditions for lag quasi-synchronization of the master system (1) and the slave systems (2). In what follows, let be a region in the phase space that contains the attractor of system (1). We first state and establish the following result. Theorem 1: Under Assumption 1, suppose that and . The trajectory of the error system (4) converges exponentially to the set (8) i.e., the master system (1) and the slave system (2) achieve lag quasi-synchronized with an error level if there exist a nonsingular matrix and a matrix measure such that

(9) Proof: Choose a Lyapunov function as Remark 1: It should be pointed out that in Assumption 1 we do not require that and , which means that the nonlinear functions satisfying Assumption 1 include a wide range of functions. For neural networks, it is natural to assume and . Without loss of generality, we will make no further explicit mention of the values of and . The purpose of this paper is to derive some simple criteria for lag quasi-synchronization of coupled delayed systems with parameter mismatch. To end this section, we introduce the following lemma which is useful in deriving sufficient conditions of lag quasi-synchronization. Lemma 1: ([41]) If

where . If there exists

(10) where is a non-singular matrix. Taking the upper-right Dini derivative of with respect to along the solution of (4) yields

, and are continuous functions and such that

Then we have (11) Recalling Assumption 1, we have where

and .

(12)

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From (11) and (12), we obtain

if there exists a matrix measure

such that

(17) Corollary 2: Under Assumption 1, suppose that and the parameter mismatches . Let satisfy . The trajectory of the error system (4) converges exponentially to the set (18) (13)

if there exists a matrix measure

such that

Let (19)

. From (9), we have that . Use Lemma 1 to obtain

Proof: Under Assumption 1, we obtain that . Notice that . We have

where Consequently, it is clear that

.

(20)

(14) from which we conclude that the error system (4) converges exponentially to the set containing the origin, where (15) which implies that the master system (1) and the slave system (2) achieve lag quasi-synchronized with an error level . This completes the proof. Remark 2: Theorem 1 provides a sufficient condition of lag quasi-synchronization associated with an error level. As goes to infinity, the error bound decreases and con, known as the verges to the infimum error level. Notice that in Theorem 1 we require that the matrix is nonsingular. By selecting , from Theorem 1 we have the following corollaries: Corollary 1: Under Assumption 1, suppose that and . The trajectory of the error system (4) converges exponentially to the set (16)

By Theorem 1, the conclusion of Corollary 2 follows immediately. This completes the proof. Remark 3: From Theorem 1, Corollary 1, and Corollary 2, one can see that mismatch system matrices and the feedback matrix influence the error level. It is usually to adjust the feedback matrix according to the extent of parameter mismatch and the predetermined error level. In the case that , the master system and the slave system become (21) and (22) The corresponding error system is

(23) where

By Theorem 1, we can conclude the following.


Corollary 3: Under Assumption 1, suppose that and . The trajectory of the error system (23) converges exponentially to the set

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. The trajectory of the error system (28) converges exponentially to the set (31)

(24) if there exists a matrix measure

such that (25)

Remark 4: Corollary 3 is the special case of Theorem 1, as in [8], [10], [32], [44] in which the criterion is still effective. We illustrate this with Example 3. The results above are derived mainly based on mismatches of system parameter matrices. We now consider the time-delay mismatch in the master system and the slave system. More specifically, consider the master system (26) where is the transmittal delay and satisfies ( is a positive constant); and initial condition; and the slave system

is the

(27) where is the transmittal delay and satisfies ( is a positive constant); and is the initial condition. Then the corresponding error system is given by

(28)

i.e., the master system (26) and the slave system (27) achieve lag quasi-synchronized with an error level if there exist a nonsingular matrix and a matrix measure such that

