Robust Gain Scheduling Synchronization Method ... - Semantic Scholar
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Robust Gain Scheduling Synchronization Method for Quadratic Chaotic Systems With Channel Time Delay Yu Liang and Horacio J. Marquez
Abstract—In this paper, a general robust gain scheduling synchronization method is proposed for quadratic chaotic systems with parametric uncertainty and unknown channel time delay. Quadratic systems contain nonlinearity of quadratic form of the systems' states and can always be transformed into linear parameter-varying (LPV) forms. Based on the LPV forms, our approach provides two different gain scheduling structures to achieve global synchronization under the case of having parametric uncertainty and time delay. The convergence of synchronization errors is demonstrated using Lyapunov stability theory. Index Terms—Chaos synchronization, gain scheduling, linear parameter varying (LPV), parametric uncertainty, time delay.
I. INTRODUCTION HE IDEA behind chaos synchronization is to use an output of a chaotic system, called the drive system, to control another chaotic system called the response system so that the state trajectories of the response system asymptotically converge to those of the drive system. Because of the critical sensitivity to initial conditions which characterizes chaotic systems, even infinitesimal changes in the initial conditions can lead to exponential divergent orbits, making chaos synchronization a challenging problem. Motivated by the pioneering work of Pecora and Carroll [2], [17], chaos synchronization has attracted much attention over the last 15 years. In [2] and [17], Pecora and Carroll proposed the possibility of synchronizing two chaotic systems by sending a signal from the master system to the slave system. Since then, several other techniques have been proposed by various authors to achieve synchronization under different conditions. Ogorzalek [15] proposed synchronization using ˇ linear coupling of chaotic systems. Grassi [6], Celikovský [3], and Feki [5] solved the synchronization problem for various chaotic systems using nonlinear observer design approaches. Yang [23] and Park [16] solved the synchronization problem for Chua’s circuit and Genesio chaotic system, respectively, using the backstepping approach. Liao [11], Boutayeb [1], and Jiang [8] developed observer-based synchronization methods with application to secure communication. In [10], the authors
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Manuscript received May 13, 2008. First published August 04, 2008; current version published March 11, 2009. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. This paper was recommended by Associate Editor M. Di Marco. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G2V4, Canada (e-mail: marquez@ece. ualberta.ca). Digital Object Identifier 10.1109/TCSI.2008.2002656
proposed a method for observer-based synchronization of “quadratic” chaotic systems, using gain scheduling. The technique proposed in [10] presents the advantages of being easy to design and applicable to a large class of chaotic systems, since several well-known chaotic systems fall under the definition of quadratic systems, as will be shown later. The two problems that severely constrain the efficacy of synchronization methods in practical applications are as follows: 1) model mismatch between the parameters of the response and drive systems and 2) channel delay affecting the measured signal from the drive system. The first problem arises from the fact that system parameters are always affected by errors, thus prompting the need for robustness considerations, understood as the ability of the synchronization scheme to perform in the presence of mismatch between the parameters of the response and drive systems, in any scheme based on synchronization of identical systems. The second problem originates from engineering applications, where there typically exists signal propagation delay in a communication channel. Moreover, the value of the delay is usually unknown. Several authors have studied conditions under which synchronization can be guaranteed despite parameter mismatch. Adaptive synchronization methods have been proposed by Suzuki [19], Lian [9], and Lu [12]. Yan [22] gave a sliding-mode control method for robust synchronization of unified chaotic systems. Time delay problems were also considered by several authors. In [21], Yalcin studied the synchronization of Luré-type systems, i.e., the feedback combination of a linear subsystem and a memoryless nonlinearity, with time delay and proposed that the synchronization can be obtained only if the delay time is a known value less than a small threshold. Xiang [20] gave an improved condition for synchronization based on Yalcin’s method. In [7], Jiang proposed a synchronization scheme from the approach of unidirectional linear error feedback coupling for chaotic systems with time delay. Although Jiang’s method can achieve synchronization without any knowledge of the delay time value, a limitation of this method is that a much complex bounded matrix is required from the drive chaotic system to form the design inequality condition. Recently, for three typical chaotic systems with parametric uncertainty and time delay, Ma [13] proposed an impulsive synchronization method. Other than Ma’s work, few have been done for the synchronization problem with both parametric uncertainty and time delay. In this paper, we consider the synchronization problem for quadratic chaotic systems with both parametric uncertainty and channel time delay. Following the approach introduced in [10], we propose a gain scheduling technique that ensures robust synchronization with respect to parameter uncertainties and
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LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
unknown time delay. The convergence of the synchronization method is proven via Lyapunov methods. The rest of this paper is organized as follows. In Section II, we introduce the quadratic chaotic systems to be considered and define the synchronization problem to be solved. In Section III, based on the gain scheduling technique, two different robust gain scheduling synchronization structures are proposed to achieve global synchronization for the quadratic chaotic systems with both parametric uncertainty and unknown channel time delay. Section IV contains two illustrative examples using the well-known chaotic Chen system and Lorenz system. Finally, Section V contains conclusions and final remarks. II. PRELIMINARIES In this section, we introduce the class of quadratic chaotic systems to be considered and define the main problem to be solved. A. Quadratic Chaotic Systems We will consider the quadratic chaotic system of the following form:
(1) represents the state; is an output to be where transmitted through a communication channel; and are constant matrices; and is a nonlinear function, which contains quadratic terms of systems’ states, i.e., and . Several well-known chaotic systems can be accommodated into this class, including the Genesio–Tesi system [4], the generalized Lorenz system [3], and the chaotic Chen and Lorenz systems considered in Section IV. B. Robust Chaos Synchronization With Time Delay In general, chaos synchronization consists of using a scalar signal taken from a chaotic system, the drive system, to control another chaotic system, the response system, so that the state trajectories of the response system asymptotically converge to those of the drive system. We will make the following assumptions. 1) Parametric uncertainty: We will assume that the drive and response systems have the same dynamics. However, to compensate for the effect of uncertainties in the system parameters, we will assume that the state equation of the drive system is affected by parameter uncertainties of the form
2) Communication delay: To compensate for delays in the communication channel, we will assume that the synchronization signal available to the response system is delayed by “ ” seconds, where is not assumed to be known.
