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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 4, MAY 2008

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Gain Scheduling Synchronization Method for Quadratic Chaotic Systems Yu Liang and Horacio J. Marquez

Abstract—A global gain scheduling synchronization method is developed in this paper for the identical synchronization of quadratic chaotic systems. The quadratic chaotic system contains nonlinearity of quadratic terms of system’s states. With chaotic states being bounded in certain regions, the quadratic chaotic system can be rewritten into the linear parameter varying (LPV) form through algebraic transformations. Then, using the gain scheduling technique, two different synchronization structures are proposed to achieve the global synchronization for the quadratic chaotic system. The convergence of the synchronization errors is guaranteed under the second Lyapunov stability theory. Generalized Lorenz systems, such as the Chen system and the Lorenz system, are illustrated as examples to demonstrate the efficiency of the proposed methods. Index Terms—Chaos synchronization, gain scheduling, generalized Lorenz system, linear matrix inequality (LMI), linear parameter varying (LPV).

I. INTRODUCTION HAOS synchronization consists of using an output of a chaotic system, called the drive system, to control another chaotic system, called the response system, so that the states of the response system converge to those of the drive system asymptotically. Chaotic dynamics are characterized by critical sensitivity on initial conditions, in which infinitesimal changes in the initial conditions can lead to an exponential divergence of orbits. Chaotic signals are also broad-spectrum and noiselike. These complex characteristics make chaos synchronization very difficult but potentially very important in enabling secure communications. In 1990, Pecora and Carroll [2], [18] observed that two identical chaotic systems can be synchronized under a master–slave structure. Motivated by their pioneering work, chaos synchronization has attracted much attention over the last 15 years, and many other techniques have been proposed by various authors to achieve synchronization under different conditions. Although a few researchers such as Kocarev et al. [9] and Parlitz et al. [17] have considered the generalized synchronization problem consisting of forcing the synchronization of two different chaotic systems, much more attention has been focused on the case of synchronization of two coupled similar or

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Manuscript received August 10, 2005; revised February 16, 2007. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. This paper was recommended by Associate Editor C.-T. Lin. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: marquez@ece. ualberta.ca). Digital Object Identifier 10.1109/TCSI.2008.916434

identical chaotic systems, in the future referred to as the identical synchronization problem. Usually, the identical synchronization problem is studied using a master-slave configuration [22], in which the coupling of the two chaotic systems is unidirectional. Under the master–slave structure, chaos synchronization can be studied in two ways. On the one hand, chaos synchronization can be tackled as a controller design problem. In this case, the principle consists of considering the error dynamics defined as the difference between the state vectors of the two chaotic systems to be synchronized and using control methods to force convergence to zero of the errors. Ogorzalek [15] considered the possibilities of synchronization using linear coupling of chaotic systems in spite of the Pecora–Carroll concept of synchronization. Park [16] and Yang et al. [23] used the popular backstepping methods to achieve chaos synchronization for certain chaotic systems. On the other hand, chaos synchronization can also be tackled as an observer design problem. A common approach for the observer design problem of chaos synchronization is to use a copy of the drive system as the response system plus a compensation term which aims at attenuating the difference between the states of the drive system and the states of the response system. Most of the existing synchronization methods can be seen as deriving from the observer design technique. Grassi and Mascolo [6], [7] considered a chaotic system as described by a linear system plus an output measurable nonlinearity, which resulted in synchronization by only designing an observer for the linear part and avoiding the difficult task of dealing with nonlinear observer design. Millerioux [13] used a similar idea to solve the synchronization problem with the difference that the linear part was replaced using a polytopic system. The disadvantage of this approach was that it required the drive system’s nonlinearity to be sent simultaneously with the output to the response system, a need that prevented this scheme from being of interest in secure communication. Many other papers considered chaos synchronization as a nonlinear observer design problem. For example, Nijmeijer [14] proposed an observer using the complex nature of the drive systems through several examples. Celikovsky and Chen [3] used coordinate transformations to obtain an observer canonical form for the generalized Lorenz system and then solved the synchronization problem. Feki [5] studied the synchronization problem for chaotic systems with special triangular form, based on the sliding mode theory. Suykens et al. [20] studied the parameter mismatch between the drive and the response system for the Lur’e systems using robust synthesis. Lian et al. [10] and Suzuki et al. [21] proposed adaptive synchronization methods for several chaotic systems with unknown parameters. With detailed application to secure communication, several observer-based synchronization structures were given such as [1], [8], and [11].

