Antisynchronization of the Hyperchaotic Systems with Uncertainty and Disturbance Using the UDE-Based Control Method

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.

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. e 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 UDEbased 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. erefore, 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.
is 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 UDEbased control method, a new UDE-based dynamic feedback control method is proposed. en, 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.

Dynamic Feedback Control Method for Antisynchronization.
Consider the following hyperchaotic system: where w ∈ R n is the state and G(w) ∈ R n is a continuous vector function.
Let system (1) be the master system; then, the corresponding slave system with v is given as where v ∈ R n is the state, G(v) ∈ R n is a continuous vector function, b ∈ R n×s is a constant matrix, and u ∈ R s is the designed controller, s ≥ 1.
Set E � w + v; then, the sum system is described as where E ∈ R n is the state and b and u are given in (2).

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.
is controllable; then, the dynamic feedback controller is designed as follows: where K � k(t)b T , K ∈ R n×n , and the feedback gain k(t) is updated by the following law:

UDE-Based Control
Method. Consider the following controlled systems with model uncertainty and disturbance: where p ∈ R n is the state, H(p) is a continuous vector function, b ∈ R n×s , s ≥ 1, ΔH(p) ∈ R n represents the model uncertainty, d(t) ∈ R n is the external disturbance vector, u is the controller to be designed, and (H(x), b) is assumed controllable. e stable linear reference model is presented as follows: where p m ∈ R n is the reference state, A m ∈ R n×n is a Hurwitz constant matrix, B m ∈ R n×s is a vector, and C ∈ R s 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 g f (t) satisfies the following condition, where , then UDE-based controller u is expressed as follows: Remark 2. Since controller in equation (9) cancels H(p) 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 lowpass filter: in general, τ � 0.001. e other is the secondary filter: where w 0 � 4π, a � 10w 0 , and b � 100w 0 .

Problem Formulation
Consider the following hyperchaotic system with both model uncertainty and external disturbance: where x ∈ R n is the state, f(x) ∈ R n is a continuous vector function, Δf(x) ∈ R n denotes system model uncertainty, and d(t) ∈ R n stands for the external disturbance.

Mathematical Problems in Engineering
Let system (12) be the master system; then, the slave system with y is given as where x ∈ R n is the state, B ∈ R n×r is a constant matrix, and u ∈ R r , r ≥ 1 is the controller to be designed, and it is assumed that (f(y) + f(x), B) is controllable. Let e � x + y, then the sum system is described as follows: e main goal of this paper is to design a controller u to meet the following condition:

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 g f (t) is designed to satisfy the following condition, where , then the dynamic feedback UDE-based controller u is designed as follows: where

k(t) is updated by the update law (5), and
where B + � (B T B) − 1 B T and * stands for the convolution operator.
Proof. Substituting controller (17) into system (14), we obtain where According to Lemma 1, the system _ e � F(x) is globally asymptotically stable. Noting condition (16), we can obtain us, system (20) is rewritten as and this system is globally asymptotically stable, which completes the proof.

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.
e new 4D hyperchaotic system with uncertainty and disturbance is given as follows: where x ∈ R 4 , f(x) ∈ R 4 is a continuous vector function, Δf(x) ∈ R 4 is the model uncertainty, and d(t) ∈ R 4 is the external disturbance, i.e., Obviously, f(− x) � − f(x). us, the antisynchronization problem of the system _ x � f(x) exists. Let system (23) be the master system, then the corresponding slave system with y is described as where y ∈ R 4 , u � u s + u ude is the controller to be designed, and Set e � x + y, then the sum system is given as follows: where e ∈ R 4 . e controlled sum system without model uncertainty and external disturbance is presented as Mathematical Problems in Engineering Our goal is to design a controller u � u s + u ude . to stabilize the system (27), i.e., lim t⟶∞ ‖e(t)‖ � 0. e first step is to design controller u s . For system (28), it is obvious that if e 2 � 0 and e 3 � 0, we can get that the following two-dimensional system is globally asymptotically stable. From Lemma 1, we can design controller u s as follows: where k(t) 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: x 1 , x 2 , x 3 , and x 4 , antisynchronize, the states of the slave system: y 1 , y 2 , y 3 , and y 4 , respectively. e second step is to design the UDE controller u ude . Let u d � Δf(x) + d(t) and F(x) � f(x) + f(y) + Bu s ; the system (27) is rewritten as  According to eorem 1, the controller u ude is designed as where B + � (B T B) − 1 B T and * stands for the convolution operator. us, u � u s + u ude is obtained.  Figure 3 shows that under the abovementioned controller, the error system is asymptotically stable. Figure 4 shows that u d and u d after a certain time tend to the same constant. Figure 5 shows that the feedback gain k(t) converges to an appropriate constant. Figure 6 shows that the system achieves antisynchronization. e numerical simulation results show that the new 4D hyperchaotic system achieves antisynchronization under the abovementioned controller. Figure 7 shows that states of the master system: x 1 , x 2 , x 3 , and x 4 also antisynchronize the states of the slave system: y 1 , y 2 , y 3 , and y 4 , respectively.

Conclusion
In this paper, the antisynchronization problem of the hyperchaotic systems has been investigated. A new UDEbased dynamic feedback control method has been proposed, and the antisynchronization of the new 4D hyperchaotic system has been realized by the obtained control method. e 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
e authors declare that there are no conflicts of interest regarding the publication of the article.    Mathematical Problems in Engineering 5