Abstract

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.

1. 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, 921]. 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 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.

2. System Description

Consider the following chaotic system:where is the state vector of system (1) and ,  , and   are the system’s parameters, which are known in advance. denotes the output variable of system (1).

Remark 1. It is well known that some chaotic systems such as the Lorenz system [22], the generalized Lorenz system [23], and the unified chaotic system [24] can be written in the form of system (1).
Let denote the length of interval or . Furthermore, we suppose that , , , , , and , where ,  ,  , and , are not both zero.

Remark 2. It is easy to see that if or , then the output of system (1) is continuous variable. Otherwise, the output is switched variable which is discontinuous.

3. 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 aswhere and are the controllers to be designed later.

Our work in this section is to design controller to make system (2) be stabilized at origin; that is, .

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), supposethen we have the following:(1)If , then (2)If and , then (3)If , , and there exists constant such that , then (4)If , , then

Proof. Note that ; from the second equation of system (2) and the expression of and it is easy to see that if then . Keep in mind that ; from the third equation of system (2) it is obvious that if then . Thus, in order to prove we only need to show that . Two cases are discussed in the following according to .
Case 1. If , , then and . From the first equation of system (2), we haveThat is,Therefore, one getsThus, we haveFrom inequality (7), it is easy to see that if , then as . Therefore we have which means that conclusion () of Theorem 4 is correct.
Case 2. If , , then and . From the first equation of system (2), we haveThat isTherefore, one getsFrom inequality (10), it is obvious that if and , then as . Therefore we obtain which implies that conclusion () of Theorem 4 is correct.
Now, we assume that ,
For , based on inequality (7), one getswhich leads toFor , based on inequality (10), we havewhich impliesIn general, for one obtains thatSince , it is easy to see that implies that . Note that ; thus we have as which implies that .
For one gets thatNote that , and we obtain thatSince and , we have as which implies that . Therefore, conclusion () of Theorem 4 is correct.
In the following, we assume that ,
For , based on inequality (15), one getsNote that in this case and as ; we know that as which implies that .
For , based on inequality (16), one getsNote that in this case and as ; we obtain that as which means that . Therefore, conclusion () of Theorem 4 is correct. This completes the proof of Theorem 4.

Remark 5. It is well known that the states of chaotic systems are bounded which means that , , and are all bounded.
In order to prove we only need to show that is bounded. In the following a simple proof shows that is bounded.
From system (2) and Theorem 4 we knowwhereSince , , and are all bounded, based on (20)–(21), one can easily derive that , , and are bounded. Furthermore, by (21) we can obtain that is bounded.
Now, in view ofwe conclude that is bounded which implies that Assumption 3 is reasonable.

4. 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:where , , and are the estimated values of , , and , respectively. , , , are intermediate variables. and , , are positive numbers to be designed later.

Prior to going any further, we make the following assumption.

Assumption 6. Suppose the state variables of systems (1) and (23) are bounded, which means that there exists nonnegative constant such that , ,
Let , denote the minimum eigenvalues of matrix , , respectively, whereThen we have the following theorem.

Theorem 7. (1) If and there exist nonnegative constants and such that , then system (23) can synchronize system (1) in the sense of
(2) If and there exist nonnegative constants and such that , then system (23) can synchronize system (1) in the sense of
(3) For .
(a) If and there exists constant such that , then system (23) can synchronize system (1) in the sense of
(b) If and there exists constant such that , then system (23) can synchronize system (1) in the sense of , where , , and    and are the minimum eigenvalues of matrixes and , respectively.

Proof. Let , , and Two cases are discussed in the following.
Case 1. If , , then , , and In this case from systems (1) and (23) we haveThus, we get the error dynamical system:For system (26), if we take Lyapunov candidatethen the derivative of (27) isThus, we haveFrom inequality (29), it is easy to see that if , then as . Therefore we have which means that conclusion () of Theorem 7 is correct.
Case 2. If , , then , , and In this case from systems (1) and (23) we haveThus, we get the error dynamical system:For system (31), if we take Lyapunov candidatethen the derivative of (32) isThus, we haveFrom inequality (34), it is easy to see that if , then as . Therefore we have which means that conclusion () of Theorem 7 is correct.
In the following, we assume that Suppose .
For , based on inequality (29) one getswhich leads toFor , based on inequality (34) we havewhich impliesIn general, for one obtains thatSince , it is easy to see that implies that . Note that ; thus we have as which implies that .
For one gets thatIf , from the inequality (40), we obtainSince and , we have as which implies that . Note that is a constant; we derive which means that . In view of we have .
If , from the inequality (40), we getAlong the same lines, one gets . Therefore, conclusion () (a) of Theorem 7 is correct. In the same way we can prove that conclusion () (b) of Theorem 7 is correct. This completes the proof of Theorem 7.Since and , we have as which implies that . Therefore, conclusion () (a) of Theorem 7 is correct. In the same way we can proof that conclusion () (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 .

5. Numerical Simulations

In this section, we will take the generalized Lorenz chaotic system as an example to illustrate the effectiveness of the proposed methods. The generalized Lorenz chaotic system is given by the following set of differential equations [17]:where is the state vector of system (44), and the system parameter satisfies System (44) is chaotic for each . Since in our control and synchronization schemes we need , we set The dynamic behavior with and is shown in Figure 1.

System (44) with output variable can be rewritten asBy comparing with system (1) it is easy to see that , , , and

Example 9 (the control of the generalized Lorenz chaotic system). In this simulation process, we suppose that and the controllers and are taken as that of Theorem 4. Then, according to Theorem 4 we have . The simulation result with , , and is shown in Figure 2. It can be seen that the orbits of system (45) approach the origin asymptotically.
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 aswhere , , and denote the parameter uncertainties, while , , and 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 upper bound is chosen as If we take and , then one can easily check that is a positive definite matrix. Its eigenvalues are easy to obtain which are , , and , respectively. Thus we can take For matrix , we suppose that and The eigenvalues of are , , and , respectively, which means that . For convenience, we let and ; then it is easy to check that which implies that condition () (a) of Theorem 7 is satisfied. According Theorem 7, we know that system (23) can synchronize system (45) in the sense of
The simulation results with , , , , , and 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 aswhere and denote the parameter uncertainties, while 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.

6. 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.

Second, the proposed model is especially suitable for secure communication. In a typical chaotic synchronization communication scheme the transmission signal is carried from the transmitter to the receiver by a chaotic signal through an analog channel. In order to mask the contents of transmission signal using chaotic signals different methods have been developed [25, 26]. However, it has been shown that most of these approaches are not secure or have a low level of security because one can extract the encoded message signal from the transmitted chaotic signal by using different unmasking techniques [27, 28]. Using our proposed synchronization scheme to transmit signal, one can add the transmitted signal to and derive