The Exponential Stabilization of a Class of nD Chaotic Systems via the Exact Solution Method

This paper treats the exponential stabilization of a class of n-D chaotic systems. A new control approach which is called the exact solution method is presented.Themost important feature of this method is that the solution of the system under consideration can be carefully designed to converge exponentially to the origin. Based on this method, the exponential stabilization of a class of n-D chaotic systems and its application in controlling chaotic system with unknown parameter are presented.The Genesio-Tesi system is taken to give the numerical simulation which is completely consistent with the theoretical analysis presented in this paper.


Introduction
The chaotic system is a special class of nonlinear system whose dynamic behavior is extremely dependent on initial conditions.Owing to the result of high sensitivity to initial conditions, the behavior of the chaotic system appears to be unpredictable and stochastic, even if the model of the considered chaotic system is deterministic.Since small differences of initial input may yield dramatically different results, the accurate prediction of long-term development of chaotic system is impossible.Now, it is generally agreed that this complex and irregular phenomenon is useful because it has many applications in some areas such as secure communications and information sciences [1].For the purpose of utilizing chaotic signals, the chaos control and chaos synchronization of dynamical systems have attracted a wide range of research activities for over two decades [2][3][4][5][6][7][8].A wide variety of approaches have been proposed for achieving chaos control and synchronization which include adaptive control method [9], sliding mode control [10,11], predictive control method [12], and backstepping method [13].
Most of the approaches for dealing with the chaos control and synchronization are based on the Lyapunov method.The Lyapunov method consists of two steps.First, design a proper Lyapunov function.Second, calculate its derivative along the trajectories of chaotic system.If the derivative of the Lyapunov candidate is negative, then the equilibrium of the considered system is globally asymptotically stable.Because it determines the stability of the system based on an energy function (called as the Lyapunov function) rather than solving the differential equations, Lyapunov method is an important method to analyze the stability of the system.The downside of this method is that it does not tell us whether the system is globally exponentially stable at the origin and what is the rate of the convergence.
In this paper, a new control approach which is called the exact solution method is presented.Contrary to the Lyapunov method, the exact solution method needs to construct an exact solution of the considered system.Based on this method, the exponential stabilization of a class of n-D chaotic systems is considered.Some novel controllers are presented to make the controlled system be exponentially stabilized.A numerical example is given to show the validity and feasibility of the proposed controller.
The main contents of this paper are described as follows.The system description is introduced in Section 2. By using the exact solution method, the stabilization of a class of n-D chaotic systems and its application in controlling chaotic system with unknown parameter are presented in Section 3. To show the effectiveness of the proposed approach, simulation results are given in Section 4. Finally, some concluding remarks are summarized in Section 5.

System Description
In this paper, the control of a class of −D chaotic systems which can be described as system (1) is considered.
The considered system is given as where  = ( 1 ,  2 , . . .,   )  ∈  ×1 is the state vector of system (1) and () is a continuous function of .

The Main Results
In this section, we consider the exponential stabilization of system (1) at the origin by using the exact solution method.In order to control chaotic behaviors in system (1), the control input  is added in the last state equation.Then, the controlled system is rewritten as where  is a controller to be designed later.Before giving the main results, we introduce the following essential definition.Definition 1.The controlled system (2) is said to be globally exponentially stable at the origin if there exist constants (> 0) and (< 0) such that |  | ≤   ,  ≥ 0,  = 1, 2, . . ., , hold for any initial values.Now, we construct an exact solution of system (2) and propose the following Theorem 2.
Theorem 2. Suppose that the controller  is chosen as Proof.For the sake of making the origin of system (2) globally exponentially stable, we suppose that where  1 ,  2 , ⋅ ⋅ ⋅ ,   are constants that are related to initial conditions.Since   < 0, it is obvious that lim →∞  1 = 0.
In view of ẋ 1 =  2 , we have Obviously, we obtain lim →∞  2 = 0. Repeating this process for  − 2 times yields Similarly, we get lim →∞   = 0.According to the last equation of (2), we derive Based on the above deduction process, we come to the conclusion that if  is chosen as (8), then system (2) has a solution: Because the solution of system ( 2) is uniqueness, we know that (9) are the unique solution of system (2) and satisfy lim →∞  1 = lim →∞  2 = ⋅ ⋅ ⋅ = lim →∞   = 0.
According to Definition 1, we know that system (2) is globally exponentially stable at the origin and the speed of convergence is relevant to , and the larger the number , the faster the rate of the convergence.This ends the proof of Theorem 2.
Remark 3. The controller in (8) By using ( 10)-( 12), it is easy to prove that  is a proper Lyapunov function.In fact, ( 16) is equivalent to The derivative of  is  (18) By (18), one can see that system (2) is globally stable at the origin, but in this case we do not know whether the state of the system converges exponentially to the origin and what is the convergence rate.

