Synchronization of a class of chaotic systems with both uncertainty and disturbance by the UDE-based control method

In this paper, synchronization of chaotic systems with both uncertainty and disturbance is investigated. Firstly, a new uncertainty and disturbance estimation (UDE)-based control method is proposed, which are composed of two controllers, one is the stabilization controller, the other is the UDE controller. Then, two examples are studied by the above method. Finally, numerical simulations verify the effeteness and correct of the theoretical results.


Introduction
The chaotic synchronization phenomenon that caused great sensation in the academic world was first discovered by Pecora and Carrol in 1990 [1]. They realized the chaotic synchronization of the two same systems with different initial conditions in the electronics experiment. Decades later, Pecora and Carrol reviewed the literature on chaotic synchronization again [2]. Chaotic synchronization refers to the fact that the unstable motion controlled by the chaotic system tends to move in angular phase or amplitude with a similar rhythm through a certain coupling relationship. Chaotic synchronization has a wide range of applications in communications encryption, information science, chaotic generator design, and chemical reactions [3,4,5,6] and the reference therein. In recent years, scholars have proposed a variety of control methods to realize the chaotic synchronization between the same chaotic systems with different initial conditions, especially for the different chaotic systems, please refer to [7,8,9,10,11,12,13,14,15,16,17,18]. Among many control methods, the adaptive feedback control method [18] has a wide range of applications due to its simple design and easy implementation. As far as we know, for chaos synchronization, most of the existing control methods [7,8,9,10,11] only deal with the chaotic systems whose systems do not include model uncertainty and external disturbance. Some methods can be used to cancel the uncertainty and disturbance, but the uncertainty and the disturbance are assumed bounded. As a matter of fact, uncertainty and the disturbance of such systems cannot be avoided, and are often very large. Therefore, to investigate chaotic synchronization of the systems with both model uncertainty and disturbance is not only necessary, but also meaningful. The UDE-based control method [22,23,24] is very effective method to deal with the systems with both uncertainty and disturbance. It has been applied to solve all kinds of control questions. However, this UDE-based control method has some limitations in applications. Such as, the designed controller design is too complex to be realized in applications, which partly motivates our present work. It is very in-This work was supported by National Natural Science Foundation of Shandong Province [ZR2018MF016] teresting to combine the adaptive feedback method [18] and the UDE-based method to derive a new UDE-based adaptive feedback control method to deal with the chaos synchronization of the systems with both uncertainty and disturbance. Thus, we investigate the chaos synchronization of chaotic systems with both uncertainty and disturbance. At first, we propose a new UDE-based adaptive feedback control method, which are composed of two controllers, one is the stabilization controller, the other is the UDE controller. Then, by the new method, two examples are studied. Finally, numerical simulations verify the effeteness and correct of the theoretical results.

Adaptive feedback control method
Consider the following chaotic system: where x ∈ R n is the state, g(x) is a continuous function.
Let the system (1) be the master system, then the corresponding controlled slave system is given as: where y ∈ R n is the state, B ∈ R n×r , r ≥ 1, u is the designed controller. Let e = y − x, the controlled error system can be written as At present, there are many methods are used to realized the chaos synchronization. Among them, the adaptive control method [18] has a wide range of applications because of its simple design and easy implementation. Here is a brief introduction.
According to the existing result [18], the follow result is presented.

UDE-based control method
The UDE-based control method [23] is suitable for both linear and nonlinear systems with both uncertainty and disturbance, we firstly introduce it in the follows. Consider the following nonlinear system: and x ∈ R n is the state, B ∈ R n×r is a constant matrix, Δh(x) ∈ R n is the model uncertainty, d(t) ∈ R n is an unknown disturbance.
The stable linear reference model is described as: where x m ∈ R n is the reference state, A m ∈ R n×n is Hurwitz, B m ∈ R n×r , C ∈ R r×1 is a piecewise continuous and uniformly bounded command to the system.

Lemma 2.2 [24]
Consider the system (6). If a designed filter g f (t) to satisfy: then the UDE-based controller u is designed as: where −1 denotes the inverse Laplace transform operator, Remark 2.1 According to the existing result [23], the following two filters are often used in applications. One is the first-order low-pass filter: in general, τ = 0.001. The other is the secondary filter: where w 0 = 4π, a = 10w 0 , b = 100w 0 .

Main results
Since controller (9) cancels h(x) in the system (6) directly, the controller is too complex to be realized in actual chaotic synchronization system. Considering the advantages of adaptive control method and UDE-based control method, we propose a new UDE-based adaptive control method, and present the following results. Consider the following master chaotic system with uncertainty and disturbance: where x ∈ R n , Δf (x) denotes system model uncertainty and d(t) is external disturbance. Then, the slave system is shown as follows: where y ∈ R n , B = (b ij ) n×r and b ij = 0 or b ij = 1 ,i = 1, 2, · · · , n, j = 1, 2, · · · , r, u is the controller. Let e = y − x, the error system is presented aṡ where e ∈ R n , and (f (y) − f (x), B) is controllable. Then, we propose a conclusion.
Theorem 3.1 Consider the error system (14). If the designed filter g f (t) satisfies the following condition: then the UDE-based controller u is: where u s = Ke, K = k(t)B T , anḋ where Proof: Substituting u in (16) to the error system (14), it results iṅ According to (15), we can get Bu ude =û d . Note that Lemma 2.1, it can be concluded that the systemė = F (x, e) is globally asymptotically stable, thus the conclusion is established.

Illustrative examples with numerical simulation
In this section, the three-dimensional Lorenz chaotic system is taken as examples.
Example 4.1 Consider the following master chaotic system with uncertainty and disturbance: where x = (x 1 , x 2 , x 3 ) T , Δf (x) denotes system model uncertainty and d(t) is external disturbance, i.e., Then, the slave system is shown as follows: where y = (y 1 , y 2 , Let e = y − x, then the error system is given as: where e = y − x ∈ R 3 , u = u s + u ude is the controller to be designed. Our goal is to design a controller u = u s + u ude to stabilize the system (25), i.e., lim t→∞ e(t) = 0.
In the first step, we design the controller u s . Observing (20) and (23), if e 2 = 0, then the following twodimensional system: is globally asymptotically stable. According to Lemma 2.2, the controller u s is designed as follows: and the feedback gain k(t) is updated by the update law (17). Next, numerical simulation is carried out with the initial conditions: x 1 (0) = 0.1, x 2 (0) = 0.2, x 3 (0) = 0.3, y 1 (0) = 1, y 2 (0) = 2, y 3 (0) = 3, k0 = −1, the numerical simulation results are shown in Figure 1, Figure 2. The second step is to design the UDE controller u ude . Let u d = Δf (x)+d(t), F (x) = f (y)−f (x)+Bu s , the system (25) is rewritten aṡ According to Lemma 2.2, the controller u ude is designed as Thus, the controller u = u s + u ude is designed.
In the following, we carry out simulation results are shown

Conclusion
In this paper, synchronization of chaotic systems with both uncertainty and disturbance has been investigated. Firstly, a new UDE-based control method has been presented, which are composed of two controllers, one is the stabilization controller, the other is the UDE controller. Then, two examples have been studied by the above methods. Finally, numerical simulations have been verified the effeteness and correct of the theoretical results.