global synchronization of chaotic systems via linear balanced feedback control

Applied Mathematics and Computation 186 (2007) 923–931

www.elsevier.com/locate/amc

Global synchronization of chaotic systems via linearbalanced feedback control

Heng-Hui Chen

Department of Mechanical Engineering, HsiuPing Institute of Technology, No. 11,

Gungye Road, Dali City, Taichung, Taiwan 412, ROC

Abstract

Global synchronization of two identical chaotic systems by the linear balanced feedback control, and the Gerschgorintheorem is studied in this paper. Linear feedback control method based on Lyapunov stability theory and extremeapproach is introduced to design controller to synchronize two identical chaotic systems. Two techniques are applied tofour-scroll chaotic systems for determining the balanced feedback gains. A numerical example is given to illuminate thedesign procedure and advantage of the results derived.� 2006 Elsevier Inc. All rights reserved.

Keywords: Chaos synchronization; Balanced feedback control

1. Introduction

Chaos synchronizations have been widely explored in a variety of fields including secure communications,optics, chemical and biological systems, etc. During the last decades, many methods have been successfullyapplied to chaos synchronization such as PC method [1], linear feedback control [2], adaptive control [3,4],backstepping design [5], active control [6], and nonlinear control [7,8], etc. For practical applications, a linearfeedback controller is more desirable due to its simple implementation in real systems. Recently, a simple gen-eric criterion is derived for global chaos synchronization by the linear feedback control based on the Lyapu-nov stabilization theory and Gerschgorin theorem [9].

The aim of this paper is to study the global chaos synchronization of the four-scroll new chaotic systems[10] by linear balanced feedback control. Linear balanced feedback control can derive the roughly equal feed-back gains based on Lyapunov stability theory and extreme approach. The advantage of our method is thatthe balanced feedback gains of the system can be obtained analytically. The proposed method will be appliedto make two identical new chaotic systems globally asymptotically synchronized. Finally, numerical simula-tions are presented to demonstrate the effectiveness of the chaos synchronization.

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.08.017

E-mail address: [email protected]

mailto:[email protected]

924 H.-H. Chen / Applied Mathematics and Computation 186 (2007) 923–931

2. Design of controller via linear balanced feedback control

Consider a chaotic system in the form of:

_x ¼ Axþ gðxÞ; ð1Þ
where x 2 Rn is the state vector, A 2 Rn·n is a constant matrix, and g(x) is a continuous nonlinear function.
From the linear feedback approach, the controlled response system is given by

_y ¼ Ayþ gðyÞ þ Kðx� yÞ; ð2Þ
where y 2 Rn denotes the state vector of the response system, and K = diag{k1,k2, . . .,kn} 2 Rn·n is a feedbackmatrix to be designed later. Assuming that the following condition is held:
gðxÞ � gðyÞ ¼Mx;yðx� yÞ; ð3Þ

where the elements in Mx,y are bounded and dependent on x and y. In fact, most of chaotic systems can bedescribed by (1) and (3), which will be further illustrated by an example in Section 3.

From (1)–(3), the error dynamics can be obtained:

_e ¼ ðA� KþMx;yÞe; ð4Þ
where e = x � y is the error state vector.
Construct a Lyapunov function:

V ðeÞ ¼ eTPe; ð5Þ
where P is a positive definite diagonal constant matrix.
The derivative of the Lyapunov function along the trajectory of system (4):

_V ¼ _eTPeþ eTP _e ¼ eT½ðA� KþMx;yÞTPþ PðA� KþMx;yÞ�e ¼ �eTQe; ð6Þ
where Q = � [(A � K + Mx,y)TP + P(A � K + Mx,y)].
First, the synchronization problem is to design a linear feedback controller that synchronizes the states ofboth the drive and response systems. One may achieve chaos synchronization by selecting a set of linear feed-back gains K = diag{k1,k2, . . .,kn} 2 Rn·n to make the matrix Q a positive definite function. Then the states ofthe response system and drive system are globally asymptotically synchronized.

Second, the problem of minimization of a function, i.e., a method minimizing the sum of the feedbackgains, is considered. At the same time, the linear balanced feedback gains are obtained.

The procedure for designing control gains is as follows:The first step is to solve the linear feedback control gains from the positive definite matrix Q. Assume all the

principal minor determinants corresponding to the symmetric matrix Q as following:

Di ¼ jQqrj ¼ mi > 0; q; r ¼ 1; 2; . . . ; i; i ¼ 1; 2; . . . ; n: ð7Þ

From (7), we obtain:

ki ¼ Siðm1;m2; . . . ;mqÞ; q ¼ 1; 2; . . . ; i; i ¼ 1; 2; . . . ; n ð8Þ
and
oki

omi6¼ 0;

okj

omi¼ 0; j ¼ 1; 2; . . . ; i� 1; i ¼ 1; 2; . . . ; n: ð9Þ

The second step is to minimize the sum of the control gains, i.e., f = Min(k1 + � � � + kn). This means that thecontrol gains are roughly equal, i.e., balanced.