Complexity
Remark 5.Many papers have investigated the control or synchronization problem of system (1).For example, papers [14,15] considered the synchronization of system (1) by using the derivative control and backstepping method, respectively.Similarly, the authors did not tell us whether the system is globally exponentially stable at the origin and what is the rate of the convergence.
In the following, we discuss the application of Theorem 2. Consider the (n+1)-D chaotic system which is given as where  = ( 1 ,  2 , . . .,   ,  +1 )  ∈  (+1)×1 is the state vector of system (19), (), () are two continuous functions of ,  ∈  is the unknown parameter, and  is the controller.Now, with the help of Theorem 2, we can consider the stabilization of system (19) and derive the following Theorem 6. Theorem 6. Suppose that the controller  in system ( 19) is chosen as and the updated law of θ is and then system ( 19) is globally stable at the origin; that is, Proof.By Theorem 2, we know that if  +1 =  0 , then we get lim →∞  1 = lim →∞  2 = ⋅⋅ ⋅ = lim →∞   = 0. Furthermore, we can see that lim →∞  0 = 0. Since we assume that  +1 =  0 , we have lim →∞  +1 = 0. Therefore, in the following, we only need to prove that lim →∞  +1 =  0 .From the last equation of system (19), we have Substituting  defined in (20) into (22), we obtain Take The derivative of  along system (23) is Obviously, we have lim →∞  +1 =  0 .This ends the proof of Theorem In what follows, we consider the special case where  is a known parameter in advance.In this case, we have the following Corollary whose proof is omitted.

Corollary 7.
If  is a known parameter and we suppose the controller  in system ( 19) is chosen as then system ( 19) is globally stable at the origin; that is,

Simulation Results
Note that most of the chaotic attractors are 3-D chaotic systems, so in this section we take a 3-D system, that is, the Genesio-Tesi system, as an example to verify the effectiveness of the proposed scheme.The Genesio-Tesi system [16] is one of the famous chaotic systems, which is given as where  is the controller,  1 > 0,  1 > 0, and  1 > 0 are constants satisfying  1  1 <  1 .System (27) is chaotic for  1 = 6,  1 = 2.92,  1 = 1.2, and  = 0.The chaotic attractor of system ( 27) is shown in Figure 1.In the following, we suppose that  1 = 6,  1 = 2.92, and  1 = 1.2 so that system (27) has a chaotic attractor.
For simplicity's sake, we suppose that  1 = −1 and  2 = −2.By Theorem 2, we have  and according to Cramer's Rule, we obtain and then we have Thus, we have Now, we suppose that  1 ,  1 , and  1 are three unknown parameters.Obviously, () =  2  1 .Based on Theorem 6, the controller  can be chosen as The updated laws are given as   From Figures 2-7, one can observe that the trajectories of system (27) under controllers (31) and (36) converge to the origin rapidly which is completely consistent with the theoretical analysis presented in this paper.

Conclusions
In this paper, we use the exact solution method to investigate the exponential stabilization of a class of n-D chaotic systems.This method possesses two advantages: firstly, the solution can be designed to converge exponentially to the origin; secondly, the speed of convergence is known, which is determined by  = max{ 1 ,  2 , . . .,   }.Therefore, by taking proper value of , the rapid convergence can be obtained.Based on this method, the exponential stabilization of a class of n-D chaotic systems and its application in controlling chaotic system with unknown parameter are presented.The simulation results reveal that the proposed novel control strategy is effective.