(32) , Theorem 2 reduces Remark 5: When to Theorem 1. Hence, Theorem 2 is more general than Theorem 1. Moreover, if , then the corresponding sufficient condition of complete lag synchronization is easily derived. Furthermore, if , then the sufficient condition of complete synchronization follows immediately [7]. Remark 6: From Theorem 2, one can see clearly that the condition (32) does not include information about the delays and . We refer this condition as a delay-independent condition. In some practical situation, the time-delays and do have the effect on the synchronization and should be taken into account. Therefore, a delay-dependent condition, which includes information about the delays, is needed. We leave this work for another paper. As mentioned in Remark 1, Assumption 1 covers a wide range of nonlinear functions and the systems under Assumption 1 include a general class of chaotic systems such as the Lorenz system, Rössler system, and Chen systems. For neural networks, which have triggered a number of works, motivated by applications such as signal processing, pattern recognition, associative memory and combinational optimization, instead of Assumption 1, we adopt the following assumption. Assumption 2: Let . There exist constants such that for any

where

(29) and

Under Assumption 2, we now state and establish the following result. Theorem 3: Under Assumption 2, suppose that and . Then the error system (4) converges exponentially to the set (33)

(30) Choosing the Lyapunov function (10), similar to the proof of Theorem 1, one can derive the following result. Theorem 2: Under Assumption 1, suppose that and

if there exist a nonsingular matrix such that

and a matrix measure

(34)

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where

and

Proof: See Appendix B. Let . The following corollary follows immediately. Corollary 4: Suppose that and Assumption 2 holds. The parameter mismatches . Let satisfy . The trajectory of the error system (4) converges to the set where (35) if there exists a matrix measure

such that

(36) and are defined in Theorem 3. where Obviously, Theorem 3 and Corollary 4 are not easy to verify. In order to have the easily applicable criterion, we now modify the condition of the nonlinear function in the following assumption. Assumption 3: Let . There exist constants such that for any

We need the following lemma to state the result under Assumption 3. Lemma 2: ([7]) Let Assumption 3 hold. Then

where

are defined in Theorem 3; and

Proof: See Appendix C. . Remark 7: The synchronization in presence of parameter mismatch is of practical significance and importance due to the fact that it is difficult to construct two absolutely identical systems in practical situation. In fact, there are two ways to deal with synchronization with respect to parameter mismatch: one is approximation synchronization, which is considered in this paper; the other is generalized synchronization, which means the two coupled systems are synchronized in a general sense [28]. Remark 8: In [34], [35], synchronization of coupled systems with parameter mismatch was discussed and the synchronization criteria were formulated in term of matrix inequalities. Different from [34], [35], in this paper, we provide some simpler sufficient conditions to ensure lag quasi-synchronization by employing matrix measure theory. Moreover, considering the case when the transmittal delay , the model in this paper is reduced to the one in [34] and [35] in which theoretical results are still applicable. We illustrate this with Example 1. Furthermore, it is indicated that the feedback gain can be greatly reduced by numerical simulations in comparison with [34] and [35] and the estimated error level is more accurate, which shows the advantage of the proposed method. B. Controller Design From the obtained results in Section III-A, it is clear that parameter mismatch and feedback matrix play a key role in estimating the synchronization error. If the parameters of coupled systems slightly mismatch, the states are close to act as lag synchronization but remain different. If the mismatch is large, the associated feedback gain matrix is designed in a proper way to achieve the desired error level. Based on the synchronization criteria, one can design the controller gain matrix. We take Corollary 1 as an example. For convenience, we modify Corollary 1 in the following Corollary 5: Under Assumption 1, suppose that and . The trajectory of the error system (4) converges exponentially to the set (39)

Theorem 4: Under Assumption 3, suppose that and . The error system (4) converges exponentially to the set (37) if there exists a matrix measure

and and

is selected such that

(40) If becomes

, the condition (40)

such that

(38) where

if the feedback gain matrix

are defined in Theorem 3;

(41) We are in a position to present the design procedure of based on the condition (40) as follows. Design Procedure of the Feedback Gain Matrix Step 1) According to the control performance, choose an appropriate error level . From , one obtains ; ; Step 2) Calculate


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Fig. 2. The synchronization error curve of coupled systems (42) and (43) with . Fig. 1. The chaotic attractor of Ikeda system (42).

