# Synchronization of a Class of Chaotic Systems with Both Uncertainty and Disturbance by the UDE-Based Control Method

Authors: Zuoxun Wang; Hongsheng Sha
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, chaotic synchronization with uncertainties and disturbances is studied. Firstly, a new control method based on UDE control is proposed, which is composed of two controllers: one is the stabilization controller in order to realize the stabilization of nominal system without uncertainties and disturbances and the other is the UDE controller in order to deal with the given controlled system with both uncertainties and disturbances. Then, two examples are studied by the above method. Finally, the correctness and effectiveness of the method are verified by numerical simulation.


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## Body

## 1. 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, they 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 and it has a wide range of applications in communications encryption, information science, chaotic generator design, and chemical reactions [3–8], etc. In recent years, scholars had proposed a variety of control methods to realize the synchronization between the same chaotic systems or the different chaotic systems with different initial conditions (refer to [9–20]). Among many control methods, the adaptive feedback control method [20] has a wide range of applications due to its simple design and easy implementation.As far as we know, for chaotic synchronization, most of the existing control methods [9–13] only deal with the chaotic systems whose systems do not include model uncertainty and external disturbance and even if 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.In order to solve this problem, the UDE controller [21–23], with its high bandwidth filter, can effectively remove the interference outside the system and it did have been widely used in the field of nonlinear control [22, 24]. However, this control method has some limitations in applications; for example, the controller’s design is too complex to be realized in applications, which partly motivates our present work. Based on this, a new adaptive feedback control method based on UDE control is proposed by combining the adaptive feedback controller with the UDE controller to deal with synchronization problems of given chaotic system with both uncertainty and disturbance.The remainder content of this paper is as follows: first of all, an adaptive feedback control method is given to solve the synchronization of given chaotic systems without uncertainty and disturbance. Then, the adaptive feedback controller is combined with the UDE controller to deal with synchronization problems of given chaotic system with both uncertainty and disturbance. At last, the simulation results verify the effectiveness and feasibility of the proposed control method.The main structure of the following article is divided into three parts:(1)
Firstly, an adaptive feedback control method is given to solve the synchronization of given chaotic systems without uncertainty and disturbance(2)
Secondly, the adaptive feedback controller is combined with the UDE controller to deal with synchronization problems of given chaotic system with both uncertainty and disturbance(3)
Finally, the simulation results verify the effectiveness and feasibility of the proposed control method


## 2. Preliminary

### 2.1. Adaptive Feedback Control Method

Consider the following nonlinear system(1)w˙=gw,where w∈Rn is the state vector and gw is a continuous function.Let system (1) be the master system, then the corresponding controlled slave system is given as follows:(2)v˙=gv+Nu,where v∈Rn is the state vector, N∈Rn×r, r≥1, and u is the designed controller.Letq=v−w, the controlled error system can be written as follows:(3)q˙=gv−gw+Nu.Definition 1.
Consider system (3). If limt⟶∞qt=0, the master system (1) will synchronize the slave system (2) by the controller u.Next, according to the existing result [20], let us briefly introduce the adaptive feedback controller.(4)us=Kq,where K=mtNT, and(5)m˙t=−oqt2,where o>0 is a constant.Lemma 1.
(see [20]). Consider the controlled error system (3), if N=Nijn×r and Nij=0 or Nij=1, i=1,2,…,n and j=1,2,…,r, and gv−gw,N is controllable, then an adaptive controller us is obtained as follows:


### 2.2. The UDE Controller

Since the UDE controller [22] is suitable for linear and nonlinear systems with uncertainties and perturbations, it is used by us to solve the problems of system uncertainty and perturbation. The following is a brief introduction of this control method.Consider the following nonlinear system(6)w˙=hw+Δhw+Nu+Dt,and w∈Rn is the state vector, N∈Rn×r is a constant matrix, Δhw∈Rn is the model uncertainty, and Dt∈Rn represents the external disturbance.The stable linear reference model is described as follows:(7)w˙m=Zmwm+NmC,where wm∈Rn is the reference state vector, Zm∈Rn×n is Hurwitz, Nm∈Rn×r, and C∈Rr×1 is continuously consistent and bounded instructions for the system.Lemma 2.
(see [23]). Consider system (6), the designed filter gAt should satisfy(8)u˜d=u^d−ud⟶0,t⟶∞,where u^d=w˙−hw−Nu∗sft and ud=Δhw+Dt, and then, the UDE controller u is designed as follows:(9)u=N+ℓ−111−Jfs∗Zmw+NmC−Kq−N+hw+ℓ−1sJfs1−Jfs∗wt,where ℓ−1 denotes the inverse Laplace transform operator, N+=NTN−1NT, ∗ is the convolution operator, and Jfs=ℓgft.Remark 1.
According to the existing result [22], the following two filters are often used in applications.
One is(10)Jfs=1τs+1,in general, τ=0.001.
And the other is(11)Jfs=as+b−w02s2+as+b,where w0=4π, a=10w0, and b=100w0.


## 2.1. Adaptive Feedback Control Method

Consider the following nonlinear system(1)w˙=gw,where w∈Rn is the state vector and gw is a continuous function.Let system (1) be the master system, then the corresponding controlled slave system is given as follows:(2)v˙=gv+Nu,where v∈Rn is the state vector, N∈Rn×r, r≥1, and u is the designed controller.Letq=v−w, the controlled error system can be written as follows:(3)q˙=gv−gw+Nu.Definition 1.
Consider system (3). If limt⟶∞qt=0, the master system (1) will synchronize the slave system (2) by the controller u.Next, according to the existing result [20], let us briefly introduce the adaptive feedback controller.(4)us=Kq,where K=mtNT, and(5)m˙t=−oqt2,where o>0 is a constant.Lemma 1.
(see [20]). Consider the controlled error system (3), if N=Nijn×r and Nij=0 or Nij=1, i=1,2,…,n and j=1,2,…,r, and gv−gw,N is controllable, then an adaptive controller us is obtained as follows:


## 2.2. The UDE Controller

Since the UDE controller [22] is suitable for linear and nonlinear systems with uncertainties and perturbations, it is used by us to solve the problems of system uncertainty and perturbation. The following is a brief introduction of this control method.Consider the following nonlinear system(6)w˙=hw+Δhw+Nu+Dt,and w∈Rn is the state vector, N∈Rn×r is a constant matrix, Δhw∈Rn is the model uncertainty, and Dt∈Rn represents the external disturbance.The stable linear reference model is described as follows:(7)w˙m=Zmwm+NmC,where wm∈Rn is the reference state vector, Zm∈Rn×n is Hurwitz, Nm∈Rn×r, and C∈Rr×1 is continuously consistent and bounded instructions for the system.Lemma 2.
(see [23]). Consider system (6), the designed filter gAt should satisfy(8)u˜d=u^d−ud⟶0,t⟶∞,where u^d=w˙−hw−Nu∗sft and ud=Δhw+Dt, and then, the UDE controller u is designed as follows:(9)u=N+ℓ−111−Jfs∗Zmw+NmC−Kq−N+hw+ℓ−1sJfs1−Jfs∗wt,where ℓ−1 denotes the inverse Laplace transform operator, N+=NTN−1NT, ∗ is the convolution operator, and Jfs=ℓgft.Remark 1.
According to the existing result [22], the following two filters are often used in applications.
One is(10)Jfs=1τs+1,in general, τ=0.001.
And the other is(11)Jfs=as+b−w02s2+as+b,where w0=4π, a=10w0, and b=100w0.


