A New Scheme on Synchronization of Commensurate Fractional-Order Chaotic Systems Based on Lyapunov Equation

This paper proposes a new fractional-order approach for synchronization of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. A simple but practical method to synchronize many familiar fractionalorder chaotic systems has been put forward. A new theorem is proposed for a class of cascade fractional-order systems and it is applied in chaos synchronization. Combined with the fact that the states of the fractional chaotic systems are bounded, many coupled items can be taken as zero items.Then, the whole system can be simplified greatly and a simpler controller can be derived. Finally, the validity of the presented scheme is illustrated by numerical simulations of the fractional-order unified system.


Introduction
Fractional calculus, with more than 300-year-old history, is generalization of ordinary differentiation and integration to arbitrary order.Until the recent decades, the fractional calculus attracted attention of researchers in various fields [1][2][3][4][5].Many systems in physics and engineering have been found to be effectively modeled as fractional-order systems [6][7][8].Fractional-order systems could also behave chaotically.Recently, it has been demonstrated that some fractionalorder differential systems, including fractional-order Lü system, fractional-order Chen system, fractional-order coupled dynamos system, fractional-order Liu system, and fractionalorder unified system [9][10][11][12][13] exhibit chaotic behaviors.
Integer order chaotic systems have been extensively studied for many decades [14,15], whereas their fractionalorder counterparts have just recently been investigated.An extensive survey of various fractional-order chaotic systems can be found in [4].Chaos control of fractional-order chaotic systems has received a lot of considerations of researchers.However, compared to integer order chaotic systems, the research results in this field are relatively rare.For the limitation of the available theoretical tools, a number of research works mainly focus on integer order chaotic systems.
Chaos control based on stability theories for linear fractional-order systems has been extensively proposed in [16,17].However, this approach provides only local stability.Active control is another approach that can be found in the literature [18,19].The approach made use of a nonlinear control law to cancel nonlinearities in the control system.Based on the stability theory for linear fractionalorder systems, a linear control law was designed to stabilize the linearized system.Reference [20] presented LMI-based stabilization method for fractional-order chaotic systems.The method required that the systems must be transformable into a linear interval fractional-order system.In [21,22], the authors proposed a sliding mode control approach to realize chaos control.By adopting a fractional-order sliding surface, the stability can be obtained via the Lyapunov stability method.However, the fractional-order sliding surface might be difficult to implement.Moreover, discontinuous control signals could induce some undesired behavior.
Many of the above controllers are nonlinear.From a practical point of view, to design a simple linear controller by a special strategy is more valuable.However, to our best knowledge, there is a limitation of the available theoretical tools that can be used for nonlinear fractional-order systems.Therefore, designing a linear controller for fractional-order chaotic systems has still remained as an open and challenging problem.
Motivated by the above discussions, in this paper, based on fractional-order Lyapunov stability theory, a new strategy is proposed to synchronize a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances.Then, a simple linear controller with low dimensions is designed via the proposed new strategy.Compared with the control approach in [20], only two simple and linear feedback controllers are designed to achieve synchronization.Compared with the control approach in [21,22], it can be implemented easily.
The rest of this paper is organized as follows: in Section 2, some preliminaries and main results are presented.In Section 3, we give the famous fractional-order unified chaotic systems to illustrate that it can be transformed to our cascade form for further simple design.Experimental analyses are presented in Section 4. Finally, conclusions are given in Section 5.

Preliminaries and Main Results
In this section, first some basic definitions and lemmas are briefly presented.Then a useful theorem is put forward to be utilized later to design a robust linear control law.Definition 1.The definition of fractional integral is described by where Γ(⋅) is the well-known Gamma function.
Lemma 4 (see [23]).Let  ∈  × be a real matrix.Then, a necessary and sufficient condition for the asymptotical stability of   () = () is where 0 <  < 1, () is the spectrum of all eigenvalues of .
In this paper, we consider the case of the synchronization of two commensurate fractional-order chaotic systems.The master system is as follows: where  = ( 1 ,  2 , . . .,   )  ∈   is the state vector, () is the parametric coefficient matrix of states , and 0 <  ≤ 1 is the fractional commensurate order.The slave system is where () is the controller to be designed.Define () = x() − ().We can get the error system: where (, ) = (x)x − ().The synchronization problem can be transformed to design a controller  such that lim →∞ ‖()‖ = 0.

