The Robust Control and Synchronization of a Class of Fractional-Order Chaotic Systems with External Disturbances via a Single Output

This paper investigates the stabilization and synchronization of a class of fractional-order chaotic systems which are affected by external disturbances. The chaotic systems are assumed that only a single output can be used to design the controller. In order to design the proper controller, some observer systems are proposed. By using the observer systems some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. Numerical examples are presented by taking the fractional-order generalized Lorenz chaotic system as an example to show the feasibility and validity of the proposed method.


Introduction
Chaos control and synchronization have attracted a great deal of attention since the innovative works proposed by Huber, Pecora, and Carroll in 1990 [1].Nowadays, owing to their potential applications in many areas such as in chemical reactions, power converters, information processing, and secure communications, various types of synchronization phenomena have been discovered, such as complete synchronization [2], combination synchronization [3], and equal combination synchronization [4].
Fractional-order calculus is a branch of mathematics that deals with derivatives and integrals of non-integer orders.It has been shown that the models presented by fractionalorder systems are more adequate than that described by integer order systems.Many systems such as viscoelastic systems, dielectric polarization, and electromagnetic waves [5] are known to display fractional-order dynamics.In recent years, a number of fractional-order chaotic systems have been investigated, such as the fractional-order economical system [6] and the fractional-order Lorenz system [7].
Similarly to integer order chaotic systems, the control and synchronization of fractional-order chaotic systems has become an active research field [8][9][10][11][12][13][14][15].It is not difficult to see that in papers [8][9][10][11][12][13][14][15] the authors have used all state variables to design controllers.However, in the real situation it is well known that only part of the variables can be used in many nonlinear systems.Therefore, it is necessary to investigate the control and synchronization of fractional-order chaotic system with a single output.
Motivated by the above discussion, in this paper we consider the stabilization and synchronization of a class of fractional-order chaotic systems via a single output.Some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived via the observer systems.The fractional-order generalized Lorenz chaotic system is taken as an example to show the feasibility and validity of the proposed method.
The rest of the paper is organized as follows.In Section 2, we introduce some preliminaries, including some definitions, lemmas, and the general form description of a class of fractional-order chaotic systems.The control and synchronization schemes of a class of fractional-order chaotic systems via a single output are presented in Sections 3 and 4, respectively.In Section 5, numerical simulations results are shown.Some conclusions are drawn in Section 6.
).Let L(()) denote the Laplace transform of a function ().Based on the definition of Laplace transform: The Laplace transform of Mittag-Leffler function with two parameters is Lemma 3 (see [17]).If  > 0, then the origin of system is globally asymptotically stable.
Proof.Based on system (8), we obtain The Laplace transform of (11) gives The Laplace transform of ( 9) is Subtracting ( 13) from ( 12) one has The above inequality is equivalent to Taking the inverse Laplace transform of (15) yields It implies that This concludes the proof of Lemma 6.

The Control Scheme
In this section, the stabilization of system ( 18) is investigated.In order to force the states of system (18) to its origin, the control input  is added to the second state equation.Thus, system (18) can be rewritten as where  is a controller to be designed later.Now, some Assumptions are introduced.
and () ≤ , by Lemma 6 the observer of system Furthermore, by Lemma 6 we have Thus, we get In view of and () ≤ , then by Lemma 6 we know the observer of system Thus, in order to design proper controller , we can propose the following observer system for variables  1 and  3 : then the origin of system ( ) is stable in the sense of lim →∞  1 = lim →∞  2 = lim →∞  3 = 0, where x1 , x3 are defined by ( ) and  > 0.
Proof.The proof of Theorem 10 is divided into two steps.In the first step, we shall show that lim →∞  2 = 0.
To this purpose, let us consider the following Lyapunov function candidate: The time derivative of (28) is where It should be noted that  ≥ 0 and  > 0. Thus, based on Lemma 3 we have lim →∞  2 = 0. Now, in the second step we prove that lim →∞  1 = lim →∞  3 = 0.By using the results obtained in the proof of step 1, it is obvious that lim →∞ ( 1  2 +  2  2 2 ) = 0. From the first equation of system (19) we derive that Since () < 0, according to Lemma 4 we know that lim →∞  1 = 0.In the same way we have lim →∞  3 = 0.This completes the proof of Theorem 10.

