# Projection Synchronization of a Class of Complex Chaotic Systems with Both Uncertainty and Disturbance

Authors: Hongsheng Sha; Guijuan Wang; Tao Hao; Zuoxun Wang
Journal: Complexity
 (2020)

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## Abstract

This paper mainly investigates the projection synchronization of complex chaotic systems with both uncertainty and disturbance. Using the linear feedback method and the uncertainty and disturbance estimation- (UDE-) based control method, the projection synchronization of such systems is realized by two steps. In the first step, a linear feedback controller is designed to control the nominal complex chaotic systems to achieve projection synchronization. An UDE-based controller is proposed to estimate the whole of uncertainty and disturbance in the second step. Finally, numerical simulations verify the feasibility and effectiveness of the control method.


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## Body

## 1. Introduction

The chaotic synchronization phenomenon that caused a great sensation in academia was firstly proposed by Pecora and Carroll in early 1990 [1]. They achieved chaotic synchronization of two identical systems with different initial conditions in electronic experiments. Until now, many types of chaotic synchronization have been discovered, such as complete synchronization, phase synchronization, lag synchronization, antisynchronization, and projection synchronization, and many other important results have been obtained (see references [2–8]). Especially, projective synchronization has received much attention due to its faster communication and proportionality between the dynamical systems. In case of projective synchronization, the master and the slave system can be synchronized up to a scaling factor and the scaling factor is a constant transformation between the driving and the response variables that can further increase the security of secure communication and the transmission speed of communication. It has potential application prospects in the field of chaotic secure communication.Many control methods about chaotic projection synchronization have been reported [9–29]. However, most controllers are complicated in structure and difficult in design. Due to the complexity of structure, many control methods are not suitable for projective synchronization control of complex chaotic systems. Among these, the linear feedback controller, because of its simple structure, easy design, and good control effect, was used to realize the projection synchronization of given complex chaotic system. Moreover, in the simulation experiment, it is also proved that the linear feedback controller has a good experimental effect.We note that most of the literature on solving the control problems of chaotic systems with external perturbations is generally complex and difficult to implement. Moreover, when designing the controller, the method to deal with the external disturbance is just simply to cancel the disturbance term from the formula of the controller, and it is not rigorous in nonlinear system control theory. In fact, in the field of nonlinear system control, the UDE-based controller can deal with many structured and unstructured robust control problems and has been applied to the engineering field in some literatures [30–32]. In the simulation experiment, we have noticed that the UDE control method, which is composed of filters with appropriate bandwidth, has an ideal processing effect on the external disturbance of the system which is finally used by us.The main contribution of this paper is to design a physical controller, which is simple in form, to realize the projection synchronization of a complex chaotic system. A linear feedback UDE-based control method is proposed by combining the linear feedback controller and the UED-based controller in two steps. A linear feedback control controller is designed for the nominal complex chaotic system in the first step. In the second step, an UDE-based controller is proposed to estimate the whole of uncertainty and disturbance. In the end, two complex chaotic systems with numerical simulations are used to verify the validity and effectiveness of the proposed theoretical results.


## 2. Preliminary

Consider the following controlled chaotic system:(1)X˙=FX+B∗U∗,where X∈Rn is the state, FX=F1X,⋯,FnXT is a continuous vector function, B∗∈Rn×l, and U∗=U1∗,⋯,Ul∗T is the controller to be designed, l≥1.Let system (1) be the master system; then, the slave system is given as follows:(2)Y˙=FY,where Y∈Rn is the state and FY=F1Y,⋯,FnYT is a continuous vector function.Lete=X−αY, where α=Diagα1,⋯,αn, and the error system is shown as follows:(3)e˙=FX−αFX+B∗U∗,where e∈Rn is the state vector.Definition 1.
Consider the controlled error system (3). If limt⟶∞et=0, then the master system (1) and the slave system (2) are called to achieve projection synchronization.
According to the results in [17], a lemma is introduced as follows.Remark 1.
The projection synchronization of system (1) is achieved if and only it is divided into the following two subsystems:(4)W˙m=AZWm,(5)Z˙=HZ,Wm,where Wm∈Rs, Z∈Rn−s, s≥1, AZ∈Rs×s is a matrix with constants and variable Z, and HZ,Wm is nonlinear continuous function.
An algorithm was also proposed in [17], by which we can solve the solutions of the projection synchronization and choose the variables Wm and Z.