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Due to the complexity of chaotic systems, all of the existing synchronization methods are only applicable to a special class of chaotic systems with particular characteristics. In this paper, we consider the identical synchronization problem for quadratic chaotic systems, based on the observer design approach. Quadratic chaotic systems, understood as systems that contain nonlinearities of quadratic terms of systems’ states, appear very frequently, and the quadratic property is very easy to verify. Many well-known chaotic systems can be classified into this class. With all of the states of a chaotic system being bounded to certain regions, quadratic chaotic systems can be rewritten into linear parameter varying (LPV) forms using straightforward algebraic transformations. Then, the gain scheduling concept can be used to achieve identical synchronization of quadratic chaotic systems. Based on this principle, two different gain scheduling synchronization structures are proposed and the global convergence of the synchronization errors is proved using the second Lyapunov stability theorem. The remainder of this paper is organized as follows. In Section II, we define the problem to be solved and introduce briefly the gain scheduling technique for completeness. In Section III, based on the gain scheduling technique, two different gain scheduling synchronization structures are proposed, and the global convergence proofs of the synchronization errors are given correspondingly. In Sections IV and V, the generalized Lorenz system, including Chen system and Lorenz system, are used as examples to illustrate the efficiency of the proposed design methods. Finally, Section VI contains conclusions and final remarks. II. PRELIMINARIES This section consists of two parts. In Section II-A, we define the class of chaotic systems to be considered and the concept of identical chaos synchronization. In Section II-B, we introduce the basic background on the gain scheduling technique that will be used in later sections. Throughout this paper, represents the field of real numbers, denotes the field of natural numbers, denotes -dimensional real vector space, and represents the set of all real ( ) matrices. A. Problem Definitions In this paper, we consider a chaotic system which can be described by (1) is the state, is the output as the transmitted where , , and are conscalar signal, is the nonlinear term. When the nonstant matrices, and only contains quadratic terms of systems’ states, linearity , we call the system (1) i.e., a quadratic chaotic system. Many chaotic systems can be classified into this class such as the following examples. 1) Example 1: Genesio–Tesi system ([4])

Fig. 1. Common synchronization structure for chaotic systems.

with and system is equal to

. Letting

, the

2) Example 2: Rössler hyperchaotic system ([6])

with , , and . This system also belongs to this class. 3) Example 3: The generalized Lorenz system ([3]). This system will be discussed in detail in the example section. The following definition [7] introduces the concept of identical chaos synchronization. Definition 1: Given two chaotic systems, the dynamics of which are described as (2) (3) , , and is a nonlinear where vector field. Systems (2) and (3) are said to be synchronized if as where represents the synchronization error . Generally, the identical chaos synchronization problem consists of using an output of a chaotic system to control another same or similar chaotic system such that the two chaotic systems are asymptotically synchronized. Fig. 1 shows the basic principle. The drive system is the original chaotic system, the response system is always a similar or exact copy of the origis a filter to be designed inal chaotic system, and the block to achieve the synchronization of the response system with the drive system. B. Gain Scheduling Technique The gain scheduling technique is a widely used technique for controller design. In this paper, we propose the use of the gain scheduling technique to solve the synchronization problem for the quadratic chaotic system (1). For completeness, we briefly introduce the gain scheduling technique in this section. According to [19], the design of a gain scheduling controller for a nonlinear system can be described as a four-step procedure, though differing technical avenues are available in each step. The first step is to compute an LPV model for the nonlinear system. Generally, the most common approach is based on Jacobian linearization of the nonlinear system about a family

LIANG AND MARQUEZ: GAIN SCHEDULING SYNCHRONIZATION METHOD FOR QUADRATIC CHAOTIC SYSTEMS

of equilibrium points. This yields a parameterized family of linearized systems and forms the basis for linearization scheduling. Another common approach is quasi-LPV scheduling, in which the system dynamics are rewritten to disguise nonlinearities as time-varying parameters that are used as scheduling variables. The second step is to design linear controllers for the linear parameter varying system model that arises in either linearization or quasi-LPV scheduling. This design process may result directly in a family of linear controllers corresponding to the linear parameter dependent plant, or there may be an interpolation process to arrive at a family of linear controllers from a set of controller designs at isolated values of the scheduling variables. The third step, also called the actual gain scheduling step, involves implementing the family of linear controllers such that the controller gains are scheduled according to the current value of the scheduling variables. The final step is performance assessment. In the best case, the analytical performance is guaranteed as a part of the design process. More typically, the local stability and performance properties of the gain scheduled controller might be subject to analytical investigation, while the nonlocal performance evaluation is based on simulation studies. The central idea of the gain scheduling technique is to make the controller gains be scheduled according to current values of scheduling variables. This makes the gain scheduling technique suitable for identical synchronization of the quadratic chaotic system, as will be shown by the following sections.

