Chaos Control and Synchronization via Switched Output Control Strategy

This paper investigates the control and synchronization of a class of chaotic systems with switched output which is assumed to be switched between the first and the second state variables of chaotic system. Some novel and yet simple criteria for the control and synchronization of a class of chaotic systems are proposed via the switched output. The generalized Lorenz chaotic system is taken as an example to show the feasibility and efficiency of theoretical results.


Introduction
Chaotic phenomenon occurs naturally in many engineering, physical, biological, and social systems [1].Although chaotic phenomenon could be beneficial in some applications, it is undesirable in many physical applications and should therefore be controlled in order to improve the system performance.For a quite long period of time, due to the high sensitivity of a chaotic system to its initial condition, people thought that chaos was not controllable, and two chaotic systems could not be synchronized.However, the OGY method [2] developed by Ott et al. and in particular the concept of the synchronization proposed by Pecora and Carroll [3] in 1990 have completely changed the situation.Control and synchronization of chaotic systems have many potential applications in physical systems, lasers, circuits, chemical reactor, ecological systems, and secure communication [1,4].For example, in the field of secure communication many chaotic systems, such as the logistic map, the Hénon map, and the piecewise linear chaotic map [5], have been used to develop chaotic ciphers due to the sensitivity to initial conditions, ergodicity, and pseudorandom behavior of chaotic systems satisfies the analogous requirements for a good cryptosystem [6].Recently, spatiotemporal chaos has been employed for hash functions [7] and Chebyshev maps have been used for key agreement protocols [8].
Owing to the potential applications of chaos control and synchronization, many efforts have been devoted by researchers to achieve the goals of chaos control and synchronization in the last two decades and, as a result, a wide variety of approaches have been proposed for the control and synchronization of chaotic systems which include adaptive control [9], active control [10], integral control [11], impulsive control [12], backstepping control [13], sampled-data control [14], and so forth.
In the literature there are many results concerning the control and synchronization of chaotic system [3,[9][10][11][12][13][14][15][16][17][18][19][20][21].However, most of the existing works dealing with controlling chaos and chaos synchronization are based on the same assumption that the state variables of chaotic systems are all available for designing the controller.As it is well known that for most nonlinear systems the state variables are often unavailable in practice.For example, in the input-output system only the output is available which means that the above requirement is not very reasonable.Thus, the investigation of chaos control and synchronization with only output states available becomes an important topic.On the other hand, the transmitted signal may be switched between different signals and even interrupted for a variety of reasons.Therefore, it is necessary and important to investigate the control and synchronization of chaotic system with switched output.So far as we know, less attention has been paid to this issue.
With the above motivations, our main aim in this paper is to investigate the control and synchronization of a class of chaotic systems with switched output.The chaotic systems 2 Complexity are assumed that only the output variable is available and the output may be switched between the first and the second state variables.Some novel criteria for the control and synchronization of a class of chaotic systems are proposed via the switched output.The generalized Lorenz chaotic system is taken as an example to show the feasibility and efficiency of theoretical results.
The paper is organized as follows.First, a brief description of a class of chaotic systems is introduced in Section 2. The control and the synchronization schemes are discussed in Sections 3 and 4, respectively.Section 5 includes several numerical examples to demonstrate the effectiveness of the proposed approach.Finally, conclusion remarks are presented in Section 6.
Remark 2. It is easy to see that if  out =  1 or  out =  2 , then the output  out of system (1) is continuous variable.Otherwise, the output  out is switched variable which is discontinuous.

The Control Scheme of a Class of Chaotic Systems
In this section, we investigate the stabilization of system (1) at origin.For the purpose of forcing the states to converge to origin, we add two controllers to system (1).The controlled system (1) with a specified output is given as where  1 and  2 are the controllers to be designed later.Our work in this section is to design controller () = ( out ()) to make system (2) be stabilized at origin; that is, lim →+∞  1 = lim →+∞  2 = lim →+∞  3 = 0. Assumption 3. Suppose the state variables of system (2) are bounded, which means that there exists nonnegative constant  such that |( Theorem 4. For system (2), suppose (3) then we have the following: ( (3) If  max  max ̸ = ∞,  < 0, and there exists constant Proof.Note that  < 0; from the second equation of system (2) and the expression of  1 and  2 it is easy to see that if lim →+∞  1 = 0 then lim →+∞  2 = 0. Keep in mind that  > 0; from the third equation of system (2) it is obvious that if lim →+∞  1 = lim →+∞  2 = 0 then lim →+∞  3 = 0. Thus, in order to prove lim →+∞  1 = lim →+∞  2 = lim →+∞  3 = 0 we only need to show that lim →+∞  1 = 0. Two cases are discussed in the following according to .
Remark 5.It is well known that the states of chaotic systems are bounded which means that  1 ,  2 , and  3 are all bounded.
In order to prove we only need to show that ẍ 1 is bounded.In the following a simple proof shows that ẍ 1 is bounded.
From system (2) and Theorem 4 we know where Since  1 ,  2 , and  3 are all bounded, based on ( 20)-( 21), one can easily derive that ẋ 1 , ẋ 2 , and  2 are bounded.Furthermore, by (21) we can obtain that u 1 is bounded.Now, in view of we conclude that ẍ 1 is bounded which implies that Assumption 3 is reasonable.