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Fig. 1. Common synchronization structure for chaotic systems with time delay.
Fig. 1 shows the synchronization scheme proposed here. In this figure, the drive system is the original chaotic system with parametric uncertainty, and is a finite time delay which is an unknown constant. The response system consists of a copy of the original chaotic system and a filter term to be designed to achieve the synchronization of the response system with the drive system. Let and be the states of the drive and response systems, respectively. Then, we will say that the state trajectory of the response system robustly synchronizes with that of the drive system if the state of the response system at time asymptot, i.e., ically approaches the state of the drive system at time (2) In [10], the authors considered synchronization of quadratic chaotic systems of the form (1) without model uncertainty and time delay by first transforming the system (1) into linear parameter-varying (LPV) form. Then, based on the gain scheduling technique, two different synchronization structures were proposed to achieve the asymptotical convergence of synchronization errors. In this paper, we extend our work to the case of robust synchronization of quadratic chaotic systems with both parametric uncertainty and unknown channel time delay.
III. ROBUST GAIN SCHEDULING SYNCHRONIZATION METHOD From the definition of robust chaos synchronization with time delay, the final objective is to make the synchronization error dynamics asymptotically stable at the origin. As mentioned in Section I, we pursue this objective extending the gain scheduling technique introduced in [10] to include delay and parameter uncertainty. We begin by transforming the system (1) into LPV form, following the approach in[10]. Denoting the maximum value by and the minimum value by of state , we obtain the time-varying parameters and as
It is clear that
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and . Then, the quadratic chaotic system (1) can be identically transformed into LPV form
denote the synchronization error. Let Deriving from the drive system (5) and the response system (6), the synchronization error dynamics is
(3) where matrices and are all constant and are matrices and parameters used as scheduling variables. and the unConsidering now the parametric uncertainty in the known channel time delay , we replace with original chaotic system (1) and its LPV form (3) to obtain the as drive system in Fig. 1 at time (7) (4)
(5) in which
From condition (2), the robust synchronization problem with time delay is then equivalent to the problem of setting the error dynamics (7) asymptotically stable at the origin. Based on the gain scheduling technique, we propose two different response systems to achieve global synchronization for the quadratic chaotic systems with both parametric uncertainty and channel time delay. A. First Synchronization Structure In this section, we present the first synchronization scheme. From the error dynamics (7), we define matrices , , and as follows:
Assumption 1: There are parameters, , in ma, the paratrix . Then, for each parameter metric uncertainty is denoted by . From Assumption 1, it is evident that we can obtain the following transformation:
in which
is a linear combination of state and . To solve the robust synchronization problem of the quadratic system(4) or (5) with time delay, the response system in Fig. 1 is given a general form
where , , , , and are constant matrices to be designed. Considering now the uncer, we propose the tain quadratic chaotic system (5) at time first gain scheduling synchronization structure of the following form:
(6)
, in which matrices to be designed.
,
, and
are varying
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(8)
LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
With this definition, the synchronization error dynamics is
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Substituting these identities in(9), we obtain the following error dynamics:
To prove the stability of the error dynamics, we consider the . For this function, we Lyapunov function have
By construction, the parameters are always greater and . Moreover, than or equal to zero all of them cannot be zero at the same time. Then, with the inequality condition in (13) satisfied, it is easy to see that
(9) The convergence of the synchronization error can then be guaranteed by the following theorem. Theorem 1: If there exists a symmetric matrix , matrices and and and , and constant matrix such that conditions(10)–(13) are satisfied simultaneously for every pair and , of and at (10) (11) (12) (13) then the synchronization structure (8) robustly synchronizes with the drive system (4) or(5) globally. Proof: Assuming that(10)–(13) are satisfied for every pair of and , it is straightforward to see that
It follows from the second Lyapunov stability theorem (see, for example, [14]) that the synchronization error dynamics is globally asymptotically stable at the origin. Thus, from the definition of robust chaos synchronization with time delay, the gain scheduling structure (8) and the quadratic chaotic system (4) or (5) are globally asymptotically synchronized. Remark 1: In most cases, after transforming the original system(1) into LPV form, there is only one or two pairs of paand appearing in (3). The rameter design procedure is easy to complete as conditions (10)–(13) can be easily solved by using linear matrix inequality (LMI) tools. This point will be illustrated in the example section. We conclude this section emphasizing that the proposed robust gain scheduling structure(8) achieves global synchronization for the quadratic chaotic system, even in the presence of parametric uncertainty and unknown time delay . B. Alternative Synchronization Structure A shortcoming of the synchronization structure (8) proposed in the previous section is that it requires knowledge of the . This means that, in varying parameters 's at time addition to the transmitted signal , some or all thestates of the drive chaotic system are also needed, thus restricting the application value of the result. In this section, we modify the previous scheme using an alternative structure. is available at Assuming that only the output signal the received end, we define
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The alternative gain scheduling synchronization structure is proposed as follows:
(15) From the definition of parameters
and , it is obvious that
(14) where
Using straightforward algebraic manipulations, we obtain
It is immediate that the definition of these varying parameand ters has the same form as that of parameters . Then, the synchronization error dynamics in this case is
where each and is a constant matrix obtained from direct algebraic transformation. Thus, the error dynamics(15) is
The convergence of the synchronization error is proven in the following theorem. , maTheorem 2: If there exists a symmetric matrix and and trices and , and constant matrix such that conditions(16)–(19) are satisfied simultaneously for every pair and , of and at (16) (17) (18) (19) Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on June 17, 2009 at 16:31 from IEEE Xplore. Restrictions apply.
LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
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where
then the synchronization structure (14) robustly synchronizes with the drive system (4) or (5) globally. Proof: The proof follows by direct application of the second Lyapunov stability theory with Lyapunov function as . Remark 2: As mentioned earlier, there are only one or two shown in the inequality condition (19). pairs of parameters With simple algebraic transformations, we can still rewrite the inequality condition such that the effect of the varying parameand is taken off, and then solve the condition ters easily by using the LMI technique. The detailed transformation and procedure will be shown in Section 4. Remark 3: In general, the alternative synchronization structure (14) is suitable for practical applications and thus preferable over (8). In the special case in which the specific state needed to construct the varying parameters 's is the same as the output signal , the first synchronization structure will be more direct and much easier to design and has much better performance. This point will be shown in Example 2 in the next section.
(21) where
Defining an output signal , and assuming that there is uncertainty on parameter , the Chen system(20) or (21) with output and parametric uncertainty at time is
IV. EXAMPLES In this section, we consider two examples to illustrate the efficiency of the proposed gain scheduling synchronization methods, using well-known chaotic systems.
(22) or, equivalently
A. Chen System The chaotic Chen system [3] is described by (23)
(20) where, accordingly in which is the state vector. When , and , the system (20) displays chaotic beis havior. Assuming that the bounded value for state , we obtain the varying parameters 's as
It is apparent that that be rewritten as follows:
and and . Then, the Chen system(20) can
Then, the first gain scheduling synchronization structure proposed in this paper is
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Subtracting (24) from (23), we obtain the following error dynamics:
(25) is the synchronization error. where Since the parameters 's are nonnegative, conditions (10)–(13) in Theorem 1 for the synchronization errors are
Fig. 2. Three synchronization errors of e(t; ) = x(t system under the first structure.
0 ) 0 x^(t) of the Chen
For the alternative synchronization structure, we have The bounded values used in this example are chosen to be and . These values were selected by simulating the system (20) with the indicated parameters for different initial conditions, and noticing that each state trajectory falls in a bounded section, which are defined approximately as , and . Using follows: the LMI technique, we obtain
(26) where
Subtracting (26) from (23), we obtain
(27) Using the definition of
and , the error dynamics (27) is
Then, with the aforementioned chosen parameters, the two systems (22) or (23) and (24) are globally asymptotically synchronized under the existence of parametric uncertainty and s, , and a constant time delay . Assuming that and the initial conditions are , Fig. 2 shows the synchronization errors. Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on June 17, 2009 at 16:31 from IEEE Xplore. Restrictions apply.
(28)
LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
Denoting now the bounded values of state the new varying parameters are
by
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,
Then, for the error dynamics (28), conditions (16)–(19) in Theorem 2 are
Fig. 3. Three synchronization errors of e(t; ) = x(t system under the second structure.
0 ) 0 x^(t) of the Chen
where
The aforesaid inequalities can also be easily solved using LMIs. The resulting matrices are
the two synchronization structures presented earlier. The results show that the proposed structures achieve synchronization despite random delay. To illustrate the significance of the robust synthesis proposed here, we consider the same system with small uncertainty on the parameter (not considered in the design). The simulation on are shown in Fig. 6 for the first results with synchronization method and Fig. 7 for the second one. The figures clearly show that, unless uncertainty is incorporated as part of the design, even a small disturbance on a parameter can cause loss of convergence.
B. Lorenz System In this example, we consider the well-known Lorenz system, which is described by (29)
Thus, with the alternative synchronization structure, the two systems(22) and (26) are also globally asymptotically synchroand s also, the simulation result is nized. With shown in Fig. 3. To further illustrate our results, the next simulations shown in Figs. 4 and 5 consider a randomly selected time delay for
is the state vector. When in which , and , the system (29) presents chaotic behavior. By simulating the Lorenz system with different initial conditions, we see that each state trajectory stays in a bounded region with state bounds that can be approximated as follows: , , and . Because the exact values will not affect the internal convergence of the proposed methods, it is enough to choose values which contain the corresponding sections. Here, we choose the following state bounds: , , and . In order to satisfy the conditions in the proposed theorems, the output is introduced. The transformation process signal and synchronization structures are similar to those used in the previous example.
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Fig. 4. Three synchronization errors of e(t; ) of the Chen system under the first structure with different .
Fig. 5. Three synchronization errors of e(t; ) of the Chen system under the second structure with different .
For the first synchronization structure, using conditions (10)–(13), we obtain
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LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
Fig. 6. Three synchronization errors of e(t; ) = x(t system with b disturbance under the first structure.
0 ) 0 x^(t) of the Chen
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0 0
Fig. 8. Three synchronization errors of e(t; ) = x(t ) x ^(t) of the Lorenz system under the first structure.
0 0
Fig. 7. Three synchronization errors of e(t; ) = x(t system with b disturbance under the second structure.
0 ) 0 x^(t) of the Chen
Fig. 9. Three synchronization errors of e(t; ) = x(t ) x ^(t) of the Lorenz system under the second structure.
Assuming that initial conditions and , and assuming that parametric unceron the parameter , simulation results under the tainty first synchronization structure are shown in Fig. 8. For the second synchronization structure, solving the conditions (16)–(19) using LMIs, we obtain
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V. CONCLUSION In this paper, we have extended the synchronization method proposed in [10] to the important case of quadratic chaotic systems with parametric uncertainty and unknown time delay. Our method consists of transforming the original system into LPV form, a transformation that always exists for quadratic chaotic systems, and then using LMIs and the gain scheduling approach to ensure synchronization. Our approach is easy to apply and can benefit from readily available software tools to solve LMI problems.