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The transformation is achieved by the quasi-LPV scheduling, which can be illustrated by the following example. Assume a four-dimensional dynamics

(6) behaves chaotically in which are constant values and is the output signal. Let and be the maximum and the minimum bound of the state . Then, we obtain the time-varying parameters as

and the system (6) can be rewritten as

(7) and , it is apparent From the definition of parameters and . Thus, the that system (7) can be identically expressed in LPV form as follows:

III. GAIN SCHEDULING SYNCHRONIZATION METHOD From the definition of identical chaos synchronization, the final objective is to make the synchronization error dynamics asymptotically stable to the origin. Here, we propose the use of the gain scheduling technique introduced above to solve the synchronization problem. To this end, the first step is to obtain an LPV model for the nonlinear chaotic system. For the quadratic chaotic system (1), it is well known that the states are all bounded to certain regions. and the Denoting the maximum value of the state by ( ), time-varying parameminimum value by ters and can be introduced by

(4) It is clear that

and . Then, the original quadratic chaotic system (1) can be identically transformed into the following LPV form:

(5) where the matrices ( trices and the parameters as the scheduling variables.

are all constant ma) are used

Remark 1: We emphasize that, for almost every known chaotic system which belongs to the quadratic chaotic systems, after the quasi-LPV transformation, only one pair of parameters and ( ) will appear in the LPV form and in (5). Then, we only have one pair of matrices which there are only one or two elements different as being or , which can be shown by the above example. and ( ) are equal to zero. All other pairs of Based on the gain scheduling technique, we propose two different response systems to achieve the global synchronization for the quadratic chaotic systems. A. Direct Gain Scheduling Synchronization Structure After obtaining the LPV form (5) of the quadratic chaotic system (1), the second step is to design a linear synchronization structure for the resulting linear system from the first step. According to Fig. 1, assume the drive system is a linear system of the form

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and Theorem 1: If there exist a symmetric matrix such that the following matrices inequalities:

(10) are satisfied simultaneously, a gain scheduling synchronization structure in the form of (8) can globally synchronize with the quadratic chaotic system (1) or (5). Proof: For the synchronization error dynamics (9), we choose the following Lyapunov function candidate:

Fig. 2. Direct gain scheduling synchronization structure for the quadratic chaotic system.

Then, we have that

For identical synchronization, we propose the response system to be a copy of the drive system as With all of the varying parameters and at any time, if the inequalities in condition (10) are satisfied simultaneously, it is easy to see that Let the error be and the filter be . can be solved by using well-established Then, the matrix as . Thus, in this linear design methods to achieve ( ), a step, for each linear system with corresponding linear synchronization structure can be designed, which results in a family of linear synchronization structures. Then, the third step is the actual gain scheduling step to implement the family of linear synchronization structures achieved in the second step such that the structure gains are scheduled according to the scheduling variables. For the quadratic chaotic system (5), we propose a direct gain scheduling synchronization structure as described by

(8) which can be shown in Fig. 2. In this figure, the response system is a similar copy of the original chaotic system and the matrices ( ) in the filter are to be designed to achieve the synchronization. . The Define the synchronization error vector as error dynamics derived from the quadratic chaotic system (5) and the synchronization structure (8) is (9) Then, the convergence of the synchronization errors can be guaranteed by the following theorem.