The Synchronization of a Class of Chaotic Systems
Suppose system (1) is the drive system.In order to synchronize system (1), we introduce the following response system: ) .
Complexity 5 Then we have the following theorem.
For  ∈ [ 0 ,  1 ), based on inequality (29) one gets which leads to For  ∈ [ 1 ,  2 ), based on inequality (34) we have which implies In general, for  ∈ [ 2 ,  2+1 ) one obtains that Since For  ∈ [ 2+1 ,  2+2 ) one gets that If  2 ≤ 0, from the inequality (40), we obtain Since  → ∞ and If  2 > 0, from the inequality (40), we get Along the same lines, one gets lim (a) of Theorem 7 is correct.In the same way we can prove that conclusion (3) (b) of Theorem 7 is correct.This completes the proof of Theorem 7.

|𝑉 (𝑡)| ≤ 𝑒
(a) of Theorem 7 is correct.In the same way we can proof that conclusion (3) (b) of Theorem 7 is correct.This completes the proof of Theorem 7.
Remark 8.It is easy to see that in our synchronization scheme we do not need to restrict  < 0.
In order to show the robust of our control scheme to parameter uncertainties and external disturbances, we add some parameter uncertainties and external disturbances to system (45).Thus the controlled system (45) is rewritten as where 0.1( 2 −  1 ), −0.1 1 , and −0.1 3 denote the parameter uncertainties, while −0.2 1  2 , 0.1 2  3 , and −0.1 1  3 represent the external disturbances.The simulation result with the same controller and initial conditions as in Figure 2 is given in Figure 3. From Figure 3 one can easily see that the asymptotically stable of the origin of system (46) is actually achieved.
Example 10 (the synchronization scheme of the generalized Lorenz chaotic system).Based on Figure 1 The simulation results with  1 (0) = 6,  2 (0) = 8,  3 (0) = 10, x1 (0) = 2, (0) = 6, and x3 (0) = 13 are presented in Figure 4. From Figure 4, it is easy to see that the drive system (45) and the response system (23) are synchronized within a few seconds.For the sake of showing the robust of our synchronization scheme to parameter uncertainties and external disturbances, we add some parameter uncertainties and external disturbances to system (45).Thus system (45) is rewritten as where 0.1( 2 −  1 ) and −0.1 1 denote the parameter uncertainties, while −0.2 1  2 represents the external disturbances.The synchronization simulation between systems (47) and (23) with the same initial conditions as in Figure 4 is presented in Figure 5. From Figure 5 it is easy to see that although in Figure 5 the synchronization between systems (47) and ( 23) is reached, its speed of synchronization is slower than that shown in Figure 4.
Remark 11.From Figures 3 and 5 one can conclude that the presented control and synchronization schemes are robust to some special parameter uncertainties and external disturbances.However, it should be pointed out that since in our approaches the adaptive control method is not used our schemes may not be robust to arbitrary uncertainties and external disturbances.

Conclusion
The control and synchronization of a class of chaotic systems with switched output is investigated in this paper.By using the switched output, which is assumed to be switched between the first and the second state variables, some novel criteria for  the control and synchronization of a class of chaotic systems are proposed.The generalized Lorenz chaotic system is taken as an example to demonstrate the efficiency of the proposed approach.It is not difficult to see that our paper has two contributions.First, we present a new model which has switched output.As it is well known that in real-life situations the transmission signals may be interrupted for various reasons which means that the output should be discontinuous variable.Since the switched output is discontinuous variable, our model is closer to the actual situation than that having continuous output variable.