Abstract
The stabilization problem of the complex chaotic system is investigated in this paper. First, a systematic method is proposed, by which a given complex chaotic system can be transformed into its equivalent real chaotic system. Then, both simple and physical controller is designed for the corresponding real chaotic system by the dynamic feedback control method, thereby the controller for the original complex chaotic system is obtained. Especially, for some complex system, the controller is obtained by the linear feedback control method. Finally, two illustrative examples with numerical simulations are used to verify the validity and effectiveness of the theoretical results.
1. Introduction
It is well known that the first chaotic system was proposed by Lorenz in 1963. From then on, many works have been done about both theoretical results and applications, see Refs. [1–15] and the references therein. Complex chaotic system whose state variables belong to complex space is another important type of chaotic dynamical system, which has been widely investigated in both theorem and applications and has become a hot topic in recent years, for details see Refs. [16–24]. Especially, the encryption effect is better due to the fact that the complex chaotic system is composed of real and imaginary numbers. Since the dynamic behavior of the complex system is more complicated than that of the real chaotic system, the control problems of such system is very difficult. Many researchers usually adopted this strategy; they firstly transfer the complex chaotic system into its corresponding real chaotic system by separating the real parts and imaginary parts of the complex state variables and then they investigate the control method of the real chaotic system. Ultimately, the control problems of such complex system were realized. However, on one hand, there is lack of a systematic method in the first step, i.e., for a specific complex chaotic system, a specific method is applied to transform it into its equivalent real chaotic system. How to find a systematic method by which the complex chaotic system can be transformed into its equivalent real chaotic system is not only important in theory but also significant in applications; thus, it stimulates our work in this paper.
On the other hand, most of the controllers designed in the aforementioned existing results are complicated; thereby, they are hard to be performed in real applications. As a matter of fact, how to design a both simple and physical controller to realize the control problems of the complex chaotic systems is also important both in theory and applications. Among the existing methods, the dynamic feedback control method and the linear feedback control method are widely applied, and thus these two methods are adopted by this study.
Motivated by the above conclusions, the stabilization problem of the complex chaotic system is studied by the dynamic feedback control method. The main contributions of this paper are given as follows:(1)A systematic method is proposed, which can be used to transform a given complex chaotic system into its equivalent real chaotic system(2)Both simple and physical controller is designed for the original complex chaotic system is obtained by the dynamic feedback control method and the linear feedback control method, respectively, and numerical simulations are performed to verify the above theoretical results
Before ending this section, we present some notations used in this paper. is the dimensional Euclidean space, is the dimensional complex space, denotes the identity matrix, denotes the set of real matrices, is the Kronecker product, and is the semitensor product (STP), i.e., let , , and be the least common multiple of and . The STP of and is defined aswhere
and represent the real part and the imaginary part of (·), respectively, and is the imaginary unit, i.e., .
2. Preliminary
Consider the following controlled chaotic system:where is the state, is continuous function with , is a constant matrix, and is the controller to be designed.
Lemma 1. (see [18]). Consider system (3). If can be stabilized, then the designed controller is of the following form:where , and the feedback gain is updated by the following equation:
3. Problem Formulation
Consider the following controlled complex chaotic system:where and are the state, ,where is the conjugate of , , is a complex constant matrix, and , , is the designed controller, i.e.,where , , , , ,that is, , , and .
Remark 1. The complex chaotic system of equation (6) is very common, which covers a lot of complex chaotic systems, such as complex Lorenz system and complex hyperchaotic Lorenz system.
The goal of this paper is to investigate the stabilization of system (6), i.e., how to design a controller to guarantee
4. Main Results
4.1. A Systematic Method which is Used to Transform a Complex System into its Equivalent Real System
In this section, a systematic method proposed, by which a complex system can be transformed into its equivalent real system.
Theorem 1. Consider the complex chaotic system (6). Its equivalent real system is described as the following form:where is the state, is continuous function with , is a constant matrix, and is the controller to be designed.