Then, based on the second Lyapunov stability theory, the synchronization error dynamics (9) is globally asymptotically stable at the origin. Thus, from the definition of chaos synchronization, the gain scheduling structure (8) and the quadratic chaotic systems (1) or (5) are globally asymptotically synchronized. Substituting into the inequality condition (10), the inequalities are then in standard linear matrix inequality (LMI) form, which can be solved by the LMI technique to ob. We can finally solve tain and by to obtain the matrix the direct gain scheduling synchronization structure (8). For the final performance assessment step, the analytical performance in this case is guaranteed as a part of the design process. Remark 2: In this synchronization structure, the plant model used in the observer needs to know the original time varying paand ( ), i.e., some or all of the states rameters are required to be known from the definition of the parameters. As mentioned before, for the purpose of communication, it is always convenient to transmit a single output signal. To achieve this goal, an alternative method is given in the next section. B. Alternative Gain Scheduling Synchronization Structure In the previously proposed gain scheduling synchronization structure, we notice that the varying parameters need to be known at all times. From their definition, that means that some or all the states of the drive chaotic system other than the transmitted signal are needed to form the synchronization structure. In practical applications such as in secure communications, this


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From the definition of the varying parameters and , we obtain

Then, using straightforward algebraic manipulations, the latter part of (12) can be transformed into

Fig. 3. Alternative gain scheduling synchronization structure for the quadratic chaotic system.

is not desirable. To solve this problem, an alternative gain scheduling synchronization structure is given. Receiving only the output signal from the original chaotic system, the alternative gain scheduling synchronization structure is as follows:

in which ( ) are constant matrices coming from the transformation. Although it is difficult to ex( ) in general explicit press matrices forms, they can be easily got in practical applications, which can be shown by the examples in the later sections. Thus, the error dynamics (12) is

(13)

(11)

Then, the convergence of the synchronization errors can be given by the following theorem. Theorem 2: If there exist a symmetric matrix and matrices such that the following inequality (14)

which can be shown in Fig. 3. In this structure, we have

These varying parameters are in the same definition form as for the parameters and ( ). If the initial conditions for the original chaotic system and the copy of chaotic and system are exactly the same, then we have ( ). The error dynamics derived from the quadratic chaotic system (1) or (5) and the alternative gain scheduling synchronization structure (11) is

is satisfied, then an alternative gain scheduling synchronization structure in the form of (11) will globally synchronize with the considered chaotic system (1) or (5). Proof: Based on the second Lyapunov stability theory, the proof of this theorem can be easily obtained using the Lyapunov . function For this synchronization structure, the inequality condition (14) can also be solved by the LMI technique with proper algebraic transformations. The analytical performance assessment in this case is also guaranteed as a part of the design process which is the best case. In Section IV, we present an example using the generalized Lorenz system which belongs to the class of quadratic chaotic systems considered here. IV. EXAMPLE OF THE GENERALIZED LORENZ SYSTEM Here, we consider the generalized Lorenz system as an illustrative example as: Definition 2 ([3]): The nonlinear system of ordinary differof the following form is called ential equations (ODEs) in the generalized Lorenz system:

(12)

(15)

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where matrix


2) real

As introduced in the previous section, the first gain scheduling synchronization structure for the generalized Lorenz system can be described by

(16)

(20)

Moreover, the generalized Lorenz system is said to be nontrivial if it has at least one solution that goes neither to zero nor to infinity nor to a limit cycle. The generalized Lorenz system given in (15) and (16) is equal to

and are to be designed. The synin which matrices chronization error dynamics is then given by

with eigenvalues

,

, and

is a (2

such that

(17)

From Theorem 1, if there exist a symmetric matrix and matrices , such that the inequalities of (10) for are satisfied, then the structure (20) achieves global identical synchronization for the generalized Lorenz system (15). Considering now the alternative synchronization structure, we obtain

For purpose of communication, we introduce the following output: (18) is a constant row vector. It is immediate that in which the generalized Lorenz system belongs to the class of quadratic chaotic systems (1). Assuming now that the bounded values for each state , and , reare spectively, we can introduce the time-varying parameters as defined in (4). Then, the generalized Lorenz system (17) with the output (18) can be transformed into the LPV form as

(21) Also, the matrices dynamics is

and

are to be designed. The error

(22) Replacing the varying parameters with their definitions, the latter part of (22) is

where written as

. Then, the error dynamics (22) can be

(19) which is equivalent to

in which

With parameters , , , and can also be transformed into Both

and and stant matrices

are constant matrices if the bounded values are known. For this LPV form, all other conin (5) are zero matrices.

, the error dynamics (22)


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in which

The error dynamics is thus

Fig. 4. (x ; x ) plot of the Chen system under the first structure. Top: the drive system. Bottom: the response system.