The third step is to study minima of function of specific variables. Then, write down the necessary condi-tions for rendering f a relative maximum or minimum as follows:

ofomi¼ 0; i ¼ 1; 2; . . . ; n: ð10Þ

By solving (10) corresponding to (9), the extreme point ðm�1;m�2; . . . ;m�nÞ is found.

H.-H. Chen / Applied Mathematics and Computation 186 (2007) 923–931 925

3. Chaos synchronization of two identical systems

Synchronization of two identical four-scroll chaotic systems is discussed in this section. Two methods arepresented: linear balanced feedback control, and the Gerschgorin theorem.

3.1. Chaos synchronization of two identical systems by linear balanced feedback control

The drive system is:

_x1 ¼ ax1 þ d1y1z1;

_y1 ¼ by1 þ d2x1z1;

_z1 ¼ cz1 þ d3x1y1;

ð11Þ

and the response system is:

_x2 ¼ ax2 þ d1y2z2 þ k1ðx1 � x2Þ;_y2 ¼ by2 þ d2x2z2 þ k2ðy1 � y2Þ;

_z2 ¼ cz2 þ d3x2y2 þ k3ðz1 � y2Þ:

ð12Þ

From (11) and (12), we can obtain the error dynamics in the form of:

_e ¼ ðA� KþMx;yÞe; ð13Þ

where

A ¼

a 0 0

0 b 0

0 0 c

2664

3775; K ¼

k1 0 0

0 k2 0

0 0 k3

2664

3775; Mx;y ¼

0 d1z2 d1y1

d2z2 0 d2x1

d3y2 d3x1 0

2664

3775; e ¼

x1 � x2

y1 � y2

z1 � z2

2664

3775:

Construct a Lyapunov function:

V ¼ eTPe; ð14Þ

where P = diag{p1,p2,p3} is a positive definite matrix.The derivative of the Lyapunov function along the trajectory of system (13):

_V ¼ eT½ðA� KþMx;yÞTPþ PðA� KþMx;yÞ�e ¼ �eTQe; ð15Þ

where Q = QT and

Q ¼

q11 �q12 �q13

�q21 q22 �q23

�q31 �q32 q33

2664

3775 ¼

2p1ðk1 � aÞ �ðp1d1 þ p2d2Þz2 �ðp1d1y1 þ p3d3y2Þ

�ðp1d1 þ p2d2Þz2 2p2ðk2 � bÞ �ðp2d2 þ p3d3Þx1

�ðp1d1y1 þ p3d3y2Þ �ðp2d2 þ p3d3Þx1 2p3ðk3 � cÞ

2664

3775:

Theorem 1. For a given set of system parameters, the two coupled unified chaotic systems (11) and (12)

are globally synchronized, if a suitable linear feedback control K is chosen such that the following conditions

hold:

ðiÞ k1 ¼ aþ m1=ð2p1Þ;ðiiÞ k2 ¼ bþ ðm2 þ U 2

3Þ=ð2m1p2Þ;ðiiiÞ k3 ¼ cþ ½m1m3 þ ðm1U 1 þ U 3U 2Þ2 þ m2U 2

2�=ð2m1m2p3Þ:ð16Þ


Proof. Define U3, U2 and U1 are the upper bounds of the absolute values of variables q12, q13 and q23,respectively. From (15), we can obtain:

_V ¼ �eTQe 6 �ETRE; ð17Þ

where E = [je1j je2j je3j]T and

R ¼2ðk1 � aÞp1 �U 3 �U 2

�U 3 2ðk2 � bÞp2 �U 1

�U 2 �U 1 2ðk3 � cÞp3

264

375:

By Sylvester’s theorem, all principal minors of R are strictly positive, i.e.,

D1 ¼ q11 ¼ m1 > 0;

D2 ¼ q11q22 � U 23 ¼ m2 > 0;

D3 ¼ ðq11q22 � U 23Þq33 � q11U 2

1 � q22U 22 � 2U 3U 2U 1 ¼ m3 > 0;

k1 ¼ aþ m1=ð2p1Þ > maxfa; 0g;k2 ¼ bþ ðm2 þ U 2

3Þ=ð2m1p2Þ > maxfb; 0g;k3 ¼ cþ ½m1m3 þ ðm1U 1 þ U 3U 2Þ2 þ m2U 2

2�=ð2m1m2p3Þ > maxfc; 0g:

ð18Þ

where U1 = jp2d2 + p3d3jUx, U2 = (jp1d1j + jp3d3j)Uy, U3 = jp1d1 + p2d2jUz; Ux, Uy and Uz are the upperbounds of the absolute values of variables x1,2, y1,2 and z1,2, respectively. h

–500

50

–50

0

50–50

0

50

xy

z

–40 –20 0 20 40–30

–20

–10

0

10

20

30

x

y

–40 –20 0 20 40–30

–20

–10

0

10

20

30

y

z

–40 –20 0 20 40–30

–20

–10

0

10

20

30

x

z

Fig. 1. The four-scroll chaotic attractor of the new system at a = 0.5, b = �10 and c = �4.


Remark 1. Define X is the maximum upper bounds of the absolute values of variables x1,2, y1,2 and z1,2, thenU1 = jp2d2 + p3d3jX, U2 = (jp1d1j + jp3d3j)X, U3 = jp1d1 + p2d2jX, and X can be specified by solving (11) and(12).

3.1.1. Minimization of functions

This section is in studying minima of the function: the sum of feedback gains is minimized, i.e.,f = Min{k1 + k2 + k3}. From (9) ok3

om36¼ 0,

okj

om3¼ 0, j = 1, 2, we know that the necessary condition of

om3¼ 0 fails

to exist. Thus, it may happen that an extreme value is taken on at a boundary point, i.e., m3 = 2ep3m2, e! 0+.A set of parameters of the four-scroll chaotic system is defined by: d1 = �1, d2 = d3 = 1, a > 0, b < 0, c < 0,

and assume thatp1 = p0, p2 = p3 = p0/2. Thus, we obtain U1 = p0X, U2 = 1.5p0X, U3 = 0.5p0X.Assume that the minima function f = Min{k1 + k2 + k3} for obtaining a set of linear balanced feedback

gains. If fm1¼ 0 and f m2

¼ 0, and fm1m1> 0; f m1m1

fm2m2> f 2

m1m2at a point pðm�1;m�2Þ, then at that point f has

a relative minimum. With

Fig. 2.X = 30

f ðk1; k2; k3Þ ¼ k1 þ k2 þ k3 ¼ f ðm1;m2; 0þÞ; ð19Þ

the necessary conditions fm1¼ fm2

¼ 0 at ðm�1;m�2Þ become

fm1¼ 1=ð2p0Þ � 5p0X 2=ð2m2

1Þ � m2=ðp0m21Þ þ p0X 2=m2 � 9p3

0X 4=ð16m21m2Þ ¼ 0;

fm2¼ 1=ðp0m1Þ � p0m1X 2=m2

2 � 3p20X 3=ð2m2

2Þ � 9p30X 4=ð16m1m2

2Þ ¼ 0;ð20Þ

0 0.1 0.2 0.3 0.4 0.5 0.6–1

0

1

2

e 1

0 0.1 0.2 0.3 0.4 0.5 0.6–1

0

1

2

e 2

0 0.1 0.2 0.3 0.4 0.5 0.6–1

0

1

2

t (sec)

e 3

Synchronization errors of two identical four-scroll chaotic systems with feedback gains (42.93, 30.61, 57.82) from Eq. (23) at.


from which there follows:

Fig. 3

m�1 ¼ 2ffiffiffi2p

p0X ; m�2 ¼ 2ffiffiffi2pþ 3=4

� �p2

0X 2 ð21Þ

and

fm1m1� 0:55=ðp0X Þ > 0; f m1m1

fm2m2� f 2

m1m2� 0:07=ðp6

0X 4Þ > 0: ð22Þ
Thus, the corresponding minimum sum of control gains is:
Minfk1 þ k2 þ k3g ¼ ðaþ bþ c0Þ þ 4:83X ; i:e:;

k1 ¼ aþffiffiffi2p

X ; k2 ¼ bþ 1:3536X ; k3 ¼ c0 þ 2:0607X ;ð23Þ

where c 0 = c + e, e! 0+.

3.2. Chaos synchronization of two identical systems by Gerschgorin theorem

From (15), based on well-known Gerschgorin theorem in matrix theory, the following results can beobtained [9].

Theorem 2. Choose P = diag{p1, p2, . . .,pn}, and let

ðAþMx;yÞTPþ PðAþMx;yÞ ¼ ½�aij� and Ri ¼Xn

j¼1;j 6¼i

j�aijj: ð24Þ

If a suitable K is chosen such that the following condition is held, then the two identical chaotic systems (11)and (12) are globally synchronized.