REFERENCES
0 ) 0 x^(t) of the
Fig. 10. Three synchronization errors of e(t; ) = x(t Lorenz system with c disturbance under the first structure.
0 0 x^(t) of the
Fig. 11. Three synchronization errors of e(t; ) = x(t ) Lorenz system with c disturbance under the second structure.
With the same initial conditions, the simulation results are shown in Fig. 9. As in the previous example, we test the efficacy of uncertainty in a parameter that is not included in our original design. Asin parameter , we obtain the suming that uncertainty simulations shown in Figs. 10 and 11. The simulations show the efficiency of the synchronization structures, as well as the need to consider robustness in the design. In this example, the state needed for varying parameters 's . In this is , which is the same as the output signal case, the first synchronization structure is also suitable for practical application. In this case, the design of the first structure is simpler, and performance is much better than using the second scheme.
[1] M. Boutayeb, M. Darouach, and H. Rafaralahy, “Generalized statespace observers for chaotic synchronization and secure communication,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 3, pp. 345–349, Mar. 2002. [2] T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEE Trans. Circuits Syst., vol. 38, no. 4, pp. 453–456, Apr. 1991. ˇ [3] S. Celikovský and G. Chen, “Synchronization of a class of chaotic systems via a nonlinear observer approach,” in Proc. IEEE Conf. Decision Control, 2002, pp. 3895–3900. [4] M. Chen, Z. Han, and Y. Shang, “General synchronization of Genesio–Tesi systems,” Int. J. Bifurc. Chaos, vol. 14, pp. 347–354, 2004. [5] M. Feki, “Observer-based exact synchronization of ideal and mismatched chaotic systems,” Phys. Lett. A, vol. 309, no. 1, pp. 53–60, Mar. 2003. [6] G. Grassi and S. Mascolo, “Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, no. 10, pp. 1011–1014, Oct. 1997. [7] G. P. Jiang, W. X. Zheng, and G. Chen, “Global chaos synchronization with channel time-delay,” Chaos Solitons Fractals, vol. 20, no. 2, pp. 267–275, Apr. 2004. [8] Z. Jiang, “A note on chaotic secure communication systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 92–96, Jan. 2002. [9] K. Lian, P. Liu, T. Chiang, and C. Chiu, “Adaptive synchronization design for chaotic systems via a scalar driving signal,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 17–27, Jan. 2002. [10] Y. Liang and H. J. Marquez, “Gain scheduling synchronization method for quadratic chaotic systems,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 55, no. 4, pp. 1097–1107, May 2008. [11] T. Liao and N. Huang, “An observer-based approach for chaotic synchronization with applications to secure communications,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 9, pp. 1144–1150, Sep. 1999. [12] J. Q. Lu and J. D. Cao, “Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters,” CHAOS, vol. 15, no. 4, pp. 1–10, Nov. 2005. [13] T. D. Ma, H. G. Zhang, and Z. L. Wang, “Impulsive synchronization for unified chaotic systems with channel time-delay and parameter uncertainty,” ACTA Physica Sinica, vol. 56, no. 7, pp. 3796–3802, 2007. [14] H. J. Marquez, Nonlinear Control Systems: Analysis and Design. Hoboken, NJ: Wiley, 2003. [15] M. J. Ogorzalek, “Taming chaos—Part I: Synchronization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 40, no. 10, pp. 693–699, Oct. 1993. [16] J. H. Park, “Synchronization of Genesio chaotic system via backstepping approach,” Chaos Solitons Fractals, vol. 27, no. 5, pp. 1369–1375, Mar. 2006. [17] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, no. 8, pp. 821–825, 1990. [18] W. J. Rugh and J. S. Shamma, “Research on gain scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, Oct. 2000. [19] Y. Suzuki, M. Iwase, and S. Hatakeyama, “A design of chaos synchronizing system using adaptive observer,” in Proc. SICE, 2002, pp. 2352–2353.
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[20] J. Xiang, Y. Li, and W. Wei, “An improved condition for master–slave synchronization of Luré systems with time delay,” Phys. Lett. A, vol. 362, no. 2/3, pp. 154–158, Feb. 2007. [21] M. E. Yalcin, J. A. K. Suykens, and J. Vandewalle, “Master–slave synchronization of Luré systems with time-delay,” Int. J. Bifurc. Chaos, vol. 11, pp. 1707–1722, 2001. [22] J. J. Yan, Y. S. Yang, T. Y. Chiang, and C. Y. Chen, “Robust synchronization of unified chaotic systems via sliding mode control,” Chaos Solitons Fractals, vol. 34, no. 3, pp. 947–954, Nov. 2007. [23] T. Yang, X. Li, and H. Shao, “Chaotic synchronization using backstepping method with application to the Chua’s circuit and Lorenz system,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 2299–2300. Yu Liang received the B.S. and M.S. degrees in automatic control from Beijing Institute of Technology, Beijing, China, in 2000 and 2003, respectively. She is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. Her research interests include observer design for nonlinear systems with specific characters and application to synchronization problem of chaotic systems.
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Horacio J. Marquez received the B.Sc. degree from Instituto Tecnológico de Buenos Aires, Buenos Aires, Argentina, in 1987 and the M.Sc.E. and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, in 1990 and 1993, respectively. From 1993 to 1996, he held visiting appointments at Royal Roads Military College, Victoria, BC, Canada, and the University of Victoria, Victoria. 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 the Department Chair. In 2008, he was a Guest Research Professor with the Université Henri Poincaré, Nancy, France. He is currently an Area Editor for the International Journal of Robust and Nonlinear Control and an Associate Editor for theJournal of the Franklin Institute. 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 was the recipient of the 2003–2004 McCalla Research Professorship awarded by the University of Alberta.