It then follows that, for this alternative gain scheduling synchronization structure (21), from Theorem 2, the stability proof and the condition for existence can be given by the following corollary. Corollary 1: If there exists a common symmetric matrix and matrices and such that the following inequalities:

are satisfied simultaneously, then the observer structure in form of (21) can achieve global identical synchronization for the original generalized Lorenz system (19) or (15). Proof: The proof is straightforward since all of the param, , , , , and are greater than or equal eters to zero and cannot be all zero at the same time. Choosing the Lyapunov candidate for the synchronization error dynamics (22) as , it is apparent that if the conditions in Corollary 1 are satisfied. Remark 3: Although the inequality condition in Corollary 1 looks quite complex and difficult, it is in fact easy to satisfy

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Fig. 5. Synchronization errors of the Chen system under the first structure.

Fig. 7. Synchronization errors of the Chen system under the second structure.

Fig. 8. (x ; x ) plot of the Lorenz system under the first structure. Top: the drive system. Bottom: the response system.

A. Chen System Fig. 6. (x ; x ) plot of the Chen system under the second structure. Top: the drive system. Bottom: the response system.

because the pair of matrices – and – . pairs

and

From [12], the well-known Chen system is described by

are very similar to the

V. SIMULATION RESULTS Here, the well-known Chen system and the Lorenz system are simulated to illustrate the efficiency of the above two synchronization structures.

When , , and , this system behaves chaotically. In our simulation, we first assume that the bounded region for is . Then, an output signal is introduced as


Fig. 9. Synchronization errors of the Lorenz system under the first structure.

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Fig. 11. Synchronization errors of the Lorenz system under the second structure.

tive definite matrix satisfying the inequality condition in Corollary 1. Then, under the same conditions, we obtain the simulation results shown in Figs. 6 and 7. Here, Fig. 6 represents the drive Chen system and the response Chen system and Fig. 7 represents the three synchronization errors. The simulation results clearly show the effectiveness of our approach. Moreover, for the Chen system, we point out that the classical master–slave method described by

cannot achieve the final synchronization. B. Lorenz System Fig. 10. (x ; x ) plot of the Lorenz system under the second structure. Top: the drive system. Bottom: the response system.

In MATLAB, all of the inequality conditions can be solved using the LMI tools. Under the first synchronization structure, the deand from Theorem 1 are signed gain values

With the initial conditions for the original chaotic system and the and , respectively, observer being we obtain the simulation results as shown in Figs. 4 and 5. Fig. 4 plots of the original Chen system and represents the the response Chen system. Fig. 5 represents the synchronization error vector between the drive and response systems. For the alternate synchronization structure, we can design the and from Corollary 1. In this simulation, gain values using the same values as for the first structure, we obtain a posi-

In [12], the celebrated Lorenz system is

When , , and , this system behaves chaotically. We proceed as in the Chen system. First, we assume the is and let the bounded region for the state . From Theorem 1, the designed gain output signal be values are

With these gain values, from Corollary 1, we can also obtain a positive definite matrix to satisfy the inequality condition. and With the initial conditions being , we obtain the following simulation results under the first structure shown in Figs. 8 and 9.

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Fig. 12. Synchronization errors of the Chen system using big regions. Left: under the first structure. Right: under the second structure.

Fig. 13. Synchronization errors of Lorenz system using big regions. Left: under the first structure. Right: under the second structure.

For the second synchronizing structure, the simulation results are given in Figs. 10 and 11. The simulation results show that the proposed gain scheduling structures successfully achieve the asymptotical synchronization for the Lorenz system. Compared with the synchronization method proposed in [3], the most important advantage of our method is that it does not require complex coordinate transformation and can be applied to much broader class of chaotic systems as opposed to the generalized Lorenz system only. Remark 4: As noticed earlier, our approach requires an estimate of the bounds of the trajectory of arbitrarily. Simulation results show that even coarse estimates of this bound have little difference on the results. Here is an example of bounded region being [ 250, 300] for Chen system. With other data being the

same, the simulation results for the two synchronization structures of the Chen system are shown in Fig. 12. For the Lorenz system, assuming that the bounded region is [ 100, 150], the simulation results are shown in Fig. 13. Comparing these results with the previous results having small regions, the performance has no significant difference. VI. CONCLUSION In this paper, we considered the identical synchronization problem for quadratic chaotic systems. Many existing chaotic systems can be classified as quadratic chaotic systems, including the generalized Lorenz system, the Genesio–Tesi system, and the Rössler hyperchaotic system. With straightforward algebraic transformations and given the bounded values