0 0.1 0.2 0.3 0.4 0.5 0.6–1

0

1

2

e 1

0 0.1 0.2 0.3 0.4 0.5 0.6–1

0

1

2

e 2

0 0.1 0.2 0.3 0.4 0.5 0.6–1

0

1

2

t (sec)

e 3

. Synchronization errors of two identical four-scroll chaotic systems with feedback gains (30.5, 35, 71) from Eq. (28) at X = 30.


ki P1

2pi

�aii þ Ri � lð Þ; i ¼ 1; 2; . . . ; n; ð25Þ

where l is a negative constant.Consider two identical new chaotic systems (11) and (12), we can obtain:

ðAþMx;yÞTPþ PðAþMx;yÞ ¼2p1a ðp1d1 þ p2d2Þz2 ðp1d1y1 þ p3d3y2Þ

ðp1d1 þ p2d2Þz2 2p2b ðp2d2 þ p3d3Þx1

ðp1d1y1 þ p3d3y2Þ ðp2d2 þ p3d3Þx1 2p3c

264

375: ð26Þ

One may then choose

k1 P ð2p1aþ U 3 þ U 2 � lÞ=ð2p1Þ;k2 P ð2p2bþ U 1 þ U 3 � lÞ=ð2p2Þ;k3 P ð2p3bþ U 2 þ U 1 � lÞ=ð2p3Þ;

ð27Þ

where U1 = jp2d2 + p3d3jUx, U2 = (jp1d1j + jp3d3j)Uy, U3 = jp1d1 + p2d2jUz; Ux, Uy and Uz are the upperbounds of the absolute values of variables x1,2, y1,2 and z1,2, respectively. According to Theorem 2, the twoidentical new chaotic systems (11) and (12) are globally asymptotically synchronized.

Assuming that l = �0.5, d1 = �1, d2 = d3 = 1, p1 = p0, p2 = p3 = p0/2, and X = Max(Ux,Uy,U z), then, weobtain U1 = p0X, U2 = 1.5p0X, U3 = 0.5p0X. Thus

k1 P aþ X � l=ð2p0Þ; k2 P bþ 1:5X � l=p0; k3 P cþ 2:5X � l=p0; ð28Þ

0 5 10 15 20 25 30–10

0

10

20

30

40

50

60

X

k 1,k2,k

3

Fig. 4. Feedback gains k1 (—), k2 (- - -), k3 (-Æ-Æ-) versus X from Eq. (23).


the corresponding sum of control gains is:

k1 þ k2 þ k3 P aþ bþ cþ 5X � 2:5l=p0; ð29Þ
where p0� l.
4. Numerical results

When a = 0.5, b = � 10, c = � 4, system (11) exhibits four-scroll chaotic behavior (see Fig. 1). It can beseen that the solutions x(t), y(t) and z(t) from the initial states x(0) = 1, y(0) = 1 and z(0) = 1 are boundedand satisfy the inequalities: �29.7 < x < 28.5, � 21 < y < 22.1, and �26.4 < z < 22.2. The maximum upperbound X can be chosen to X = 30.

From (23), we can obtain the balanced feedback gains as k1 = 42.93, k2 = 30.61, k3 = 57.82, andk1 + k2 + k3 = 131.36 which can achieve global synchronization between two identical four-scroll chaotic sys-tems. The numerical results are shown in Fig. 2. Based on Gerschgorin’s theorem, the feedback gains ask1 = 30.5, k2 = 35, k3 = 71, and k1 + k2 + k3 = 136.5, the inequality (28) holds. According to Theorem 2,the two identical chaotic systems (11) and (12) are globally synchronized as shown in Fig. 3. In Figs. 4 and5, the feedback gains are increasing function of X, and the sum and variation of feedback gains from (23)are smaller than the other.

5. Conclusion

In this paper, a balanced gain condition is derived for the global synchronization of two general chaoticsystems by linear balanced feedback control. Suitable balanced control gains can be designed according

0 5 10 15 20 25 30–10

0

10

20

30

40

50

60

70

80

X

k 1,k2,k

3

Fig. 5. Feedback gains k1 (—), k2 (- - -), k3 (-Æ-Æ-) versus X from Eq. (28).


to the given condition to ensure the global chaos synchronization. In addition, a simple criterion based onGerschgorin theorem is also used to achieve chaos synchronization between two identical four-scroll chaoticsystems. Finally, Numerical simulations are performed to show the effectiveness and distinction of these twomethods.

Acknowledgment

This research was supported by the National Science Council, Republic of China, under Grant numberNSC 93-2218-E-164-001.

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global synchronization of chaotic systems via linear balanced feedback control

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