### 2.1. Linear Feedback Control-Like Method for Chaos Projection Synchronization

Note that the subsystemW˙m=AZWm is a linear system with respect to variable Wm if the variable Z is considered a constant. Thus, the linear feedback control method is very suitable to be adopted to solve the projective synchronization problem of a given nominal complex chaotic system (i.e., there is no both uncertainty and disturbance). We briefly introduce the linear control method next.Lemma 1.
Consider the following controlled system:(6)W˙m=AZWm+B1U,where Wm,Z,AZ are given in equations (4) and (5) and B1∈Rs×r; then, the linear feedback controller is designed as follows:(7)U=KZWm,where KZ satisfies the matrix AZ+B1KZ which is Hurwitz no matter what Z is.


### 2.2. UDE-Based Control Method

It is well known that model uncertainty and external disturbance are inevitable in actual control problem, and the UDE-based control method [32] is an effective tool to deal with that problem.Consider the following system:(8)x˙=fx+ud+bu,where x∈Rn is the state, ud=Δfx+dt is the whole of model uncertainty and external disturbance, b∈Rn×k is a constant matrix, k≥1, and u∈Rk is the controller to be designed.The stable linear reference model is given as(9)x˙m=Amxm+BmC,where xm∈Rn is the reference state, Am∈Rn×n is the Hurwitz matrix, Bm∈Rn×k, and C∈Rk×1 is a piecewise continuous and uniformly bounded command to the system.Lemma 2 (see [32]).
Consider system (8). If the designed filter gft satisfies the following condition:(10)u˜d=u^d−ud⟶0,where u^d=x˙−fx−bu∗gft, then the UDE-based controller u is designed as(11)u=b+−fx+ℓ−111−Gfs∗Amx+BmC−Ke−b+ℓ−1sGfs1−Gfs∗xt,where ℓ−1 denotes the inverse Laplace transform operator, b+=bTb−1bT, ∗ is the convolution operator, and Gfs=ℓgft.Remark 2.
According to the existing result in [32], the following two filters are often used. One is the first-order low-pass filter:(12)Gfs=1τs+1.
The other is the secondary filter:(13)Gfs=as+c−w02s2+as+c,where w0=4π, a=10w0, and c=100w0.


## 2.1. Linear Feedback Control-Like Method for Chaos Projection Synchronization

Note that the subsystemW˙m=AZWm is a linear system with respect to variable Wm if the variable Z is considered a constant. Thus, the linear feedback control method is very suitable to be adopted to solve the projective synchronization problem of a given nominal complex chaotic system (i.e., there is no both uncertainty and disturbance). We briefly introduce the linear control method next.Lemma 1.
Consider the following controlled system:(6)W˙m=AZWm+B1U,where Wm,Z,AZ are given in equations (4) and (5) and B1∈Rs×r; then, the linear feedback controller is designed as follows:(7)U=KZWm,where KZ satisfies the matrix AZ+B1KZ which is Hurwitz no matter what Z is.


