Abstract
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. The most 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.
1. 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 [2–8]. 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.
2. 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 aswhere is the state vector of system (1) and is a continuous function of .
3. 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 aswhere 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 and such that , 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 asand then system (2) is globally exponentially stable at the origin, where , for , and
Proof. For the sake of making the origin of system (2) globally exponentially stable, we suppose thatwhere are constants that are related to initial conditions. Since , it is obvious that . In view of , we haveObviously, we obtain .
Repeating this process for times yieldsSimilarly, we get . According to the last equation of (2), we deriveBased 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 .
In order to use to represent , by using Cramer’s Rule, the term in (9) can be obtained as follows:Substituting into (8), we get Let By (9), one can easily derive that there exists a constant such thatAccording 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) is strongly dependent on , and different will lead to different controller
Remark 4. Let
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 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.
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 aswhere 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 asand the updated law of isand then system (19) is globally stable at the origin; that is, , where is the estimated value of and
Proof. By Theorem 2, we know that if , then we get . Furthermore, we can see that Since we assume that , we have Therefore, in the following, we only need to prove that From the last equation of system (19), we haveSubstituting defined in (20) into (22), we obtainTakeThe derivative of along system (23) isObviously, we have This ends the proof of Theorem 6.
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 asthen system (19) is globally stable at the origin; that is, .
4. 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 aswhere is the controller, , and are constants satisfying . System (27) is chaotic for , , , and The chaotic attractor of system (27) is shown in Figure 1. In the following, we suppose that , , and so that system (27) has a chaotic attractor.
Case 1. , , and are known parameters in advance.
For simplicity’s sake, we assume that , , and ; by Theorem 2, we haveandwhere
After simple calculation, we getPutting them into (29) yieldsThe chaotic trajectories of system (27) with controller (31) and , , and are shown in Figures 2–4.
Case 2. , , and are unknown parameters in advance.
For simplicity’s sake, we suppose that and . By Theorem 2, we haveand according to Cramer’s Rule, we obtainand then we haveThus, we haveNow, we suppose that , , and are three unknown parameters. Obviously, Based on Theorem 6, the controller can be chosen asThe updated laws are given as
The simulation results with , , , and are shown in Figures 5–10. Figures 5–7 show the time response of states , , and of system (27) with controller (36). Figures 8–10 display the time response of states , , and of system (27) with controller (36).
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.
5. 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 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China under Grants nos. 11761050 and 11361043, the Natural Science Foundation of Jiangxi Province under Grant no. 20161BAB201008, and the Graduate Innovative Foundation of Jiangxi Province under Grant no. YC2017-S059.