# Stabilization of a Class of Complex Chaotic Systems by the Dynamic Feedback Control

Authors: Zhi Liu; Rongwei Guo
Journal: Complexity
 (2020)

---

## 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.


---

## Body

## 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 resultsBefore ending this section, we present some notations used in this paper.ℝn is the n dimensional Euclidean space, ℂn is the n dimensional complex space, In denotes the n×n identity matrix, ℳm×n denotes the set of m×n real matrices, ⊗ is the Kronecker product, and ∝imes is the semitensor product (STP), i.e., let Mm×n∈ℳm×n, Np×q∈Np×q, and t=lcmn,p be the least common multiple of n and p. The STP of M and N is defined as(1)M∝N=M⊗It/nN⊗It/p∈ℳmt/n×qt/p,where(2)M⊗N=M11NM12N⋯M1nN⋮⋮⋮⋮Mm1NMm2N⋯MmnN∈ℳmp×np.Re⋅ and Im⋅ represent the real part and the imaginary part of (·), respectively, and i is the imaginary unit, i.e., i2=−1.


## 2. Preliminary

Consider the following controlled chaotic system:(3)w˙=Gw+bv,where w∈ℝn is the state, Gw∈ℝn is continuous function with G0=0, b∈ℝn×r is a constant matrix, and v∈ℝr is the controller to be designed.Lemma 1.
(see [18]). Consider system (3). If Gw,b can be stabilized, then the designed controller v is of the following form:(4)v=Ktw,where K=ktbT, and the feedback gain kt is updated by the following equation:(5)k˙t=−w2.


## 3. Problem Formulation

Consider the following controlled complex chaotic system:(6)p˙=fp+bu,where(7)p=zx,z=z1z2⋮zm,x=xm+1xm+2⋮xn,z∈ℂm and x∈ℝn−m are the state, m≥1,(8)fp=fx,z,z¯=Mxz+HixNz,z¯,x,where z¯ is the conjugate of z, Mx∈ℝm×m, Hi∈ℂm×n−m is a complex constant matrix, and Nx,z,z¯∈ℝn−m, b∈ℝn×r, u∈ℂr is the designed controller, i.e.,(9)Hi=h1ih2i⋮hmi,b=b100b2,where hji∈ℂn−m, j=1,…,m, b1∈ℝm×s, b2∈ℝn−m×r−s, 1≤s≤r,(10)u=uzux,uz=u1u2⋮uk,ux=uk+1uk+2⋮ur,that is, uz∈ℂk, ux∈ℝr−k, and 1≤k<r.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 u to guarantee(11)limt⟶∞pt=0.


## 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:(12)y˙=Fy+BU,where y∈ℝm+n is the state, Fy∈ℝm+n is continuous function with F0=0, B∈ℝm+n×k+r is a constant matrix, and U∈ℝk+r is the controller to be designed.Proof.
Letzj=y2j−1+y2j×i, j=1,…,m, and y2m+l=xm+l, 1≤l≤n−m; then, the equivalent real system (12) is obtained, i.e.,(13)y=yzyx,yz=y1y2⋮y2m,yx=y2m+1y2m+2⋮ym+n,Fy=F1yF2y⋮Fm+ny,F1yF2y⋮F2my=M∗x×yz+H∗×yx,M∗x=Mx⊗I2,H∗=Reh1iImh1i⋮RehmiImhmi,B=B100b2,B1=b1⊗I2,U=UzUx,Uz=Reu1Imu1⋮ReukImuk,U2=ux.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 U is designed. Moreover, the controller u of system (6) is obtained by(14)u=1i∝U=U1+U2×iU3+U4×i⋮U2r−1+U2r×i.Remark 3.
For a given general controlled complex chaotic system of the following form:(15)z˙=fz+bu,where z∈ℂn is the state, fz∈ℂn is continuous function with f0=0, and b∈ℝn×r and u∈ℂr are the designed controller, r≥1.
By Theorem1, its equivalent real system is obtained as follows:(16)y˙=Fy+BU,where y∈ℝ2n is the state, Fy∈ℝ2n is continuous function with F0=0, B∈ℝ2n×2r is a constant matrix, and U∈ℝ2r is the controller to be designed, i.e.,(17)y=Rez1Imz1⋮ReznImzn,Fy=Ref1zImf1z⋮RefnzImfnz,B=b⊗I2,U=Reu1Imu1⋮ReurImur.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:(18)p˙=fp+ud+bu,where p, fp, b, and u are given in equations (7)–(10), respectively,(19)ud=Δfp+dt=ud1ud2⋮udn,where Δfp∈ℂn stands for the model uncertainty and dt∈ℝn is the external disturbance. Its equivalent real system is described as the following form:(20)y˙=Fy+BU+UD,where y, Fy, B, and U are the same as those in Theorem 1, and(21)UD=Reud1Imud1⋮ReudnImudn.