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Robust Gain Scheduling Synchronization Method for Quadratic Chaotic Systems With Channel Time Delay Yu Liang and Horacio J. Marquez
Abstract—In this paper, a general robust gain scheduling synchronization method is proposed for quadratic chaotic systems with parametric uncertainty and unknown channel time delay. Quadratic systems contain nonlinearity of quadratic form of the systems' states and can always be transformed into linear parameter-varying (LPV) forms. Based on the LPV forms, our approach provides two different gain scheduling structures to achieve global synchronization under the case of having parametric uncertainty and time delay. The convergence of synchronization errors is demonstrated using Lyapunov stability theory. Index Terms—Chaos synchronization, gain scheduling, linear parameter varying (LPV), parametric uncertainty, time delay.
I. INTRODUCTION HE IDEA behind chaos synchronization is to use an output of a chaotic system, called the drive system, to control another chaotic system called the response system so that the state trajectories of the response system asymptotically converge to those of the drive system. Because of the critical sensitivity to initial conditions which characterizes chaotic systems, even infinitesimal changes in the initial conditions can lead to exponential divergent orbits, making chaos synchronization a challenging problem. Motivated by the pioneering work of Pecora and Carroll [2], [17], chaos synchronization has attracted much attention over the last 15 years. In [2] and [17], Pecora and Carroll proposed the possibility of synchronizing two chaotic systems by sending a signal from the master system to the slave system. Since then, several other techniques have been proposed by various authors to achieve synchronization under different conditions. Ogorzalek [15] proposed synchronization using ˇ linear coupling of chaotic systems. Grassi [6], Celikovský [3], and Feki [5] solved the synchronization problem for various chaotic systems using nonlinear observer design approaches. Yang [23] and Park [16] solved the synchronization problem for Chua’s circuit and Genesio chaotic system, respectively, using the backstepping approach. Liao [11], Boutayeb [1], and Jiang [8] developed observer-based synchronization methods with application to secure communication. In [10], the authors
T
Manuscript received May 13, 2008. First published August 04, 2008; current version published March 11, 2009. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. This paper was recommended by Associate Editor M. Di Marco. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G2V4, Canada (e-mail: marquez@ece. ualberta.ca). Digital Object Identifier 10.1109/TCSI.2008.2002656
proposed a method for observer-based synchronization of “quadratic” chaotic systems, using gain scheduling. The technique proposed in [10] presents the advantages of being easy to design and applicable to a large class of chaotic systems, since several well-known chaotic systems fall under the definition of quadratic systems, as will be shown later. The two problems that severely constrain the efficacy of synchronization methods in practical applications are as follows: 1) model mismatch between the parameters of the response and drive systems and 2) channel delay affecting the measured signal from the drive system. The first problem arises from the fact that system parameters are always affected by errors, thus prompting the need for robustness considerations, understood as the ability of the synchronization scheme to perform in the presence of mismatch between the parameters of the response and drive systems, in any scheme based on synchronization of identical systems. The second problem originates from engineering applications, where there typically exists signal propagation delay in a communication channel. Moreover, the value of the delay is usually unknown. Several authors have studied conditions under which synchronization can be guaranteed despite parameter mismatch. Adaptive synchronization methods have been proposed by Suzuki [19], Lian [9], and Lu [12]. Yan [22] gave a sliding-mode control method for robust synchronization of unified chaotic systems. Time delay problems were also considered by several authors. In [21], Yalcin studied the synchronization of Luré-type systems, i.e., the feedback combination of a linear subsystem and a memoryless nonlinearity, with time delay and proposed that the synchronization can be obtained only if the delay time is a known value less than a small threshold. Xiang [20] gave an improved condition for synchronization based on Yalcin’s method. In [7], Jiang proposed a synchronization scheme from the approach of unidirectional linear error feedback coupling for chaotic systems with time delay. Although Jiang’s method can achieve synchronization without any knowledge of the delay time value, a limitation of this method is that a much complex bounded matrix is required from the drive chaotic system to form the design inequality condition. Recently, for three typical chaotic systems with parametric uncertainty and time delay, Ma [13] proposed an impulsive synchronization method. Other than Ma’s work, few have been done for the synchronization problem with both parametric uncertainty and time delay. In this paper, we consider the synchronization problem for quadratic chaotic systems with both parametric uncertainty and channel time delay. Following the approach introduced in [10], we propose a gain scheduling technique that ensures robust synchronization with respect to parameter uncertainties and
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LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
unknown time delay. The convergence of the synchronization method is proven via Lyapunov methods. The rest of this paper is organized as follows. In Section II, we introduce the quadratic chaotic systems to be considered and define the synchronization problem to be solved. In Section III, based on the gain scheduling technique, two different robust gain scheduling synchronization structures are proposed to achieve global synchronization for the quadratic chaotic systems with both parametric uncertainty and unknown channel time delay. Section IV contains two illustrative examples using the well-known chaotic Chen system and Lorenz system. Finally, Section V contains conclusions and final remarks. II. PRELIMINARIES In this section, we introduce the class of quadratic chaotic systems to be considered and define the main problem to be solved. A. Quadratic Chaotic Systems We will consider the quadratic chaotic system of the following form:
(1) represents the state; is an output to be where transmitted through a communication channel; and are constant matrices; and is a nonlinear function, which contains quadratic terms of systems’ states, i.e., and . Several well-known chaotic systems can be accommodated into this class, including the Genesio–Tesi system [4], the generalized Lorenz system [3], and the chaotic Chen and Lorenz systems considered in Section IV. B. Robust Chaos Synchronization With Time Delay In general, chaos synchronization consists of using a scalar signal taken from a chaotic system, the drive system, to control another chaotic system, the response system, so that the state trajectories of the response system asymptotically converge to those of the drive system. We will make the following assumptions. 1) Parametric uncertainty: We will assume that the drive and response systems have the same dynamics. However, to compensate for the effect of uncertainties in the system parameters, we will assume that the state equation of the drive system is affected by parameter uncertainties of the form
2) Communication delay: To compensate for delays in the communication channel, we will assume that the synchronization signal available to the response system is delayed by “ ” seconds, where is not assumed to be known.
