Synchronization of Uncertain Fractional-Order Hyperchaotic Systems via Unidirectional Linear Error Feedback Coupling Scheme

A simple method for synchronization of uncertain fractional-order hyperchaotic systems is proposed in this paper. The method makes use of a unidirectional linear coupling approach due to its simple configuration and ease of implementation. To determine the coupling parameters, the synchronization error dynamics is first formulated as a fractional-order linear interval system. Then, the parameters are obtained by solving a linear matrix inequality (LMI) stability condition for stabilization of fractional-order linear interval systems. Thanks to the existence of an LMI solution, the convergence of the synchronization errors is guaranteed. The effectiveness of the proposed method is numerically illustrated by the uncertain fractional-order hyperchaotic Lorenz system.


Introduction
Chaos synchronization has attracted great attention due to its superior potential applications, for example, in communication and optics.It has been studied since the pioneering work of Pecora and Carroll [1] was published.Currently, studies of chaos control and synchronization have more focus on hyperchaotic systems because of their rich chaos behaviors.
Although the notion of fractional calculus dates from the 17th century [2], its practical applications have just recently been investigated.In recent year, it has been demonstrated that fractional-order systems could behave chaotically or hyperchaotically.Examples of such systems include the fractional-order Chua system [3], the fractional-order Chen system [4], the fractional-order hyperchaotic Lorenz system [5], and the fractional-order hyperchaotic Chen system [6].
Nowadays, whereas chaos synchronization of conventional integer-order chaotic and hyperchaotic systems has been extensively studied, chaos synchronization of fractionalorder chaotic and hyperchaotic systems is still considered as a challenging research topic.Examples of existing methods for chaos synchronization of fractional-order chaotic and hyperchaotic systems are an active control method [7,8], a sliding-mode control method [9,10], a robust control method [11,12], an adaptive control method [13], and a tracking control-based method [14,15].The sliding-mode control, robust control, and adaptive control are commonly used methods to cope with the nonlinear systems that have some uncertain parameters.
This paper extends the robust control approach presented in [12] to synchronization of uncertain fractionalorder hyperchaotic systems.A unidirectional linear error feedback coupling scheme is adopted here due to its simple configuration and ease of implementation in real systems.The method makes use of an LMI stability condition of fractionalorder interval linear systems.An obtained solution to the LMI condition yields the values of the coupling parameters and guarantees the stability of the synchronization error dynamics.
The rest of the paper is organized as follows.In the next section, some preliminaries are presented.The main results are given in Section 3. In Section 4, the uncertain fractionalorder hyperchaotic Lorenz system that is used as an illustrative example is described.Synchronization of the system is also given in this section.Section 5 presents numerical results that illustrate the effectiveness of the method.Finally, conclusions are drawn in the last section.

Preliminaries
There are several definitions of fractional derivative [2].The frequently found definitions for fractional derivatives are Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions.In this paper, the Caputo definition is adopted.
A fractional-order LTI interval system with no input is described as [16]    = Ȃ , where  = [ 1 ,  2 , . . .,   ]  is the state vector,  is the fractional commensurate order, and the system matrix Ȃ is interval uncertain satisfying The following notations are used in the following lemma: where    is the -column vector with the th element being 1 and all the other being 0.

Main Results
Consider a fractional-order hyperchaotic system of order  (0 <  < 1) described by where X is the state vector.Two systems in synchronization are called the master system and the slave system, respectively.By utilizing a unidirectional linear error feedback coupling scheme, the master system and the slave system are constructed, respectively, as where the subscripts  and  stand for the master and the slave, respectively, and  = diag(  ) ( = 1, 2, . . ., ) is the coupling matrix,   ( = 1, 2, . . ., ) are called the coupling parameters that will be designed.By defining the synchronization error vector as E = X  − X  and using (7), the synchronization error system can be written as where (E, X  , X  ) = (X  ) − (X  ).Next, assume that the error system (8) can be expressed in a linear interval system as where Ȃ and Ȃ = Ȃ −  are the interval matrices of the uncoupled and coupled systems, respectively.

Chaos Synchronization
The uncertain fractional-order hyperchaotic Lorenz system considered here as the illustrative example is given by where  > 0,  > 0,  > 0, and  are the nominal parameters of the system, Δ, Δ, Δ, and Δ are additive uncertainties whose magnitudes are bounded by   ,   ,   , and   , respectively, and 0 <  < 1 is the fractional commensurate order.When  = 10,  = 8/3,  = 28,  = −1, and  = 0.98, the nominal system of ( 22) has a chaotic attractor as shown in Figure 1.From ( 22), the master and slave systems can be expressed as where the lower scripts  and  stand for the master and slave, respectively, and   ( = 1, 2, 3, 4) are the coupling parameters that are designed such that the two systems are synchronized.
By defining the synchronization errors as  1 =   −   ,  2 =   −   ,  3 =   −   , and  4 =   −   .Using (23), the error system can be expressed as Thus, the error system (24) can be written in the linear interval system form (9) as where E = [ 1  2  3  4 ]  ,  = diag( 1 ,  2 ,  3 ,  4 ), and the lower and upper boundaries of Ȃ are Assuming that the LMI condition in Theorem 2 is fulfilled, the asymptotic convergence of the synchronization errors is guaranteed.Therefore, the slave system is asymptotically synchronized with the master system.

Numerical Studies
The nominal parameters of the systems are assigned as  = 10,  = 8/3,  = 28, and  = −1.The uncertainty bounds are assumed as   = 2,   = 16/30,   = 5.6, and   = 0.2 (i.e., 20% of the nominal values).The fractional order is  = 0.98.The values of , , , and  estimated through simulations are found to be 30, 30, 50, and 250, respectively.By solving LMI condition (10) of the error system (25), the following solution is obtained: The state responses of the nominal master and slave systems are shown in Figure 2. Note that the controller is activated at time = 5 sec.The numerical method used in all simulations is an Adams-type predictor-corrector method.The reader is referred to [17] for the details.Moreover, the tests with changes of the parameters are conducted.The state responses of the master and slave systems obtained after increasing all parameters by 20% are shown in Figure 3 and the ones obtained after decreasing by 20% are shown in Figure 4.In all cases, the linearly unidirectional coupling control law can effectively synchronize the slave system to the master system as desired.

Conclusions
This paper has presented a method to achieve synchronization of unidirectional coupled uncertain fractional-order hyperchaotic systems.The coupling parameters are obtained by solving an LMI condition, which is straightforward to achieve.Thanks to the existence of an LMI solution, the convergence of the synchronization errors is secured.The method has been successfully applied to the uncertain fractional-order hyperchaotic Lorenz system to illustrate the effectiveness.

Figure 1 :
Figure 1: Phase portrait of the fractional-order hyperchaotic Lorenz system.

Figure 2 :
Figure 2: Synchronization of the hyperchaotic Lorenz systems with the nominal parameters (solid line represents slave system, dotted line represents master system).

Figure 3 :
Figure 3: Synchronization of the hyperchaotic Lorenz systems with all parameters increased by 20% (solid line represents slave system, dotted line represents master system).

Figure 4 :
Figure 4: Synchronization of the hyperchaotic Lorenz systems with all parameters decreased by 20% (solid line represents slave system, dotted line represents master system).