## 3. Main Results

Consider the advantages of the adaptive control method and UDE control method, a new result of the combination of UDE controller and adaptive feedback controller is proposed.Consider the following master chaotic system with both uncertainty and disturbance(12)w˙=Aw+ΔAw+Dt,where w∈Rn, ΔAw denotes system model uncertainty, and Dt is external disturbance.Then, the slave system is shown as follows:(13)v˙=Av+Nu,where v∈Rn is the state vector, N=bijn×r and bij=0 or bij=1, i=1,2,…,n, j=1,2,…,r, and u is the controller to be designed.Letq=v−w, the error system is represented as follows:(14)q˙=Av−Aw+ΔAw+Dt+Nu,where q∈Rn and Av−Aw,N is controllable.Then, we propose a conclusion.whereFw,q=Av−Aw+Nus.Theorem 1.
Consider the error system (14), if the designed filter gAt satisfies the following condition:(15)u˜d=u^d−ud,t⟶∞,where u^d=x˙−Aw−ud∗gft and ud=ΔAw+Dt; and then, the UDE-based controller u is shown as follows:(16)u=us+uude,where us=Kq, K=mtNT, and(17)m˙t=−oet2,(18)uude=N+ℓ−1Jf1−Jfs∗Fw,q−N+ℓ−1sJfs1−JAs∗qt,Proof.
Substitutingu in (16) to the error system (14), it results in q˙=Av−Aw+Nus+Nuude−ud=Fw,q+Nuude−ud.
According to (15), it can be calculated as Buude=u^d, and according to Lemma 1, it can be concluded that the system q˙=Fw,q is globally asymptotically stable. Thus, the conclusion is established.