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Fig. 1. Common synchronization structure for chaotic systems with time delay.
Fig. 1 shows the synchronization scheme proposed here. In this figure, the drive system is the original chaotic system with parametric uncertainty, and is a finite time delay which is an unknown constant. The response system consists of a copy of the original chaotic system and a filter term to be designed to achieve the synchronization of the response system with the drive system. Let and be the states of the drive and response systems, respectively. Then, we will say that the state trajectory of the response system robustly synchronizes with that of the drive system if the state of the response system at time asymptot, i.e., ically approaches the state of the drive system at time (2) In [10], the authors considered synchronization of quadratic chaotic systems of the form (1) without model uncertainty and time delay by first transforming the system (1) into linear parameter-varying (LPV) form. Then, based on the gain scheduling technique, two different synchronization structures were proposed to achieve the asymptotical convergence of synchronization errors. In this paper, we extend our work to the case of robust synchronization of quadratic chaotic systems with both parametric uncertainty and unknown channel time delay.
III. ROBUST GAIN SCHEDULING SYNCHRONIZATION METHOD From the definition of robust chaos synchronization with time delay, the final objective is to make the synchronization error dynamics asymptotically stable at the origin. As mentioned in Section I, we pursue this objective extending the gain scheduling technique introduced in [10] to include delay and parameter uncertainty. We begin by transforming the system (1) into LPV form, following the approach in[10]. Denoting the maximum value by and the minimum value by of state , we obtain the time-varying parameters and as
It is clear that
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and . Then, the quadratic chaotic system (1) can be identically transformed into LPV form
denote the synchronization error. Let Deriving from the drive system (5) and the response system (6), the synchronization error dynamics is
(3) where matrices and are all constant and are matrices and parameters used as scheduling variables. and the unConsidering now the parametric uncertainty in the known channel time delay , we replace with original chaotic system (1) and its LPV form (3) to obtain the as drive system in Fig. 1 at time (7) (4)
(5) in which
From condition (2), the robust synchronization problem with time delay is then equivalent to the problem of setting the error dynamics (7) asymptotically stable at the origin. Based on the gain scheduling technique, we propose two different response systems to achieve global synchronization for the quadratic chaotic systems with both parametric uncertainty and channel time delay. A. First Synchronization Structure In this section, we present the first synchronization scheme. From the error dynamics (7), we define matrices , , and as follows:
Assumption 1: There are parameters, , in ma, the paratrix . Then, for each parameter metric uncertainty is denoted by . From Assumption 1, it is evident that we can obtain the following transformation:
in which
is a linear combination of state and . To solve the robust synchronization problem of the quadratic system(4) or (5) with time delay, the response system in Fig. 1 is given a general form
where , , , , and are constant matrices to be designed. Considering now the uncer, we propose the tain quadratic chaotic system (5) at time first gain scheduling synchronization structure of the following form:
(6)
, in which matrices to be designed.
,
, and
are varying
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(8)
LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
With this definition, the synchronization error dynamics is
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Substituting these identities in(9), we obtain the following error dynamics:
To prove the stability of the error dynamics, we consider the . For this function, we Lyapunov function have
By construction, the parameters are always greater and . Moreover, than or equal to zero all of them cannot be zero at the same time. Then, with the inequality condition in (13) satisfied, it is easy to see that
(9) The convergence of the synchronization error can then be guaranteed by the following theorem. Theorem 1: If there exists a symmetric matrix , matrices and and and , and constant matrix such that conditions(10)–(13) are satisfied simultaneously for every pair and , of and at (10) (11) (12) (13) then the synchronization structure (8) robustly synchronizes with the drive system (4) or(5) globally. Proof: Assuming that(10)–(13) are satisfied for every pair of and , it is straightforward to see that
It follows from the second Lyapunov stability theorem (see, for example, [14]) that the synchronization error dynamics is globally asymptotically stable at the origin. Thus, from the definition of robust chaos synchronization with time delay, the gain scheduling structure (8) and the quadratic chaotic system (4) or (5) are globally asymptotically synchronized. Remark 1: In most cases, after transforming the original system(1) into LPV form, there is only one or two pairs of paand appearing in (3). The rameter design procedure is easy to complete as conditions (10)–(13) can be easily solved by using linear matrix inequality (LMI) tools. This point will be illustrated in the example section. We conclude this section emphasizing that the proposed robust gain scheduling structure(8) achieves global synchronization for the quadratic chaotic system, even in the presence of parametric uncertainty and unknown time delay . B. Alternative Synchronization Structure A shortcoming of the synchronization structure (8) proposed in the previous section is that it requires knowledge of the . This means that, in varying parameters 's at time addition to the transmitted signal , some or all thestates of the drive chaotic system are also needed, thus restricting the application value of the result. In this section, we modify the previous scheme using an alternative structure. is available at Assuming that only the output signal the received end, we define
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The alternative gain scheduling synchronization structure is proposed as follows:
(15) From the definition of parameters
and , it is obvious that
(14) where
Using straightforward algebraic manipulations, we obtain
It is immediate that the definition of these varying parameand ters has the same form as that of parameters . Then, the synchronization error dynamics in this case is
where each and is a constant matrix obtained from direct algebraic transformation. Thus, the error dynamics(15) is
The convergence of the synchronization error is proven in the following theorem. , maTheorem 2: If there exists a symmetric matrix and and trices and , and constant matrix such that conditions(16)–(19) are satisfied simultaneously for every pair and , of and at (16) (17) (18) (19) Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on June 17, 2009 at 16:31 from IEEE Xplore. Restrictions apply.
LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
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where
then the synchronization structure (14) robustly synchronizes with the drive system (4) or (5) globally. Proof: The proof follows by direct application of the second Lyapunov stability theory with Lyapunov function as . Remark 2: As mentioned earlier, there are only one or two shown in the inequality condition (19). pairs of parameters With simple algebraic transformations, we can still rewrite the inequality condition such that the effect of the varying parameand is taken off, and then solve the condition ters easily by using the LMI technique. The detailed transformation and procedure will be shown in Section 4. Remark 3: In general, the alternative synchronization structure (14) is suitable for practical applications and thus preferable over (8). In the special case in which the specific state needed to construct the varying parameters 's is the same as the output signal , the first synchronization structure will be more direct and much easier to design and has much better performance. This point will be shown in Example 2 in the next section.
(21) where
Defining an output signal , and assuming that there is uncertainty on parameter , the Chen system(20) or (21) with output and parametric uncertainty at time is
IV. EXAMPLES In this section, we consider two examples to illustrate the efficiency of the proposed gain scheduling synchronization methods, using well-known chaotic systems.
(22) or, equivalently
A. Chen System The chaotic Chen system [3] is described by (23)
(20) where, accordingly in which is the state vector. When , and , the system (20) displays chaotic beis havior. Assuming that the bounded value for state , we obtain the varying parameters 's as
It is apparent that that be rewritten as follows:
and and . Then, the Chen system(20) can
Then, the first gain scheduling synchronization structure proposed in this paper is
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Subtracting (24) from (23), we obtain the following error dynamics:
(25) is the synchronization error. where Since the parameters 's are nonnegative, conditions (10)–(13) in Theorem 1 for the synchronization errors are
Fig. 2. Three synchronization errors of e(t; ) = x(t system under the first structure.
0 ) 0 x^(t) of the Chen
For the alternative synchronization structure, we have The bounded values used in this example are chosen to be and . These values were selected by simulating the system (20) with the indicated parameters for different initial conditions, and noticing that each state trajectory falls in a bounded section, which are defined approximately as , and . Using follows: the LMI technique, we obtain
(26) where
Subtracting (26) from (23), we obtain
(27) Using the definition of
and , the error dynamics (27) is
Then, with the aforementioned chosen parameters, the two systems (22) or (23) and (24) are globally asymptotically synchronized under the existence of parametric uncertainty and s, , and a constant time delay . Assuming that and the initial conditions are , Fig. 2 shows the synchronization errors. Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on June 17, 2009 at 16:31 from IEEE Xplore. Restrictions apply.
(28)
LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
Denoting now the bounded values of state the new varying parameters are
by
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,
Then, for the error dynamics (28), conditions (16)–(19) in Theorem 2 are
Fig. 3. Three synchronization errors of e(t; ) = x(t system under the second structure.
0 ) 0 x^(t) of the Chen
where
The aforesaid inequalities can also be easily solved using LMIs. The resulting matrices are
the two synchronization structures presented earlier. The results show that the proposed structures achieve synchronization despite random delay. To illustrate the significance of the robust synthesis proposed here, we consider the same system with small uncertainty on the parameter (not considered in the design). The simulation on are shown in Fig. 6 for the first results with synchronization method and Fig. 7 for the second one. The figures clearly show that, unless uncertainty is incorporated as part of the design, even a small disturbance on a parameter can cause loss of convergence.
B. Lorenz System In this example, we consider the well-known Lorenz system, which is described by (29)
Thus, with the alternative synchronization structure, the two systems(22) and (26) are also globally asymptotically synchroand s also, the simulation result is nized. With shown in Fig. 3. To further illustrate our results, the next simulations shown in Figs. 4 and 5 consider a randomly selected time delay for
is the state vector. When in which , and , the system (29) presents chaotic behavior. By simulating the Lorenz system with different initial conditions, we see that each state trajectory stays in a bounded region with state bounds that can be approximated as follows: , , and . Because the exact values will not affect the internal convergence of the proposed methods, it is enough to choose values which contain the corresponding sections. Here, we choose the following state bounds: , , and . In order to satisfy the conditions in the proposed theorems, the output is introduced. The transformation process signal and synchronization structures are similar to those used in the previous example.
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Fig. 4. Three synchronization errors of e(t; ) of the Chen system under the first structure with different .
Fig. 5. Three synchronization errors of e(t; ) of the Chen system under the second structure with different .
For the first synchronization structure, using conditions (10)–(13), we obtain
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LIANG AND MARQUEZ: SYNCHRONIZATION METHOD FOR CHAOTIC SYSTEMS WITH CHANNEL TIME DELAY
Fig. 6. Three synchronization errors of e(t; ) = x(t system with b disturbance under the first structure.
0 ) 0 x^(t) of the Chen
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0 0
Fig. 8. Three synchronization errors of e(t; ) = x(t ) x ^(t) of the Lorenz system under the first structure.
0 0
Fig. 7. Three synchronization errors of e(t; ) = x(t system with b disturbance under the second structure.
0 ) 0 x^(t) of the Chen
Fig. 9. Three synchronization errors of e(t; ) = x(t ) x ^(t) of the Lorenz system under the second structure.
Assuming that initial conditions and , and assuming that parametric unceron the parameter , simulation results under the tainty first synchronization structure are shown in Fig. 8. For the second synchronization structure, solving the conditions (16)–(19) using LMIs, we obtain
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V. CONCLUSION In this paper, we have extended the synchronization method proposed in [10] to the important case of quadratic chaotic systems with parametric uncertainty and unknown time delay. Our method consists of transforming the original system into LPV form, a transformation that always exists for quadratic chaotic systems, and then using LMIs and the gain scheduling approach to ensure synchronization. Our approach is easy to apply and can benefit from readily available software tools to solve LMI problems.