of the chaotic states, quadratic chaotic systems can be rewritten into LPV forms. Then, two different gain scheduling synchronization structures were proposed to achieve the global identical synchronization. The convergence of the synchronization errors was guaranteed under the second Lyapunov stability theory. Using the generalized Lorenz system as a general example, we give the design procedure and stability proof for the two proposed gain scheduling synchronization methods. The well-known Chen system and Lorenz system are simulated to successfully illustrate the efficiency of the two methods. Compared with the classical master–slave structure and the generalized Lorenz canonical form method, the proposed methods are easier to design and can be applied to a broader class of chaotic systems. REFERENCES [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. 2, pp. 345–349, Feb. 2002. [2] T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEE Trans. Circuits Syst., vol. 38, no. 3, pp. 453–456, Mar. 1991. [3] S. Celikovsky and G. Chen, “Synchronization of a class of chaotic systems via a nonlinear observer approach,” in Proc. IEEE Conf. Dec. Control, 2002, pp. 3895–3900. [4] M. Chen, Z. Han, and Y. Shang, “General synchronization of GenesioTesi 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, pp. 53–60, 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. 9, pp. 1011–1014, Sep. 1997. [7] G. Grassi and S. Mascolo, “Design of nonlinear observers for hyperchaos synchronization using a scalar signal,” in Proc. IEEE Int. Symp. Circuits Syst., 1998, vol. 3, pp. 283–286. [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] L. Kocarev, U. Parlitz, T. Stojanovski, and L. Panovski, “Generalized synchronization of chaos,” in Proc. IEEE Int. Symp. Circuits Syst., 1996, vol. 3, pp. 116–119. [10] 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. [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. 10, pp. 1144–1150, Oct. 1999. [12] J. Lu, G. Chen, and D. Cheng, “A new chaotic system and beyond: The generalized Lorenz system,” Int. J. Bifurc. Chaos, vol. 14, pp. 1507–1537, 2004. [13] G. Millerioux and J. Daafouz, “Polytopic observer for global synchronization of systems with output measurable nonlinearities,” Int. J. Bifurc. Chaos, vol. 13, pp. 703–712, 2003.

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[14] H. Nijmeijer, “On synchronization of chaotic systems,” in Proc. IEEE 36th Conf. Decision Control, 1997, pp. 384–388. [15] M. J. Ogorzalek, “Taming chaos– Part I: Synchronization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 40, no. 5, pp. 693–699, May 1993. [16] J. H. Park, “Synchronization of Genesio chaotic system via backstepping approach,” Chaos, Solitons Fractals, vol. 27, pp. 1369–1375, 2006. [17] U. Parlitz, L. Junge, and L. Kocarev, “Chaos synchronization,” in New Directions in Nonlinear Observer Design. Berlin, Germany: Springer-Verlag, 1999, vol. IV, pp. 511–525. [18] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, pp. 821–825, 1990. [19] W. J. Rugh and J. S. Shamma, “Research on gain scheduling,” Automatica., vol. 36, pp. 1401–1425, 2000. [20] J. A. K. Suykens, P. F. Curran, and L. O. Chua, “Robust synthesis for master-slave synchronization of Lur’s systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 7, pp. 841–850, Jul. 1999. [21] Y. Suzuki, M. Iwase, and S. Hatakeyama, “A design of chaos synchronizing system using adaptive observer,” Proc. SICE, pp. 2352–2353, 2002. [22] T. Ushio, “Synthesis of synchronized chaotic systems based on observers,” Int. J. Bifurc. Chaos, vol. 9, pp. 541–546, 1999. [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 the Beijing Institute of Technology, Beijing, China, in 2000 and 2003, respectively, and she is currently working toward the Ph.D. degree in electrical and computer engineering at the University of Alberta, Edmonton, AB, Canada. Her research interests are in observer design for nonlinear systems with specific characters and application to synchronization problem of chaotic systems.

Horacio J. Marquez received the B.Sc. degree from the Instituto Tecnologico de Buenos Aires (Argentina), in 1987 and the M.Sc.E and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, AB, Canada, in 1990 and 1993, respectively. From 1993 to 1996, he held visiting appointments with the Royal Roads Military College and the University of Victoria, Victoria, BC, Canada. Since 1996, he has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, where he is currently a Professor and Department Chair. He is 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 University of Alberta McCalla Research Professorship.