## 2.2. UDE-Based Control Method

It is well known that model uncertainty and external disturbance are inevitable in actual control problem, and the UDE-based control method [32] is an effective tool to deal with that problem.Consider the following system:(8)x˙=fx+ud+bu,where x∈Rn is the state, ud=Δfx+dt is the whole of model uncertainty and external disturbance, b∈Rn×k is a constant matrix, k≥1, and u∈Rk is the controller to be designed.The stable linear reference model is given as(9)x˙m=Amxm+BmC,where xm∈Rn is the reference state, Am∈Rn×n is the Hurwitz matrix, Bm∈Rn×k, and C∈Rk×1 is a piecewise continuous and uniformly bounded command to the system.Lemma 2 (see [32]).
Consider system (8). If the designed filter gft satisfies the following condition:(10)u˜d=u^d−ud⟶0,where u^d=x˙−fx−bu∗gft, then the UDE-based controller u is designed as(11)u=b+−fx+ℓ−111−Gfs∗Amx+BmC−Ke−b+ℓ−1sGfs1−Gfs∗xt,where ℓ−1 denotes the inverse Laplace transform operator, b+=bTb−1bT, ∗ is the convolution operator, and Gfs=ℓgft.Remark 2.
According to the existing result in [32], the following two filters are often used. One is the first-order low-pass filter:(12)Gfs=1τs+1.
The other is the secondary filter:(13)Gfs=as+c−w02s2+as+c,where w0=4π, a=10w0, and c=100w0.


## 3. Main Results

In this section, the UDE-based linear feedback control method is proposed in two steps. In the first step, the linear feedback control method is proposed for the nominal system. The UDE-based control method is given in the second step.


### 3.1. Linear Feedback Control Method for Projection Synchronization

Consider the following nominal system:(14)X˙=FX+BU,where X∈Rn is the state vector, FX=F1X,⋯,FnXT is a continuous function, B∈Rn×r,r≥1 , U=U1,⋯,UrT is the linear feedback controller to be designed, and FX,B is assumed to be controllable.If the projection synchronization of system (14) exists, then it can be divided into the following two subsystems:(15)W˙m=AZWm+B1U,(16)Z˙=HZ,Wm,where Wm,Z,AZ, HZ,Wm are given in equations (4) and (5), respectively, B1∈Rs×r is given in equation (6), and AZ,B1 is also controllable.The corresponding slave system is presented as follows:(17)W˙s=AZWs,where HZ,Wm is given in equation (15), Wm∈Rr, Z∈Rn−r, and AZ is a constant matrix.Lete=Wm−βWS be the error state, where the scalar β≠0,1, and the error system is obtained as follows:(18)e˙=AZe+B1U.Theorem 1.
Consider error system (18). If AZ,B1 is controllable no matter what Z is, then the linear feedback controller U is designed as follows:(19)U=KZe,where KZ satisfies the matrix AZ+B1KZ which is Hurwitz no matter what Z is; then, error system (18) is globally asymptotically stable. That is, the master system (15) and the slave system (17) achieve the projection synchronization.Proof.
Since the matrixAZ+B1KZ is Hurwitz no matter what Z is, error system (18) is globally asymptotically stable; therefore, the master system (15) and the slave system (17) achieve the projection synchronization.


### 3.2. UDE-Based Control Method for Projection Synchronization

In this section, the UDE controller is proposed to cancel the uncertainty and disturbance of the complex chaotic system.Consider the following controlled master system:(20)W˙m=AZWm+B1V+Ud,where Wm,Z,AZ, HZ,Wm are given in equations (4) and (5), respectively, B1∈Rs×r is given in equation (6), AZ,B1 is controllable, Ud=ΔAZ+Dt, ΔAZ represents the uncertainty and Dt represents the disturbance, and V is the controller to be designed, in which(21)V=U+uude.The corresponding salve system is(22)W˙s=AZWs.Lete=Wm−βWs be the error state vector, where β≠0,1; then, the corresponding error system is shown as follows:(23)e˙=AZe+ud+B1V.The controllerV is designed in two steps:Step one: according to Theorem1, the linear feedback controller U is designed for the nominal system.Step two: the controlleruude is proposed according to the following theorem.Theorem 2.
Consider error system (23). If the designed filter gft satisfies the following condition:(24)u˜d=u^d−ud⟶0,where u^d=e˙−AZ,e−B1uude∗gft, then the UDE-based controller u is designed as(25)uude=B1+ℓ−1Gf1−Gfs∗AZ,e−ℓ−1sGfs1−Gfs∗et,where AZ,e=AZ+B1KZe, B1+=B1TB−1B1T, ℓ−1 is the inverse Laplace transform, ∗ is the convolution sign, and Gfs=ℓgft.Proof.
SubstitutingV in (21) into system (23) results in(26)e˙=AZe+ud+B1V=AZ+B1KZe+Buude−ud.
According to condition (24), it leads to(27)B1uude=−u^d.
Thus,(28)e˙=Az+BV+ud=AZ,e+u˜dis globally asymptotically stable, which completes the proof.


