Abstract
In this paper, we investigate the antisynchronization problem of a class of hyperchaotic systems with both model uncertainty and external disturbance. Firstly, combining the dynamic feedback control method and the uncertainty and disturbance estimation (UDE)-based control method, we propose a new UDE-based dynamic feedback control method. Secondly, we take the 4D hyperchaotic system as an example and realize the antisynchronization problem of such system. Finally, the effectiveness and correctness of the proposed method is verified by numerical simulation.
1. Introduction
Chaotic behavior has been extensively analyzed in many fields, e.g., engineering, medicine, ecology, biology, and economics, even in social science. In 1990, Pecora and Carroll first discovered the phenomenon of chaotic synchronization [1]. Many types of synchronization problems have been proposed successively, including complete synchronization, generalized synchronization, phase synchronization, lag synchronization, and projective synchronization, see References [2–8]. Later, antisynchronization or antiphase synchronization [9] was proposed. So far, many methods have been proposed to realize chaotic antisynchronization, such as active control, adaptive control, linear feedback control, sliding mode control, and time-delay feedback approach, see References [10–22] and the references therein. Among them, the dynamic feedback control method [19] has a wide range of applications because of its simple design and easy implementation. Thus, this dynamic feedback control method is used in this paper.
It should be pointed out that among the abovementioned chaotic and hyperchaotic systems, model uncertainty and external disturbance are not considered. Unfortunately, this is not the case in practice. The UDE-based control method [23] is a good method to deal with the model uncertainty and external disturbance, and it has the following two advantages: one is that the system model or a disturbance model is not known completely; the other is that both structured (or unstructured) uncertainties and external disturbances are robust against. Being an effective robust control strategy, the UDE-based control has found widespread applications in various systems. Naturally, it is of interest to apply the UDE-based control to chaotic and hyperchaotic systems with both model uncertainty and external disturbance. However, to the best of the authors’ knowledge, this problem has not been addressed in the existing literature. Therefore, the main goal of this paper is to develop a new UDE-based dynamic feedback control method to realize the antisynchronization problem of the chaotic and hyperchaotic systems.
This paper mainly studies the antisynchronization problem of chaotic and hyperchaotic systems with both model uncertainty and external disturbance. Combining the dynamic dynamic feedback control method and the UDE-based control method, a new UDE-based dynamic feedback control method is proposed. Then, by the obtained new method, the antisynchronization problem of the 4D hyperchaotic system is realized and the effectiveness of the method is verified by the numerical simulation.
2. Preliminary
2.1. Dynamic Feedback Control Method for Antisynchronization
Consider the following hyperchaotic system:where is the state and is a continuous vector function.
Let system (1) be the master system; then, the corresponding slave system with is given aswhere is the state, is a continuous vector function, is a constant matrix, and is the designed controller, .
Set ; then, the sum system is described aswhere is the state and and are given in (2).
Definition 1. Consider the sum system (3). If ; then, we call the master system (1) and slave system (2) achieve antisynchronization.
Remark 1. According to the results in [21], the antisynchronization of system (1) exists only and only if .
At present, there are many methods for the antisynchronization problem. Among them, the dynamic feedback control method has a wide range of applications because of its simple design and easy implementation. Here is a brief introduction.
According to the existing result in [19–22], we present the following conclusion.
Lemma 1. Consider system (3), where and or , , , where is controllable; then, the dynamic feedback controller is designed as follows:where , , and the feedback gain is updated by the following law:
2.2. UDE-Based Control Method
Consider the following controlled systems with model uncertainty and disturbance:where is the state, is a continuous vector function, , , represents the model uncertainty, is the external disturbance vector, is the controller to be designed, and is assumed controllable.
The stable linear reference model is presented as follows:where is the reference state, is a Hurwitz constant matrix, is a vector, and is a command.
According to the existing results in [23], the UDE-based control method is presented as follows.