Journal of Control Science and Engineering 3
Proof.Since the subsystem is already globally asymptotically stable at  1 = 0, we only need to consider the subsystem From assumption (2) we know that lim →∞ ( 1 ) = 0. Consider the fact that ( 1 ,  2 ) is a bounded matrix.From Lemma 3 we know that lim This means that the second item in the system can be neglected when  → ∞.To study the asymptotical stability problem of system ( 9), we only need to consider the following system: Here, we take three steps to illustrate that system (15) is globally asymptotically stable at the equilibrium ( 1 ,  2 ) = (0, 0).Firstly, let us consider the following equation: where  is one of the eigenvalues of ( 1 ,  2 ) and  is a nonzero eigenvector of .Take the conjugate transpose on either side of ( 16) and we can get Combine ( 16) with ( 17) and we can get Secondly, because ( 1 ,  2 ) has the structure we can know that the matrix ( 1 ,  2 ) satisfies the continuous Lyapunov equation where  =  is the real symmetric identity matrix and is a Hermitian matrix.Moreover, we can have Namely, ( 1 ,  2 ) + (( 1 ,  2 ))  is also a Hermitian matrix.
Lastly, according to the properties of positive definite and negative semidefinite matrix, we can get two inequalities Consequently, combining (18), we can have Obviously, any eigenvalue  of the coefficient matrix ( 1 ,  2 ) satisfies the following inequality: From Lemma 4, we know that system ( 15) is globally asymptotically stable at the equilibrium ( 1 ,  2 ) = (0, 0).This completes the proof.

Illustrative Example
In the first part of section, the fractional-order unified chaotic system and its synchronization are transformed to our cascade form and a simpler controller is derived.Then its robustness analysis will be introduced.

Fractional-Order Unified Chaotic System and Its Synchronization.
Similar to the classical unified chaotic system, the fractional-order unified system could be considered as the system that bridges the gap among the fractionalorder Lorenz system, the fractional-order Lü system, and the fractional-order Chen system.The fractional-order unified chaotic system [13] is described by where    /   =    * ( = 1, 2, 3).Its order is denoted by  = ( 1 ,  2 ,  3 ) subject to 0 <  1 ,  2 ,  3 ≤ 1, and  ∈ [0, 1].Some examples of the chaotic attractor of system (26) with  1 =  2 =  3 = 0.95 are shown in Figures 1 and 2.
Two systems in synchronization are called the master system and the slave system, respectively.From (26), the master system and the slave system can be expressed, respectively, as slave system where  1 ,  2 are the control signals used to drive the slave system to follow the master system.Denote the synchronization error as  =  − .Our aim is to design a controller () = ( 1 ,  2 )  such that controlled system (28) asymptotically synchronizes master system (27) in the sense that lim Subtracting ( 27) from ( 28), then the synchronization error equation can be obtained as Here, we take two steps to design a linear controller to globally asymptotically stabilize error system (30).
Step 1. From the proof process of Theorem 5 we know that if the matrix (⋅) satisfies assumption (3) in Theorem 5, system (15) is asymptotically stable.If we can design a controller  1 such that it has a similar form to (15),  1 will be stable.Let  1 = −(25 + 10) 2 and the first subsystem of (28) becomes Obviously, for each  ∈ [0, 1], it is globally asymptotically stable at  1 = 0.
Step 2. A similar idea to  1 is used in the design of the controller  2 .Let  2 = − 2 ( > 29 − 1).Consider the remaining subsystem of (30) and substitute  2 = − 2 ( > 29 − 1) into the remaining subsystem.Then we can have Equation ( 32) can be rewritten by the matrix form as where Because master system ( 27) is a chaotic system, its states are bounded.This means that there exists a positive constant  > 0 such that So the matrix  is also a bounded matrix.Obviously, () has a special form that satisfies assumption (3) in Theorem 5. Form Theorem 5, error system (30) is globally asymptotically stable at (0, 0, 0), and slave system (28) will synchronize with master system (27).

Experimental Analyses
In this section, the predictor-corrector method is used to obtain the solutions of fractional-order differential equations with step size 0.001.If the step size is too large, the simulation results will not converge.Many authors in other references have adopted the step size as 0.001.And we have also adopted variable step size to do the simulation and we found the simulation results were better.In the simulation process for the fractional-order unified system, we let  1 =  2 =  3 = 0.95 and choose the initial states of the master and slave system as ( 1 (0),  2 (0), (0))  = (1, 2, 3)  , ( 1 (0),  2 (0), (0))  = (−1, −2, 6)  .
When  = 0 system (26) is a fractional-order Lorenz chaotic system.When  = 0.8 system (26) is a fractional-order L ü chaotic system and when  = 1 system (26) is a fractional-order Chen chaotic system.The simulation results for synchronization of the fractionalorder Lorenz, Lü, and Chen chaotic systems with known parameters are shown in Figures 3-5, respectively.We chose the uncertain parameters (Δ 1 , Δ  the trajectories of the closed loop slave system can synchronize the trajectories of the master system.These results of the simulation verify the effectiveness of the proposed scheme.

Conclusions
This paper discusses a linear control scheme for synchronizing a class of cascade fractional-order chaotic systems.Based on continuous Lyapunov equation, the stability of the closed loop system is proved.This synchronization approach is simple, global, and theoretically rigorous.Numerical simulations have been used to clarify the effectiveness of the proposed control laws.It should be noted that the introduced fractional linear controller is applicable for a large class of commensurate fractional-order chaotic systems in the presence of model uncertainties and eternal disturbances.
the rest of the paper,