REFERENCES
0 ) 0 x^(t) of the
Fig. 10. Three synchronization errors of e(t; ) = x(t Lorenz system with c disturbance under the first structure.
0 0 x^(t) of the
Fig. 11. Three synchronization errors of e(t; ) = x(t ) Lorenz system with c disturbance under the second structure.
With the same initial conditions, the simulation results are shown in Fig. 9. As in the previous example, we test the efficacy of uncertainty in a parameter that is not included in our original design. Asin parameter , we obtain the suming that uncertainty simulations shown in Figs. 10 and 11. The simulations show the efficiency of the synchronization structures, as well as the need to consider robustness in the design. In this example, the state needed for varying parameters 's . In this is , which is the same as the output signal case, the first synchronization structure is also suitable for practical application. In this case, the design of the first structure is simpler, and performance is much better than using the second scheme.
[1] M. Boutayeb, M. Darouach, and H. Rafaralahy, “Generalized statespace observers for chaotic synchronization and secure communication,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 3, pp. 345–349, Mar. 2002. [2] T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEE Trans. Circuits Syst., vol. 38, no. 4, pp. 453–456, Apr. 1991. ˇ [3] S. Celikovský and G. Chen, “Synchronization of a class of chaotic systems via a nonlinear observer approach,” in Proc. IEEE Conf. Decision Control, 2002, pp. 3895–3900. [4] M. Chen, Z. Han, and Y. Shang, “General synchronization of Genesio–Tesi systems,” Int. J. Bifurc. Chaos, vol. 14, pp. 347–354, 2004. [5] M. Feki, “Observer-based exact synchronization of ideal and mismatched chaotic systems,” Phys. Lett. A, vol. 309, no. 1, pp. 53–60, Mar. 2003. [6] G. Grassi and S. Mascolo, “Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, no. 10, pp. 1011–1014, Oct. 1997. [7] G. P. Jiang, W. X. Zheng, and G. Chen, “Global chaos synchronization with channel time-delay,” Chaos Solitons Fractals, vol. 20, no. 2, pp. 267–275, Apr. 2004. [8] Z. Jiang, “A note on chaotic secure communication systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 92–96, Jan. 2002. [9] K. Lian, P. Liu, T. Chiang, and C. Chiu, “Adaptive synchronization design for chaotic systems via a scalar driving signal,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 17–27, Jan. 2002. [10] Y. Liang and H. J. Marquez, “Gain scheduling synchronization method for quadratic chaotic systems,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 55, no. 4, pp. 1097–1107, May 2008. [11] T. Liao and N. Huang, “An observer-based approach for chaotic synchronization with applications to secure communications,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 9, pp. 1144–1150, Sep. 1999. [12] J. Q. Lu and J. D. Cao, “Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters,” CHAOS, vol. 15, no. 4, pp. 1–10, Nov. 2005. [13] T. D. Ma, H. G. Zhang, and Z. L. Wang, “Impulsive synchronization for unified chaotic systems with channel time-delay and parameter uncertainty,” ACTA Physica Sinica, vol. 56, no. 7, pp. 3796–3802, 2007. [14] H. J. Marquez, Nonlinear Control Systems: Analysis and Design. Hoboken, NJ: Wiley, 2003. [15] M. J. Ogorzalek, “Taming chaos—Part I: Synchronization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 40, no. 10, pp. 693–699, Oct. 1993. [16] J. H. Park, “Synchronization of Genesio chaotic system via backstepping approach,” Chaos Solitons Fractals, vol. 27, no. 5, pp. 1369–1375, Mar. 2006. [17] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, no. 8, pp. 821–825, 1990. [18] W. J. Rugh and J. S. Shamma, “Research on gain scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, Oct. 2000. [19] Y. Suzuki, M. Iwase, and S. Hatakeyama, “A design of chaos synchronizing system using adaptive observer,” in Proc. SICE, 2002, pp. 2352–2353.
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[20] J. Xiang, Y. Li, and W. Wei, “An improved condition for master–slave synchronization of Luré systems with time delay,” Phys. Lett. A, vol. 362, no. 2/3, pp. 154–158, Feb. 2007. [21] M. E. Yalcin, J. A. K. Suykens, and J. Vandewalle, “Master–slave synchronization of Luré systems with time-delay,” Int. J. Bifurc. Chaos, vol. 11, pp. 1707–1722, 2001. [22] J. J. Yan, Y. S. Yang, T. Y. Chiang, and C. Y. Chen, “Robust synchronization of unified chaotic systems via sliding mode control,” Chaos Solitons Fractals, vol. 34, no. 3, pp. 947–954, Nov. 2007. [23] T. Yang, X. Li, and H. Shao, “Chaotic synchronization using backstepping method with application to the Chua’s circuit and Lorenz system,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 2299–2300. Yu Liang received the B.S. and M.S. degrees in automatic control from Beijing Institute of Technology, Beijing, China, in 2000 and 2003, respectively. She is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. Her research interests include observer design for nonlinear systems with specific characters and application to synchronization problem of chaotic systems.
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Horacio J. Marquez received the B.Sc. degree from Instituto Tecnológico de Buenos Aires, Buenos Aires, Argentina, in 1987 and the M.Sc.E. and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, in 1990 and 1993, respectively. From 1993 to 1996, he held visiting appointments at Royal Roads Military College, Victoria, BC, Canada, and the University of Victoria, Victoria. 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 the Department Chair. In 2008, he was a Guest Research Professor with the Université Henri Poincaré, Nancy, France. He is currently an Area Editor for the International Journal of Robust and Nonlinear Control and an Associate Editor for theJournal of the Franklin Institute. 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 was the recipient of the 2003–2004 McCalla Research Professorship awarded by the University of Alberta.
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