### 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 Fy,B can be stabilized, then the controller U is designed of the following form:(22)U=Kty,where K=ktBT, and kt is updated by(23)k˙t=−y2.Remark 4.
According to equation (14), the controller u 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:(24)y=YX,Fy=AXYGX,Y,B=BY0,i.e.,(25)Y˙=AXY+BYU,X˙=GX,Y,with(26)X˙=GX,0,is globally asymptotically stable. If Fy,B is controllable, then the designed controller U is of the following form:(27)U=KXY,where KX meets the matrix BYKX+AX is Hurwitz whatever X is.Proof.
SinceFy,B is controllable, thus AX,BY is also controllable whatever X is. According to the pole assignment theory, the controller U given in (27) is as requested, which completes the proof.


## 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:(12)y˙=Fy+BU,where y∈ℝm+n is the state, Fy∈ℝm+n is continuous function with F0=0, B∈ℝm+n×k+r is a constant matrix, and U∈ℝk+r is the controller to be designed.Proof.
Letzj=y2j−1+y2j×i, j=1,…,m, and y2m+l=xm+l, 1≤l≤n−m; then, the equivalent real system (12) is obtained, i.e.,(13)y=yzyx,yz=y1y2⋮y2m,yx=y2m+1y2m+2⋮ym+n,Fy=F1yF2y⋮Fm+ny,F1yF2y⋮F2my=M∗x×yz+H∗×yx,M∗x=Mx⊗I2,H∗=Reh1iImh1i⋮RehmiImhmi,B=B100b2,B1=b1⊗I2,U=UzUx,Uz=Reu1Imu1⋮ReukImuk,U2=ux.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 U is designed. Moreover, the controller u of system (6) is obtained by(14)u=1i∝U=U1+U2×iU3+U4×i⋮U2r−1+U2r×i.Remark 3.
For a given general controlled complex chaotic system of the following form:(15)z˙=fz+bu,where z∈ℂn is the state, fz∈ℂn is continuous function with f0=0, and b∈ℝn×r and u∈ℂr are the designed controller, r≥1.
By Theorem1, its equivalent real system is obtained as follows:(16)y˙=Fy+BU,where y∈ℝ2n is the state, Fy∈ℝ2n is continuous function with F0=0, B∈ℝ2n×2r is a constant matrix, and U∈ℝ2r is the controller to be designed, i.e.,(17)y=Rez1Imz1⋮ReznImzn,Fy=Ref1zImf1z⋮RefnzImfnz,B=b⊗I2,U=Reu1Imu1⋮ReurImur.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:(18)p˙=fp+ud+bu,where p, fp, b, and u are given in equations (7)–(10), respectively,(19)ud=Δfp+dt=ud1ud2⋮udn,where Δfp∈ℂn stands for the model uncertainty and dt∈ℝn is the external disturbance. Its equivalent real system is described as the following form:(20)y˙=Fy+BU+UD,where y, Fy, B, and U are the same as those in Theorem 1, and(21)UD=Reud1Imud1⋮ReudnImudn.