Lemma 2. (see [23]). Consider system (6) and the reference system (7). If the designed filter satisfies the following condition,where and , then UDE-based controller is expressed as follows:where , , is the inverse Laplace transform operator, is the convolution operator, and .
Remark 2. Since controller in equation (9) cancels in system (6) directly, thus this controller is too complex to be used in actual chaotic antisynchronization system.
Remark 3. According to the existing result in [23], two kinds of filters are introduced. One is the first-order low-pass filter:in general, .
The other is the secondary filter:where , , and .
3. Problem Formulation
Consider the following hyperchaotic system with both model uncertainty and external disturbance:where is the state, is a continuous vector function, denotes system model uncertainty, and stands for the external disturbance.
Let system (12) be the master system; then, the slave system with is given aswhere is the state, is a constant matrix, and is the controller to be designed, and it is assumed that is controllable.
Let , then the sum system is described as follows:
The main goal of this paper is to design a controller to meet the following condition:
4. Main Results
In this section, we investigate the antisynchronization problem of the hyperchaotic systems with both uncertainty and external disturbance and present the following result.
Theorem 1. Consider system (14). If a filter is designed to satisfy the following condition,where and , then the dynamic feedback UDE-based controller is designed as follows:where is updated by the update law (5), andwhere and stands for the convolution operator.
Proof. Substituting controller (17) into system (14), we obtainwhere and .
According to Lemma 1, the system is globally asymptotically stable. Noting condition (16), we can obtainThus, system (20) is rewritten asand this system is globally asymptotically stable, which completes the proof.
5. An Illustrative Example with Numerical Simulation
In this section, we take the new 4D hyperchaotic system as an example to apply our theoretical results.
Example 1. The new 4D hyperchaotic system with uncertainty and disturbance is given as follows:where , is a continuous vector function, is the model uncertainty, and is the external disturbance, i.e.,Obviously, . Thus, the antisynchronization problem of the system exists.
Let system (23) be the master system, then the corresponding slave system with is described aswhere , is the controller to be designed, andSet , then the sum system is given as follows:where .
The controlled sum system without model uncertainty and external disturbance is presented asOur goal is to design a controller to stabilize the system (27), i.e., .
The first step is to design controller .
For system (28), it is obvious that if , we can get that the following two-dimensional systemis globally asymptotically stable.
From Lemma 1, we can design controller as follows:where is updated by the update law (5).
For system (28), the numerical simulation is carried out with the initial conditions: . Figure 1 shows that under the abovementioned controller, the sum system is asymptotically stable, i.e., the master-slave system achieves antisynchronization. Figure 2 shows that the feedback gain converges to an appropriate constant. Figure 3 shows that states of the master system: , antisynchronize, the states of the slave system: , respectively.
The second step is to design the UDE controller .
Let ; the system (27) is rewritten asAccording to Theorem 1, the controller is designed aswhere and stands for the convolution operator.
Thus, is obtained.
For system (27), the numerical simulation is carried out with the initial conditions: . Figure 3 shows that under the abovementioned controller, the error system is asymptotically stable. Figure 4 shows that and after a certain time tend to the same constant. Figure 5 shows that the feedback gain converges to an appropriate constant. Figure 6 shows that the system achieves antisynchronization. The numerical simulation results show that the new hyperchaotic system achieves antisynchronization under the abovementioned controller. Figure 7 shows that states of the master system: also antisynchronize the states of the slave system: , respectively.
6. Conclusion
In this paper, the antisynchronization problem of the hyperchaotic systems has been investigated. A new UDE-based dynamic feedback control method has been proposed, and the antisynchronization of the new 4D hyperchaotic system has been realized by the obtained control method. The correctness and effectiveness of the abovementioned theoretical methods have been verified by numerical simulation.
Data Availability
All figures are made by Matlab.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of the article.
Acknowledgments
This work was supported by the Natural Science Foundation of Shandong Province (ZR2018MF016).
Supplementary Materials
The supplementary file supplied includes MATLAB Programs. (Supplementary Materials)