Abstract
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.
1. 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 fractional-order 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 [8–15]. It is not difficult to see that in papers [8–15] 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.
2. Preliminaries and System Description
2.1. Fractional-Order Integral and Derivative
In this subsection, some basic definitions with respect to Caputo’s fractional derivative are introduced. Also, some useful lemmas are proposed.
Definition 1 (see [16]). Caputo’s fractional derivative of function is defined by where .
Definition 2 (see [16]). The Mittag-Leffler function with two parameters is given bywhere and
If , then
Let denote the Laplace transform of a function . Based on the definition of Laplace transform: , we have The Laplace transform of Mittag-Leffler function with two parameters is
Lemma 3 (see [17]). If , then the origin of system is globally asymptotically stable.
Lemma 4 (see [18]). If and , then the origin of system is globally asymptotically stable.
If , then based on Lemma 4 we can easily get the following Corollary 5.
Corollary 5. Suppose . If and , then the origin of system is globally asymptotically stable.
Lemma 6. Consider the following two systems: andwhere and , and then we have where are initial values of and , respectively.
Proof. Based on system (8), we obtainThe Laplace transform of (11) givesThe 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.
Remark 7. System (9) is called as the observer of system (8).
2.2. System Description
In this paper we consider the class of chaotic systems which are described bywhere is the state vector of system (18) and is the order of fractional derivatives. and are system’s parameters. represents the model uncertainty or the external disturbance. is the measured output signal which can be used to design the controller.
3. 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 aswhere is a controller to be designed later.
Now, some Assumptions are introduced.
Assumption 8. Suppose is bounded which means that there exists constants such that
Assumption 9. Suppose there exist two constants such that
Since and , by Lemma 6 the observer of system is Furthermore, by Lemma 6 we have Thus, we get In view of and , then by Lemma 6 we know the observer of system is Thus, in order to design proper controller , we can propose the following observer system for variables and : where and are the estimated values and , respectively.
Theorem 10. Suppose Assumptions 8 and 9 are satisfied. For system (19), if we choose such that then the origin of system (19) is stable in the sense of , where are defined by (26) and .
Proof. The proof of Theorem 10 is divided into two steps. In the first step, we shall show that
To this purpose, let us consider the following Lyapunov function candidate: The time derivative of (28) is By Lemma 6, we obtain where
Substituting inequality (30) into (29), we have By using (27), we haveNote that . Therefore, we have Thus, there exists a finite time such that when we get Thus, when we obtainIt should be noted that and Thus, based on Lemma 3 we have
Now, in the second step we prove that . By using the results obtained in the proof of step 1, it is obvious that From the first equation of system (19) we derive that Since , according to Lemma 4 we know that . In the same way we have . This completes the proof of Theorem 10.
Remark 11. The constant 1 in controller (27) is designed to eliminate the affect caused by Note that , and therefore the constant 1 can be replaced by any positive number. Since , thus the factor in controller (27) determines the speed of convergence. In general, the larger the number the faster the rate of the convergence.
4. 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 aswhere is the state vector of system (35), is the order of fractional derivatives. 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 12. Suppose are all bounded which means that there exists constants such that
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 and : where are the estimated values and , respectively.
Theorem 13. Suppose Assumptions 9 and 12 are satisfied. If we choose such that then the synchronization between drive system (18) and response system (35) will occur in the sense of , where are defined by (37) and .
Proof. The proof of Theorem 13 is similar to that of Theorem 10. In the first step, we shall show that 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 haveThe following proof is similar to that of Theorem 10 and omitted here. This ends the proof of Theorem 13.
5. 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 is the state vector of system and is the system parameter which satisfies It is well known that the systems display chaotic behavior for each [19].
Base on system (45), the fractional-order generalized Lorenz chaotic systems are given aswhere represents the model uncertainty or the external disturbance. The chaos attractor with is shown in Figure 1.
By comparing system (46) with system (18), it yields that Since systems display chaotic behavior for each , so we suppose in systems (46). Thus, it is easy to see that In the simulation process, we take such that system (46) is chaotic.
Example 1. 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.
Let ; then we get The parameter is taken as Suppose the observer system is (26) and the controller is selected as (27); then according to Theorem 10 we know that the origin of system (47) is stable in the sense of . The simulation results with and are shown in Figures 2 and 3. Figure 2 is the time response of states of system (47) and Figure 3 is the time response of states of system (26). From Figure 2 it is easy to see that although there is disturbance in system (47) the states of system (47) approach as , respectively, which means that the origin of system (47) is asymptotic stable. Figure 3 shows that which means that in this case and can recover the information of and .
Example 2. 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 aswhere is the controller to be designed later and is the model uncertainty or the external disturbance.
In our simulation, we take , thus which means that The parameter is taken as Suppose the observer system is (37) and the controller is designed as (38); then by Theorem 13 we know the synchronization between the drive system (46) and the response system (48) will be achieved. Numerical simulations with , and are shown in Figure 4. Figure 4 is the time evolution of errors between system (46) and system (48) with controller (38). From Figure 4 it is easy to see that although there are disturbances both in the drive system and the response system, the synchronization errors converge quickly to zero which means that the synchronization between system (46) and response system (48) is reached.
6. 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: 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. 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.
Data Availability
If necessary, the numerical simulation codes can be uploaded at any time.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China under Grant 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.