## 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 Fy,B can be stabilized, then the controller U is designed of the following form:(22)U=Kty,where K=ktBT, and kt is updated by(23)k˙t=−y2.Remark 4.
According to equation (14), the controller u 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:(24)y=YX,Fy=AXYGX,Y,B=BY0,i.e.,(25)Y˙=AXY+BYU,X˙=GX,Y,with(26)X˙=GX,0,is globally asymptotically stable. If Fy,B is controllable, then the designed controller U is of the following form:(27)U=KXY,where KX meets the matrix BYKX+AX is Hurwitz whatever X is.Proof.
SinceFy,B is controllable, thus AX,BY is also controllable whatever X is. According to the pole assignment theory, the controller U 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:(28)p˙=fp+bu,where(29)p=zx,z=z1z2,x=x3,i.e., m=2, n=3, and(30)fp=fx,z,z¯=Mxz+HixNx,z,z¯,b=b100b2=b1=010,Mx=−1010110−x3−1,Hi=0,Nx,z,z^=−2x3+12z¯1z2+z1z¯2.According to Theorem1, the equivalent real system is obtained as follows:(31)y˙=Fy+BU,where(32)y=yzyx,yz=y1y2y3y4,yx=y5,Fy=F1yF2y⋮F5y,F1yF2yF3yF4y=Mx⊗I2×yz=−1001000−10010110−y50−100110−y50−1y1y2y3y4,B=B1=b1⊗I2=000010010000,U=Uz=U1U2,i.e.,(33)y˙1=−10y1+10y3,y˙2=−10y2+10y4,y˙3=110−y5y1−y3+U1,y˙4=110−y5y2−y4+U2,y˙5=−2y5+y1y3+y2y4.Note that ify3=y4=0, then the following subsystem(34)y˙1=−10y1,y˙2=−10y2,y˙5=−2y5,is globally asymptotically stable; thus, Fy,B can be stabilized.According to Theorem2, the controller U is designed as follows:(35)U=Kty=kt0010000010y=kty3y4=kty3kty4,where K=ktBT and k˙t=−y2. Thus,(36)u=1i∝U=1i∝kty3kty4=ktz2.Numerical simulation is carried out with the initial conditions:y0=−5,3,−2,−6,7,k0=−1. Figure 1 shows y1,y2,andy3 are asymptotically stable, and Figure 2 shows y4andy5 are asymptotically stable, which implies the zt and xt are stabilized. Figure 3 shows the feedback gain kt approaches to constant.Figure 1
y1,y2,andy3 are asymptotically stable.Figure 2
y4andy5 are asymptotically stable.Figure 3
kt approaches to constant.According to Theorem3, the controller U is designed as follows:(37)U=KXY=00X−1100000X−110Y=X−110Y1X−110Y2,where(38)Y=Y1Y2Y3Y4=y1y2y3y4,X=y5.Thus,(39)u=1i∝U=1i∝x3−110y1x3−110y2=x3−110z1.Numerical simulation is carried out with the initial conditions:y0=−5,3,−2,−6,7. Figure 4 shows y1,y2,andy3 are asymptotically stable, and Figure 5 shows y4andy5 are asymptotically stable, which implies the zt and xt are stabilized.Figure 4
y1,y2,andy3 are asymptotically stable.Figure 5
y4andy5 are asymptotically stable.Example 2.
The controlled complex hyperchaotic Lorenz system [26], which is presented as follows:(40)p˙=fp+bu,where(41)p=zx,z=z1z2,x=x3x4,that is, m=2, n=4, and(42)fp=fx,z,z¯=Mxz+HixNx,z,z¯,b=b100b2=b1=0100,Mx=−141445−x3−1,Hi=01+i00,Nx,z,z¯=−5x3+12z¯1z2+z1z¯2−5.5x4+12z¯1z2+z1z¯2.According to Theorem1, the equivalent real system is obtained as follows:(43)y˙=Fy+BU,where(44)y=yzyx,yz=y1y2y3y4,yx=y5y6,Fy=F1yF2y⋮F6y,H∗=01010000,F1yF2yF3yF4y=Mx⊗I2×yz+H∗×yx=−1401400−1401445−y50−10045−y50−1y1y2y3y4+01010000y5y6=−14y1−y3+y5−14y2−y4+y645−y5y1−y345−y5y2−y4B=B1=b1⊗I2=000010010000,U=Uz=U1U2,i.e.,(45)y˙1=−14y1−y3+y5,y˙2=−14y2−y4+y6,y˙3=45−y5y1−y3+U1,y˙4=45−y5y2−y4+U2,y˙5=−5y5+y1y3+y2y4,y˙6=−5.5y6+y1y3+y2y4.Notice that ify3=y4=01/2, then the following system(46)y˙1=−14y1,y˙2=−14y2,y˙5=−5y5,y˙6=−5.5y6,is globally asymptotically stable; thus, Fy,B can be stabilized.According to Theorem2, the controller U is designed as follows:(47)U=Kty=kt0010000010y=kty3y4=kty3kty4,where K=ktBT and k˙t=−y2.Therefore,(48)u=1i∝U=1i∝kty3kty4=ktz2.Numerical simulation is performed with the initial conditions:y0=−5,3,−2,−6,7,−8,k0=−1. Figure 6 shows y1,y2,andy3 are asymptotically stable, and Figure 7 shows y4,y5,andy6 are asymptotically stable, which means that the zt and xt are stabilized. Figure 8 shows the feedback gain kt tends to constant.Figure 6
y1,y2,andy3 are asymptotically stable.Figure 7
y4,y5,andy6 are asymptotically stable.Figure 8
kt 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.



---

*Source: 4938149-2020-05-30.xml*