The Synchronization Scheme
In this section the synchronization scheme of a class of fractional-order chaotic systems is presented via the observer based method.Suppose system ( 18) is the drive system; in order to synchronize system (18) the corresponding response system with controller  is constructed as where  = ( 1 ,  2 ,  3 )  ∈  3×1 is the state vector of system (35),  ∈ (0,1) is the order of fractional derivatives. 1 () represents the model uncertainty or the external disturbance.  is the measured output signal which can be used to design the controller. is the controller to be designed later.
Let us define the synchronization error as  =  − , then the dynamics of synchronization error between systems ( 18) and (35) can be described by Assumption .Suppose (),  1 () are all bounded which means that there exists constants  > 0 such that | 1 () − ()| ≤ .
The objective of the current synchronization problem is to design an appropriate control signal  such that, for any initial conditions of the drive and response systems, the synchronization errors converge to zero.For this end, similar to system (26) we proposed the following observer for variables  1 ,  3 and  1 ,  3 : then the synchronization between drive system ( ) and response system ( ) will occur in the sense of lim →∞  1 = lim →∞  2 = lim →∞  3 = 0, where x1 , x3 , ŷ1 , ŷ3 are defined by ( ) and  > 0.
Proof.The proof of Theorem 13 is similar to that of Theorem 10.In the first step, we shall show that lim To this purpose, let us consider the following Lyapunov function candidate: The time derivative of (40) is  By Lemma 6, we obtain where Substituting inequality (42) into (41), we have By using (38), we have The following proof is similar to that of Theorem 10 and omitted here.This ends the proof of Theorem 13.

Numerical Simulations
In this section we take the fractional-order generalized Lorenz chaotic systems as an example to verify and demonstrate the effectiveness of the proposed control scheme.The integer-order generalized Lorenz chaotic systems can be described as [19] where  = ( 1 ,  2 ,  3 )  ∈  3×1 is the state vector of system (32) and  is the system parameter which satisfies  ∈ { | −232 <  < −11.6} ∪ { |  > −11.6}.It is well known that the systems (32) display chaotic behavior for each 0 ≤  ≤ 29 [19].
Base on system (45), the fractional-order generalized Lorenz chaotic systems are given as where () represents the model uncertainty or the external disturbance.The chaos attractor with  = 0.995, () = 0,  = 2 is shown in Figure 1.
Example .The control of the fractional-order generalized Lorenz chaotic systems.
The controlled system, based on system (46), is given as where  is the controller to be designed later.
Example .The synchronization between two identical fractional-order generalized Lorenz chaotic systems.Let system (46) be the drive system; then the corresponding response system is given as where  is the controller to be designed later and  1 () is the model uncertainty or the external disturbance.

Conclusions
The observer-chaos-based stabilization and synchronization of a class of fractional-order chaotic systems with a single output are investigated in this paper.In the literature, there are some papers that consider the stabilization and synchronization of chaotic systems via the observer -based method.There are two main differences between our paper and the published papers: (1) it is easy to see that in the published paper the constructed observer system can exactly recover the information of the unavailable states.One can use the recovery states to design the controller easily.However, the observer system presented in this paper cannot recover the information of the unavailable states exactly; it just provides the upper bound of the unavailable states.Thus, to design the proper controller is a more difficult task.
(2) In the literature, most of the papers that concerned the observer-based scheme do not consider the effects of external disturbances.However, the external disturbances are taken into consideration in this paper.By using the observer system, some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived.The fractional-order generalized Lorenz chaotic system is taken as an example to show the feasibility of the designed method.
x3 are the estimated values  1 and  3 , respectively.