## 4. Illustrative Examples with Numerical Simulation

In this section, the three-dimensional Lorenz chaotic system and 4D hyperchaotic system are taken as examples, respectively.(19)w˙=Aw+ΔAw+Dt,where w=w1,w2,w3T, ΔAw denotes system model uncertainty, and Dt is external disturbance, i.e.,(20)Aw=A1wA2wA3w=10w2−w1,28w1−w2−w1w3−83w3+w1w2,(21)ΔAw=0,−0.1w12,0,Dt=0,0.1sint0,Example 1.
Consider the following master chaotic system with both uncertainty and disturbanceThen, the slave system is shown as follows:(22)v˙=Av+Nu,where v=v1,v2,v3T and Av−Av,N is controllable, i.e.,(23)Av=A1vA2vA3v=10v2−v1,28v1−v2−v1v3,−83v3+v1v2,(24)N=010.Letq=v−w, the error system is shown as follows:(25)q˙=Av−Aw+Nu−ΔAw−Dt,where q=v−w∈R3 and u=us+uude is the controller to be designed. Our goal is to design a controller u=us+uude to stabilize system (25) with two control steps.In the first step, the controllerus is to be designed as follows. By observing (20) and (23), if q2=0, the following two-dimensional system(26)q˙1=−28q1,q˙3=−83q3+w2q1is globally asymptotically stable.According to Lemma2, the controller us can be designed as follows:(27)us=mtNTq=mt010Tq,and the feedback gain mt is updated by the update law (17). Next, the numerical simulation is carried out with the initial conditions (for the convenience of the reader, we replace w. with x., v. with y., and m. with k., and the specific initial values are shown as follows: x10=0.1,x20=0.2,x30=0.3,y10=1,y20=2,y30=3,k0=−1); then, the numerical simulation results are shown as follows.Figure1 shows that the error system is asymptotically stable, and Figure 2 shows that the feedback gain kt converges to a negative constant.Figure 1
The error system is asymptotically stable.Figure 2
The feedback gainkt converges to a negative constant.The second step is to design the UDE controlleruude. Let ud=ΔAw+Dt,Fx=Av−Aw+Nus, and system (25) is rewritten as follows:(28)w˙=Fw+Nuude−ud.According to Lemma2, the controller uude is designed as(29)uude=N+w˙−Aw∗gft.Thus, the controlleru=us+uude is completely designed. In the following, the simulation results is carried out as follows.Figure3 shows that the error system is also asymptotically stable, Figure 4 shows that the states x1,x2,x3 synchronize the states y1,y2,y3, respectively, Figure 5 shows that the feedback gain kt converges to a negative constant, and Figure 6 shows that u^d tends to ud as t⟶∞.Figure 3
The error system is also asymptotically stable.Figure 4
The statesx1,x2,x3 synchronize the states y1,y2,y3, respectively.Figure 5
The feedback gainkt converges to a negative constant.Figure 6
u^d tends to ud as t⟶∞.Example 2.
Consider the controlled 4D hyperchaotic system with both uncertainty and disturbance as follows:(30)w˙=Aw+ΔAw+Dt,where w=w1,w2,w3,w4T. (31)Aw=A1wA2wA3wA4w=15w2−w1,10w2−w1w3+w4,w1w2−5w3,w3−w4,ΔAw=0,−0.1w12,0,0,Dt=0,0.1sint,0,0,
Let system (30) be the master system, then the slave system is given as follows:(32)v˙=fv+Nu,where v=v1,v2,v3,v4T.(33)Av=A1vA2vA3vA4v=15v2−v110v2−v1v3+v4v1v2−5v3v3−v4,N=00100100.
Letq=v−w, the error system is represented as follows:(34)q˙=fv−fw+Nu−ΔAw−Dt,where u=us+uude is the controller to be designed.Our goal is to design a controlleru=us+uude to stabilize system (34) with two control steps.In the first step, the controllerus is to be designed as follows.By observing the error system (34), i.e.,(35)q˙1=15q2−q1,q˙2=10q2−q1q3+q4−w3q1−w1q3,q˙3=q1q2−5q3+w1q2+w2q1,q˙4=q3−q4.By observing the conclusion results of (35), we note that, if q2=0,q3=0, then the following subsystem(36)q˙1=−15q1,q˙4=−q4is globally asymptotically stable.Therefore, according to Lemma1, the controller us can be designed as follows:(37)us=mtNTq=mt01000010q,and the feedback gain mt is updated by the update law (17).Then, the simulation result is carried out with the initial condition (for the convenience of the reader, we replacew. with x., v. with y., and m. with k.: x10=1,x20=−2,x30=3,x40=−4,y10=5,y20=−6,y30=7,y40=−8,k0=−1).Figure7 shows that the error system is asymptotically stable, and Figure 8 shows that the feedback gain kt converges to a negative constant.Figure 7
The error system is asymptotically stable.Figure 8
The feedback gainkt converges to a negative constant.In the second step, the controlleruude is designed as follows. Let ud=ΔAw+Dt,Fw=Av−Aw+Nus, system (30) is rewritten as follows:(38)w˙=Fw+Nuude−ud.According to Lemma2, the controller uude can be designed as follows:(39)uude=N+ℓ−1JAs1−JAs∗Fw,q−N+ℓ−1sJAs1−JAs∗qt,where ℓ−1 denotes the inverse Laplace transform operator, N+=NTN−1NT, ∗ is the convolution operator, and JAs=ℓgAt. Thus, the controller u=us+uude is completely designed. Next, the simulation is carried out with the initial condition (for the convenience of the reader, we replace w. with x., v. with y., and m. with k.: x10=1,x20=−2,x30=3,x40=−4,y10=5,y20=−6,y30=7,y40=−8,k0=−1), and the numerical simulation results are shown in Figure 9–12.Figure 9
The error is also asymptotically stable.Figure 10
The statesx1,x2,x3,x4 synchronize the states y1,y2,y3,y4, respectively.Figure 11
The feedback gainkt converges to a negative constant.Figure 12
u^d tends to ud as t⟶∞.Figure9 shows that the error system is also asymptotically stable, Figure 10 shows that the states x1,x2,x3 synchronize the states y1,y2,y3, respectively, Figure 11 shows that the feedback gain kt converges to a negative constant, and Figure 12 shows that u^d tends to ud as t⟶∞.


## 5. 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 is composed of two controllers: one is the stabilization controller and the other is the UDE controller. Secondly, two examples have been studied by the above methods. Finally, numerical simulations have verified the effectiveness and correctness of the theoretical results.



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