(43)

It is known that the chaotic attractor of system (42) is contained in the set , and parameter mismatches satisfy . Thus, one obtains that and . Let . Consequently, , the same as [34]. Notice that and . Applying Corollary 2, one has , showing that is the minimum feedback gain required to achieve the predetermined error level 0.0146. However, the feedback gain needs to be 14.0996 to achieve the error level 0.0146 in [34]. In [35], is required for an error level 0.02645 under intermittent control, the ratio of which is about 97.5%. By simply computing, the average feedback gain is about 10.2375 corresponding to the error level 0.02645. Set . It is known that the synchronization error between master–slave systems does not converge to zero, as shown in Fig. 2. It fluctuates from to 0.015. Next, we will show the accuracy of the error estimation. With a given feedback gain , we calculate the error level by Corollary 2, then compare the estimated value with the simulated one, and repeat the procedure as increases. In order to compare the results in [34], first, we choose the average of absolute values of synchronization errors at 3500 points from to as the simulated errors in the setting of the iteration step 0.01 and increasing from 3.6 to 15. Fig. 3 reports the simulated errors and the corresponding theoretical error levels with different . It is shown that the estimated error levels in this paper are smaller, which are more accurate in comparison with Fig. 3 in [34], On the other hand, instead of taking the average synchronization errors as the simulated result, we select the maximal absolute value of synchronization errors from to as the simulated error, as shown in Fig. 4. As increases, the estimated error level is very close to the simulated one, which verifies the accuracy of theoretical results. We now give delayed neural networks to demonstrate the effectiveness of the theoretical results of lag quasi-synchronization. Example 2: Consider the following delayed neural network [42]:

and . It is assumed that there is where no delay in the coupling term, that is, .

(44)

Step 3) Determine the feedback gain matrix depending on selection of the matrix measure: a) If , then take . , then take b) If , where represents the maximum eigenvalue of a symmetric matrix. c) If , then take . Remark 9: If , based on (41), the above design procedure is applicable by replacing Step 3) with the following Step 3) : Step 3) Determine such that

Remark 10: It should be pointed out that is usually of . Therefore, the proposed method can deal the form with the uncertainty of the feedback gain matrix as shown in Example 2. IV. NUMERICAL SIMULATIONS In this section, three examples are presented to show the effectiveness of the proposed scheme. To show the advantage of the criteria based on matrix measure, a scalar Ikeda oscillator, the same model as in [34], [35], is investigated in the context of quasi-synchronization with in the following example. Example 1: The dynamics of Ikeda oscillator is described by (42) and System (42) exhibits chaotic behavior when as shown in Fig. 1. The corresponding slave system is given by

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Fig. 5, with initial values . The two attractors have the similar shapes but different orientations. The same structure accounts for the similar shapes of the attractors, and differences in initial conditions and parameter matrices lead to different orientations. Generalized lag synchronization does not exist in these two systems although they share the similar images as shown in Fig. 5. In fact, whether generalized lag synchronization occurs depends on the feedback matrix . In our previous work [28], the existence of generalized synchronization between two different chaotic systems was analyzed in detail. A similar discussion can be followed to study generalized lag synchronization between coupled neural network systems (44) and (45). More information on generalized synchronization can be found for instance in [26]–[28] to cite a few. One observes that

Fig. 3. The comparison between the estimated error level and the numerical simulated error of coupled Ikeda systems (42) and (43) in the average sense as increases.

Fig. 4. The comparison between the estimated error level and the maximal simulated error of coupled Ikeda systems (42) and (43) as increases.

where , and . Assumption 3 holds with . The slave system is of the form [42] (45)

Take and , and notice that and . Thus and . By . Set numerical simulation, we obtain , that is the error level . Following the controller design procedure step 1), we have . Asis positive diagonal. Thus sume that . Based on Theorem 4, holds. Calculate by Step 2). Then based on Step 3) , we have . Choose . By Theorem 4, lag synchronization is achieved with an error level 0.5. Fig. 6 shows time evolutions of state trajectories of neural networks (44) and (45). One can find that approximately retards by a time delay . The corresponding error curves between and are illustrated in Fig. 7, and temporal evolution of is depicted in Fig. 8. It is observed that after a peacts periodically. Therefore, it is reasonable to take riod, the maximal value of over instead of . Let . Fig. 9 depicts the estimated error level and the corresponding simulated error, which is obtained by taking the maximal 1-norm value of synchronization errors from to with a given feedback gain. It is shown that with the increase of the feedback gain, both the estimated and simulated error decrease and tend to be close, which indicates the effectiveness of the error estimation. The third example is to specialize the results in this paper to lag quasi-synchronization of Chua’s circuits, which can be implemented in hardware [43], [44]. Example 3: A Chua’s circuit is described by

where

(46) .