Proof. Let , , and , ; then, the equivalent real system (12) is obtained, i.e.,
Remark 2. Since the complex chaotic system (6) is equivalent to the corresponding real system (12). Thus, the stabilization problem of system (12) is investigated and the controller is designed. Moreover, the controller of system (6) is obtained by
Remark 3. For a given general controlled complex chaotic system of the following form:where is the state, is continuous function with , and and are the designed controller, .
By Theorem 1, its equivalent real system is obtained as follows:where is the state, is continuous function with , is a constant matrix, and is the controller to be designed, i.e.,
If the complex chaotic system in which there exist model uncertainty and external disturbance, we present the following result.
Corollary 1. Consider the following complex chaotic system with both model uncertainty and external disturbance:where , , , and are given in equations (7)–(10), respectively,where stands for the model uncertainty and is the external disturbance. Its equivalent real system is described as the following form:where , , , and are the same as those in Theorem 1, and
4.2. Stabilization of the Complex Chaotic System
In this section, the stabilization of the complex chaotic system is investigated by the dynamic feedback control method and the linear feedback control method, respectively, and the conclusions are presented.
Theorem 2. Consider system (12). If can be stabilized, then the controller is designed of the following form:where , and is updated by
Remark 4. According to equation (14), the controller for system (6) is obtained; thus, the stabilization of system (6) is realized.
If system (12) has some special structure, the stabilization problem of such system can be realized by the linear feedback control method, and the result is proposed.
Theorem 3. Consider system (12) of the following form:i.e.,withis globally asymptotically stable. If is controllable, then the designed controller is of the following form:where meets the matrix is Hurwitz whatever is.
Proof. Since is controllable, thus is also controllable whatever is. According to the pole assignment theory, the controller given in (27) is as requested, which completes the proof.
5. Illustrative Examples with Numerical Simulations
In this section, we shall take two complex chaotic systems for example to show how to apply the obtained theoretical results, and then numerical simulations are performed to verify the effectiveness and the validity of the aforementioned theoretical results.
Example 1. The controlled complex Lorenz chaotic system [25], which is presented as follows:wherei.e., , , and
According to Theorem 1, the equivalent real system is obtained as follows:wherei.e.,
Note that if , then the following subsystemis globally asymptotically stable; thus, can be stabilized.
According to Theorem 2, the controller is designed as follows:where and . Thus,
Numerical simulation is carried out with the initial conditions: . Figure 1 shows are asymptotically stable, and Figure 2 shows are asymptotically stable, which implies the and are stabilized. Figure 3 shows the feedback gain approaches to constant.
According to Theorem 3, the controller is designed as follows:where
Thus,
Numerical simulation is carried out with the initial conditions: . Figure 4 shows are asymptotically stable, and Figure 5 shows are asymptotically stable, which implies the and are stabilized.
Example 2. The controlled complex hyperchaotic Lorenz system [26], which is presented as follows:wherethat is, , , and
According to Theorem 1, the equivalent real system is obtained as follows:wherei.e.,
Notice that if , then the following systemis globally asymptotically stable; thus, can be stabilized.
According to Theorem 2, the controller is designed as follows:where and .
Therefore,
Numerical simulation is performed with the initial conditions: . Figure 6 shows are asymptotically stable, and Figure 7 shows are asymptotically stable, which means that the and are stabilized. Figure 8 shows the feedback gain tends to constant.
6. Conclusions
In conclusion, the stabilization problem of the complex chaotic system has been studied in this paper. First, a systematic method has been proposed, which is applied to transform the complex chaotic system into its equivalent real chaotic system. Then, both simple and physical controllers have been designed for the complex chaotic system by the dynamic feedback control method and the linear feedback control method, respectively. Finally, two illustrative examples with numerical simulations have been performed to verify the validity and effectiveness of the theoretical results.
Data Availability
No data were used in this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Shandong Province Key Research and Development Program (Grant no. 2019GGX101056) and Natural Science Foundation of Shandong Province (Grant no. ZR2018MF016).