## 3.1. Linear Feedback Control Method for Projection Synchronization

Consider the following nominal system:(14)X˙=FX+BU,where X∈Rn is the state vector, FX=F1X,⋯,FnXT is a continuous function, B∈Rn×r,r≥1 , U=U1,⋯,UrT is the linear feedback controller to be designed, and FX,B is assumed to be controllable.If the projection synchronization of system (14) exists, then it can be divided into the following two subsystems:(15)W˙m=AZWm+B1U,(16)Z˙=HZ,Wm,where Wm,Z,AZ, HZ,Wm are given in equations (4) and (5), respectively, B1∈Rs×r is given in equation (6), and AZ,B1 is also controllable.The corresponding slave system is presented as follows:(17)W˙s=AZWs,where HZ,Wm is given in equation (15), Wm∈Rr, Z∈Rn−r, and AZ is a constant matrix.Lete=Wm−βWS be the error state, where the scalar β≠0,1, and the error system is obtained as follows:(18)e˙=AZe+B1U.Theorem 1.
Consider error system (18). If AZ,B1 is controllable no matter what Z is, then the linear feedback controller U is designed as follows:(19)U=KZe,where KZ satisfies the matrix AZ+B1KZ which is Hurwitz no matter what Z is; then, error system (18) is globally asymptotically stable. That is, the master system (15) and the slave system (17) achieve the projection synchronization.Proof.
Since the matrixAZ+B1KZ is Hurwitz no matter what Z is, error system (18) is globally asymptotically stable; therefore, the master system (15) and the slave system (17) achieve the projection synchronization.


## 3.2. UDE-Based Control Method for Projection Synchronization

In this section, the UDE controller is proposed to cancel the uncertainty and disturbance of the complex chaotic system.Consider the following controlled master system:(20)W˙m=AZWm+B1V+Ud,where Wm,Z,AZ, HZ,Wm are given in equations (4) and (5), respectively, B1∈Rs×r is given in equation (6), AZ,B1 is controllable, Ud=ΔAZ+Dt, ΔAZ represents the uncertainty and Dt represents the disturbance, and V is the controller to be designed, in which(21)V=U+uude.The corresponding salve system is(22)W˙s=AZWs.Lete=Wm−βWs be the error state vector, where β≠0,1; then, the corresponding error system is shown as follows:(23)e˙=AZe+ud+B1V.The controllerV is designed in two steps:Step one: according to Theorem1, the linear feedback controller U is designed for the nominal system.Step two: the controlleruude is proposed according to the following theorem.Theorem 2.
Consider error system (23). If the designed filter gft satisfies the following condition:(24)u˜d=u^d−ud⟶0,where u^d=e˙−AZ,e−B1uude∗gft, then the UDE-based controller u is designed as(25)uude=B1+ℓ−1Gf1−Gfs∗AZ,e−ℓ−1sGfs1−Gfs∗et,where AZ,e=AZ+B1KZe, B1+=B1TB−1B1T, ℓ−1 is the inverse Laplace transform, ∗ is the convolution sign, and Gfs=ℓgft.Proof.
SubstitutingV in (21) into system (23) results in(26)e˙=AZe+ud+B1V=AZ+B1KZe+Buude−ud.
According to condition (24), it leads to(27)B1uude=−u^d.
Thus,(28)e˙=Az+BV+ud=AZ,e+u˜dis globally asymptotically stable, which completes the proof.