. The coupling term vanishes, which means First, let that neural network systems (44) and (45) are independent of each other. They have different chaotic attractors as shown in

with the nonlinear characteristics , and parameters . Hence, Chua’s circuit can be interpreted as


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Fig. 5. Chaotic attractors of systems (44) and (45). (a) System (44). (b) System (45).

Fig. 8. Time evolution of Fig. 6. Time evolution of each variable of coupled neural networks (44) and (45).

of coupled neural networks (44) and (45).

model (1) where

As shown in Fig. 10, Chua’s circuit possesses the chaotic behavior. Perturbation is considered in parameter matrices and . The slave system is with the same structure as (46), but has the different parameter matrices, that is,

Fig. 7. Lag synchronization errors of coupled neural networks (44) and (45).

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Fig. 9. Estimated error levels and the corresponding simulated errors of coupled neural networks (44) and (45) as increases.

Fig. 11. The slave system with perturbed parameters in the form of Chua’s circuit.

Fig. 10. Double scroll attractor of Chua’s circuit.

remains zero. The slave system exhibits limit cycle behavior, as shown in Fig. 11. One can see that as system parameters change, the dynamics of the master system and the slave system are totally different, as shown in Figs. 10 and 11. With the delayed coupling scheme proposed in this paper, lag quasi-synchronization can be achieved with a predetermined error level. Choose the coupling delay and the coupling matrix for simplicity. The error signal is defined as . Take . Thus and . By numerical simulation, one has . Set . Thus the error level is 0.12242, that is, . By Corollary 3, , one obtains . Take . It is concluded that the slave system approximately follows the master system with a time delay , as shown in Fig. 12. Fig. 13 depicts the evolution of , as well as the estimated error level. Numerical results show that lag quasi-synchronization is achieved between the master system and the slave system in

Fig. 12. Time evolutions of the states of Chua’s circuits .

and

a predetermined error level by the proposed feedback scheme. However, it is observed that the theoretical result is still conservative, as seen in Fig. 13. How to improve the accuracy of the synchronization error estimation is deserved to study further in the near future. V. CONCLUSION We have proposed a lag quasi-synchronization scheme for coupled delayed systems with parameter mismatch and obtained some simple synchronization criteria by means of matrix measure and a generalized Halanay inequality. We have also provided three examples to show the effectiveness of the synchronization criteria. It should be mentioned that sufficient conditions of the synchronization criteria have not included information about


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(47) Fig. 13. Time response curve of . Chua’s circuits with

and the estimated error level of

time delays, which means that the synchronization criteria are delay-independent ones. When transmittal delays are known and small, the delay-independent synchronization criteria may be conservative. So we will investigate the effects of the transmittal delays on synchronization of coupled delayed systems and derive some delay-dependent synchronization criteria in a near future study.

By Assumption 2, one obtains

(48) Consequently, from (47) and (48), we have

APPENDIX A THE DEFINITION OF MATRIX MEASURE Definition 2: [45] The matrix measure of a real square matrix is defined as (49) The matrix measures corresponding to the norms are given below (see [45, p. 26]).

The rest of the proof is similar to the proof in Theorem 1. Due to the page limitation, it is omitted. This completes the proof. APPENDIX C PROOF OF THEOREM 4 Use the property of matrix measure (see [45, p. 22]) to obtain

By Lemma 2, we have APPENDIX B PROOF OF THEOREM 3 Choosing the Lyapunov function (10) and taking the upperwith respect to along the soright Dini derivative of lution of (4), we have

Referring to (34) with the proof.

, we arrive at (38). This completes

ACKNOWLEDGMENT The authors would like to thank Prof. Xiang Li, Associate Editor, and anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper.

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[26] L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett., vol. 76, pp. 1816–1819, 1996. [27] H. Abarbanel, N. Rulkov, and M. Sushchik, “Generalized synchronization of chaos: The auxiliary system approach,” Phys. Rev. E, vol. 53, pp. 4528–4535, 1996. [28] W. He and J. Cao, “Generalized synchronization of chaotic systems: An auxiliary system approach via matrix measure,” Chaos, vol. 19, p. 013118, 2009. [29] V. Astakhov, M. Hasler, T. Kapitaniak, A. Shabunin, and V. Anishchenko, “Effect of parameter mismatch on the mechanism of chaos synchronization loss in coupled systems,” Phys. Rev. E, vol. 58, pp. 5620–5628, 1998. [30] G. Johnson, D. Mar, T. Carroll, and L. Pecora, “Synchronization and imposed bifurcations in the presence of large parameter mismatch,” Phys. Rev. Lett., vol. 80, pp. 3956–3959, 1998. [31] J. A. Suykens, P. F. Curran, and L. O. Chua, “Robust nonlinear synchronization of chaotic Lur’e Systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, pp. 891–904, Oct. 1997. [32] J. A. Suykens, P. F. Curran, and L. O. Chua, “Robust synthesis for master–slave synchronization of Lur’e systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 7, pp. 841–850, Jul. 1999. [33] C. Li, G. Chen, X. Liao, and Z. Fan, “Chaos quasisynchronization induced by impulses with parameter mismatches,” Chaos, vol. 16, p. 023102, 2006. [34] T. Huang, C. Li, and X. Liao, “Synchronization of a class of coupled chaotic delayed systems with parameter mismatch,” Chaos, vol. 17, p. 033121, 2007. [35] T. Huang, C. Li, W. Yu, and G. Chen, “Synchronization of delayed chaotic systems with parameter mismatches by using intermittent linear state feedback,” Nonlinearity, vol. 22, pp. 569–584, 2009. [36] H. Zhang, T. Ma, G. Huang, and Z. Wang, “Robust global exponential synchronization of uncertain chaotic delayed neural networks via dualstage impulsive control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 831–844, Jun. 2010. [37] K. Yuan and J. Cao, “Synchronization of master–slave systems with mixed time delays and mismatched parameters,” in Proc. 27th Chinese Control Conf., 2008, pp. 540–543. [38] Q. Song, J. Cao, and F. Liu, “Synchronization of complex dynamical networks with nonidentical nodes,” Phys. Lett. A, vol. 374, pp. 544–551, 2010. [39] Z. Duan and G. Chen, “Global robust stability and synchronization of networks with Lorenz-type nodes,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 8, pp. 678–683, Aug. 2009. [40] D. Hill and J. Zhao, “Global synchronization of complex dynamical networks with non-identical nodes,” in Proc. 47th IEEE Conf. Decision Control, 2008, pp. 817–822. [41] L. Wen, Y. Yu, and W. Wang, “Generalized Halanay inequalities for dissipativity of Volterra functional differential equations,” J. Math. Anal. Appl., vol. 347, pp. 169–178, 2008. [42] H. Lu, “Chaotic attractors in delayed neural networks,” Phys. Lett. A, vol. 298, pp. 109–116, 2002. [43] L. Torres and L. Aguire, “Inductorless Chua’s circuit,” Electron Lett., vol. 36, no. 3, pp. 1915–1916, 2000. [44] M. Porfiri and F. Fiorilli, “Global pulse synchronization of chaotic oscillators through fast-switching: Theory and experiments,” Chaos, Solitons Fractals, vol. 41, pp. 245–262, 2009. [45] M. Vidyasagar, Nonlinear System Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1993. Wangli He (M’10) received the B.S. degree in information and computing science and Ph.D. degree in applied mathematics from Southeast University, Nanjing, China, in 2005 and 2010, respectively. Since January 2010, she has been a Postdoctoral Research Fellow with the Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, School of Information Science and Engineering, East China University of Science and Technology, Shanghai, China. From July 2010 to July 2011, she has also been a Visiting Postdoctoral Research Fellow with the Centre for Intelligent and Networked Systems and the School of Information and Communication Technology, Central Queensland University, Australia. Her current research interests include chaos synchronization and control, stability theory, neural networks, complex network synchronization, bifurcation analysis, and consensus in multiagent systems.