## 4. Illustrative Example with Numerical Simulation

In this section, one example with numerical simulations is used to demonstrate the effectiveness and validity of the proposed results.Consider the following complex Lorenz system:(29)x˙1=10x1−x2,x˙2=110x1−x1x3−x2,x˙3=−2x3+12x¯1x2+x1x¯2,where x1=x1r+jx1i,x2=x2r+jx2i are complex variables, x3 is a real variable, j2=−1 represents imaginary unit, and x¯1 and x¯2 are complex conjugate variables of x1, x2, respectively.Separating the real and imaginary parts of complex variablesx1,x2 in system (29), i.e., setting X1=x1r,X2=x1i,X3=x2r,X4=x2i, and X5=x3 representing x5, a new real-variable system is shown as follows:(30)X˙1=10X3−X1,X˙2=10X4−X2,X˙3=110X1−X1X5−X3,X˙4=110X2−X2X5−X4,X˙5=−2X5+X1X3+X2X4.


### 4.1. The Existence of Projection Synchronization of the Complex Lorenz System

According to the results in [17], for system (30), the results are obtained as follows:(31)F1αX−α1F1αX=10α3−α1X3≡0,F2αX−α2F2αX=10α4−α2X4≡0,F3αX−α3F3αX=α1−α1α5X1−α3−α1α5X3≡0,F4αX−α4F4αX=α2−α2α5X2−α4−α2α5X4≡0,F5αX−α5F5αX=α5−α1α3X5−α1α3−α2α4X1X3≡0.It results in(32)α1=α3,(33)α2=α4,(34)α1=α1α5,α3=α1α5,(35)α2=α2α5,α4=α2α5,(36)α5=α1α3,α1α3=α2α4.It is easy to obtain thatα=Diagβ,β,β,β,1 is the one solution of equations (32)–(35), where β≠1 is a nonzero scalar.Thus, the master system (30) is divided into the following two subsystems:(37)W˙m=AZWm,(38)Z˙=HZ,Wm,where(39)Wm=X1⋮X4,Z=X5,AZ=−1001000−10010110−Z0−100110−Z0−1,(40)HZ,Wm=−2Z+Wm1Wm3+Wm2Wm4.


### 4.2. The UDE-Based Linear Feedback Controller Design

The UDE-based linear feedback controller is designed by the following two steps.Step one:(41)W˙m=AZWm+B1U.Wm,Z,AZ are given in equations (38) and (39), respectively, and(42)B1=10010000.Then, the corresponding slave system is given as follows:(43)W˙s=AZWs.AZ,Z are given in equations (38) and (39), respectively.Lete=Wm−βWs, where β=2, and the uncontrolled error system is given as follows:(44)e˙1=10e3−e1,e˙2=10e4−e2,e˙3=110−X5e1−e3,e˙4=110−X5e2−e4.Note that ife1=0 and e2=0, the following system:(45)e˙3=−e3,e˙4=−e4,is globally asymptotically stable.Thus,AZ,B1 is controllable. According to Theorem 1, the linear feedback controller U is obtained as follows:(46)U=KZe=−10e3−10e4.Numerical simulation is given, and the initial values of the master-slave systems of given complex Lorenz system are chosen as follows:X10=0.1,X20=0.2,X30=0.3,X40=−0.3,X50=1.1, Y10=−1,Y20=−1,Y30=−1,Y40=−1.From Figures1 and 2, we observed that under linear feedback control, the error system between the master system and slave system is globally asymptotically stable. Through the observation of Figures 3–5, it is found that the master system and slave system achieve the projection synchronization. That is, the controlled master system and slave system have the same phase portrait, but the axis is different.Step two: consider the following master system with both model uncertainty and external disturbance:(47)W˙m=AZWm+B1V+Ud,whereAZ is given in equation (39), HZ,Wm is presented in equation (40), B1 is given in equation (42), and Ud is the whole of model uncertainty and external disturbance.Figure 1
e
1
,
e
2 are asymptotically stable.Figure 2
e
3
,
e
4 are asymptotically stable.Figure 3
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 4
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 5
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)The salve system is(48)W˙s=AZWs,where AZ is given in equation (39).Lete=Wm−βWs; then, the error system is shown as follows:(49)e˙=AZe+Ud+B1V,where(50)V=U+uude,where U is given in equation (46).According to Theorem2, the UDE-based controller uude is designed as follows:(51)uude=uude1uude2=ℓ−1Gf1−Gfs∗−10e1−sGfs1−Gfs∗e1ℓ−1Gf1−Gfs∗−10e2−sGfs1−Gfs∗e2,where ℓ−1 is the inverse Laplace transform, ∗ is the convolution sign, Gfs=ℓgft, and the design of the filter gft is given in Lemma 2.Numerical simulation results are given with the following conditions:X10=0.1,X20=0.2,X30=0.3,X40=−0.3,X50=1.1, Y10=−1,Y20=−1,Y30=−1,Y40=−1, β=2.Case 1:(52)Ud=0.1X1X2+1000.2X3X4+400.Case 2:(53)Ud=0.1X1X2+0.1sint0.2X3X4+0.3sint.It can be seen from Figures6–9 that the error system is asymptotically stable. Through the observation of Figures 10–13, it is found that the master system and slave system achieve the projection synchronization. That is, the controlled master system and slave system have the same phase portrait, but the axis is different. Figure 14 shows that u^d1 tends to ud1, and Figure 15 shows that u^d2 tends to ud2. Similarly, we found that u^d1 tends to ud1 and u^d2 tends to ud2 from Figures 16 and 17.Figure 6
e
1
,
e
2 are asymptotically stable.Figure 7
e
3
,
e
4 are asymptotically stable.Figure 8
The phase portrait of master subsystem and the slave subsystem.Figure 9
The phase portrait of master subsystem and the slave subsystem.Figure 10
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 11
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 12
u
^
d
1 tends to ud1.
(a)
(b)Figure 13
u
^
d
2 tends to ud2.
(a)
(b)Figure 14
e
1
,
e
2 are asymptotically stable.Figure 15
e
3
,
e
4 are asymptotically stable.Figure 16
u
^
d
1 tends to ud1.Figure 17
u
^
d
2 tends to ud2.


## 4.1. The Existence of Projection Synchronization of the Complex Lorenz System

According to the results in [17], for system (30), the results are obtained as follows:(31)F1αX−α1F1αX=10α3−α1X3≡0,F2αX−α2F2αX=10α4−α2X4≡0,F3αX−α3F3αX=α1−α1α5X1−α3−α1α5X3≡0,F4αX−α4F4αX=α2−α2α5X2−α4−α2α5X4≡0,F5αX−α5F5αX=α5−α1α3X5−α1α3−α2α4X1X3≡0.It results in(32)α1=α3,(33)α2=α4,(34)α1=α1α5,α3=α1α5,(35)α2=α2α5,α4=α2α5,(36)α5=α1α3,α1α3=α2α4.It is easy to obtain thatα=Diagβ,β,β,β,1 is the one solution of equations (32)–(35), where β≠1 is a nonzero scalar.Thus, the master system (30) is divided into the following two subsystems:(37)W˙m=AZWm,(38)Z˙=HZ,Wm,where(39)Wm=X1⋮X4,Z=X5,AZ=−1001000−10010110−Z0−100110−Z0−1,(40)HZ,Wm=−2Z+Wm1Wm3+Wm2Wm4.