Feng Qian received the B.Sc. degree in chemical automation and meters from Nanjing Institute of Chemical Technology, Nanjing, China, in 1982, and the M.S. and Ph.D. degrees in automation from East China Institute of Chemical Technology, Shanghai, China, in 1988 and 1995, respectively. From 1999 to 2001, he was the director of Automation Institute, East China University of Science and Technology, China, where he served as the head of Scientific and Technical Department from 2001 to 2006. He is the current Vice President of East China University of Science and Technology, the director of Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, China, and the director of Process System Engineering Research Centre, Ministry of Education, China. His research interests include modeling, control, optimization, and integration of petrochemical complex industrial process and the industrial applications, neural networks theory, real-time intelligent control technology, and their applications on the ethylene, PTA, PET, and refining industries. Prof. Qian is a Member of the China Instrument and Control Society, Chinese Association of Higher Education, and China’s PTA Industry Association.

Qing-Long Han received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Eng. and Ph.D. degrees in information science (electrical engineering) from East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. From September 1997 to December 1998, he was a Postdoctoral Researcher Fellow in LAII-ESIP, Université de Poitiers, France. From January 1999 to August 2001, he was a Research Assistant Professor in the Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville. In September 2001 he joined Central Queensland University, Australia, where he is currently Professor in the School of Information and Communication Technology, Associate Dean (Research and Innovation) in the Faculty of Arts, Business, Informatics and Education, and Director, Centre for Intelligent and Networked Systems. In March 2010, he was appointed Chang Jiang (Yangtze River) Scholar Chair Professor, East China University of Science and Technology, Ministry of Education, China. He has held a Visiting Professor position in LAII-ESIP, Univer-

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sité de Poitiers, France, a Chair Professor position at Hangzhou Dianzi University, as well as a Guest Professor position in Huazhong University of Science and Technology, East China University of Science and Technology, Zhejiang University of Technology, and Nanjing Normal University. His research interests include time-delay systems, networked control systems, neural networks, complex systems, and software development processes. He has published more than 160 refereed papers in prestigious journals and leading conferences, one research-based book (monograph), one research-based book chapter, and edited one conference proceeding and two special issues on time-delay systems. His research work has been cited more than 1930 times with h-index of 23 according to ISI/SCI database on 8 November 2010; and more than 2700 times with h-index of 27 according to SCOPUS database on 8 November 2010. Prof. Han has held editorial positions for seven international journals. He has organized two international conferences and served as a member of international program committee for more than ten international conferences.

Jinde Cao (M’07-CSM’07) received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, all in mathematics/applied mathematics, in 1986, 1989, and 1998, respectively. He was with Yunnan University from 1989 to 2000. Since 2000, he has been with the Department of Mathematics, Southeast University, Nanjing, China. From 2001 to 2002, he was a Postdoctoral Research Fellow with the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. He was a Visiting Research Fellow and a Visiting Professor with the School of Information Systems, Computing and Mathematics, Brunel University, Middlesex, U.K., from 2006 to 2008. He is the author or coauthor of more than 160 research papers and five edited books. His current research interests include nonlinear systems, neural networks, complex systems, complex networks, stability theory, and applied mathematics. Dr. Cao was an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS from 2006 to 2009. He is an Associate Editor of the Journal of the Franklin Institute, Mathematics and Computers in Simulation, Neurocomputing, International Journal of Differential Equations, Discrete Dynamics in Nature and Society, and Differential Equations and Dynamical Systems. He is a Reviewer of Mathematical Reviews and Zentralblatt-Math.