## 4.2. The UDE-Based Linear Feedback Controller Design

The UDE-based linear feedback controller is designed by the following two steps.Step one:(41)W˙m=AZWm+B1U.Wm,Z,AZ are given in equations (38) and (39), respectively, and(42)B1=10010000.Then, the corresponding slave system is given as follows:(43)W˙s=AZWs.AZ,Z are given in equations (38) and (39), respectively.Lete=Wm−βWs, where β=2, and the uncontrolled error system is given as follows:(44)e˙1=10e3−e1,e˙2=10e4−e2,e˙3=110−X5e1−e3,e˙4=110−X5e2−e4.Note that ife1=0 and e2=0, the following system:(45)e˙3=−e3,e˙4=−e4,is globally asymptotically stable.Thus,AZ,B1 is controllable. According to Theorem 1, the linear feedback controller U is obtained as follows:(46)U=KZe=−10e3−10e4.Numerical simulation is given, and the initial values of the master-slave systems of given complex Lorenz system are chosen as follows:X10=0.1,X20=0.2,X30=0.3,X40=−0.3,X50=1.1, Y10=−1,Y20=−1,Y30=−1,Y40=−1.From Figures1 and 2, we observed that under linear feedback control, the error system between the master system and slave system is globally asymptotically stable. Through the observation of Figures 3–5, it is found that the master system and slave system achieve the projection synchronization. That is, the controlled master system and slave system have the same phase portrait, but the axis is different.Step two: consider the following master system with both model uncertainty and external disturbance:(47)W˙m=AZWm+B1V+Ud,whereAZ is given in equation (39), HZ,Wm is presented in equation (40), B1 is given in equation (42), and Ud is the whole of model uncertainty and external disturbance.Figure 1
e
1
,
e
2 are asymptotically stable.Figure 2
e
3
,
e
4 are asymptotically stable.Figure 3
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 4
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 5
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)The salve system is(48)W˙s=AZWs,where AZ is given in equation (39).Lete=Wm−βWs; then, the error system is shown as follows:(49)e˙=AZe+Ud+B1V,where(50)V=U+uude,where U is given in equation (46).According to Theorem2, the UDE-based controller uude is designed as follows:(51)uude=uude1uude2=ℓ−1Gf1−Gfs∗−10e1−sGfs1−Gfs∗e1ℓ−1Gf1−Gfs∗−10e2−sGfs1−Gfs∗e2,where ℓ−1 is the inverse Laplace transform, ∗ is the convolution sign, Gfs=ℓgft, and the design of the filter gft is given in Lemma 2.Numerical simulation results are given with the following conditions:X10=0.1,X20=0.2,X30=0.3,X40=−0.3,X50=1.1, Y10=−1,Y20=−1,Y30=−1,Y40=−1, β=2.Case 1:(52)Ud=0.1X1X2+1000.2X3X4+400.Case 2:(53)Ud=0.1X1X2+0.1sint0.2X3X4+0.3sint.It can be seen from Figures6–9 that the error system is asymptotically stable. Through the observation of Figures 10–13, it is found that the master system and slave system achieve the projection synchronization. That is, the controlled master system and slave system have the same phase portrait, but the axis is different. Figure 14 shows that u^d1 tends to ud1, and Figure 15 shows that u^d2 tends to ud2. Similarly, we found that u^d1 tends to ud1 and u^d2 tends to ud2 from Figures 16 and 17.Figure 6
e
1
,
e
2 are asymptotically stable.Figure 7
e
3
,
e
4 are asymptotically stable.Figure 8
The phase portrait of master subsystem and the slave subsystem.Figure 9
The phase portrait of master subsystem and the slave subsystem.Figure 10
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 11
The phase portrait of master subsystem and the slave subsystem.
(a)
(b)Figure 12
u
^
d
1 tends to ud1.
(a)
(b)Figure 13
u
^
d
2 tends to ud2.
(a)
(b)Figure 14
e
1
,
e
2 are asymptotically stable.Figure 15
e
3
,
e
4 are asymptotically stable.Figure 16
u
^
d
1 tends to ud1.Figure 17
u
^
d
2 tends to ud2.


## 5. Conclusion

In conclusion, the projective synchronization of a class of complex chaotic systems with both uncertainty and disturbance has been solved. First, the linear feedback control method is proposed for the nominal system (without uncertainty and disturbance), and projection synchronization of such system has been realized. Then, the UDE-based linear feedback control method is presented by two steps, by which the projection synchronization of the complex chaotic systems with both uncertainty and disturbance has been completed. Finally, an experimental simulation example has been used to verify the feasibility and effectiveness of the obtained results.



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*Source: 3742876-2020-07-14.xml*