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Finite time synchronization of chaotic systems Shihua Li * , Yu-Ping Tian Department of Automatic Control, Southeast University, Nanjing 210096, PR China Accepted 29 April 2002 Abstract Using finite time control techniques, continuous state feedback control laws are developed to solve the synchro- nization problem of two chaotic systems. We demonstrate that these two chaotic systems can be synchronized in finite time. Examples of Duffing systems, Lorenz systems are presented to verify our method. Ó 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction Chaos synchronization has been of broad interest in recent years [1–7]. In [1–5], active control methods were used to solve the synchronization problem of chaotic systems such as Rossler systems, Lorenz systems, Duffing systems, Chen systems. The main idea of synchronization is to use the states of the master system to control the slave system so that the states of the slave system follows the states of the master system asymptotically. The master system and the slave system may have identical or completely different structures. The convergence of the synchronization procedure in [1–5] is exponential with infinite settling time. To achieve faster convergence in control systems, an effective method is using finite time control techniques. Finite time stability means the optimality in settling time [8]. Moreover, the finite time control techniques have demonstrated better robustness and disturbance rejection properties [9]. The problem we want to discuss here is that can these chaotic systems mentioned above be synchronized in finite settling time? The answer is certain, which we will demonstrate in the following sections. So, our goal in this paper is to develop feedback control laws to make the synchronization procedure converging in finite time, i.e., the states of the slave system follow the states of the master system in finite time. Discontinuous or open- loop control techniques from finite time control theory may be available to solve this problem. However, considering the convenience of continuous controllers in practical implementation and the robustness of closed-loop feedbacks to system uncertainties, the finite time control technique we discussed here is based on continuous state feedbacks [8,9]. 2. Finite time synchronization of second order chaotic systems Some results of finite time control techniques using continuous feedbacks are given in [8,9], and are rephrased by the following two lemmas. Lemma 1 ([8,9]). The system dx dt ¼ u ð1Þ * Tel.: +86-25-3794168; fax: +86-25-7712719. E-mail address: [email protected] (S. Li). 0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII:S0960-0779(02)00100-5 Chaos, Solitons and Fractals 15 (2003) 303–310 www.elsevier.com/locate/chaos

Finite time synchronization of chaotic systems

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Page 1: Finite time synchronization of chaotic systems

Finite time synchronization of chaotic systems

Shihua Li *, Yu-Ping Tian

Department of Automatic Control, Southeast University, Nanjing 210096, PR China

Accepted 29 April 2002

Abstract

Using finite time control techniques, continuous state feedback control laws are developed to solve the synchro-

nization problem of two chaotic systems. We demonstrate that these two chaotic systems can be synchronized in finite

time. Examples of Duffing systems, Lorenz systems are presented to verify our method.

� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Chaos synchronization has been of broad interest in recent years [1–7]. In [1–5], active control methods were used to

solve the synchronization problem of chaotic systems such as Rossler systems, Lorenz systems, Duffing systems, Chen

systems. The main idea of synchronization is to use the states of the master system to control the slave system so that

the states of the slave system follows the states of the master system asymptotically. The master system and the slave

system may have identical or completely different structures. The convergence of the synchronization procedure in [1–5]

is exponential with infinite settling time.

To achieve faster convergence in control systems, an effective method is using finite time control techniques. Finite

time stability means the optimality in settling time [8]. Moreover, the finite time control techniques have demonstrated

better robustness and disturbance rejection properties [9]. The problem we want to discuss here is that can these chaotic

systems mentioned above be synchronized in finite settling time? The answer is certain, which we will demonstrate in the

following sections.

So, our goal in this paper is to develop feedback control laws to make the synchronization procedure converging in

finite time, i.e., the states of the slave system follow the states of the master system in finite time. Discontinuous or open-

loop control techniques from finite time control theory may be available to solve this problem. However, considering

the convenience of continuous controllers in practical implementation and the robustness of closed-loop feedbacks to

system uncertainties, the finite time control technique we discussed here is based on continuous state feedbacks [8,9].

2. Finite time synchronization of second order chaotic systems

Some results of finite time control techniques using continuous feedbacks are given in [8,9], and are rephrased by the

following two lemmas.

Lemma 1 ([8,9]). The system

dxdt

¼ u ð1Þ

* Tel.: +86-25-3794168; fax: +86-25-7712719.

E-mail address: [email protected] (S. Li).

0960-0779/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0960-0779 (02 )00100-5

Chaos, Solitons and Fractals 15 (2003) 303–310www.elsevier.com/locate/chaos

Page 2: Finite time synchronization of chaotic systems

can be globally stabilized in finite time under the feedback control law

u ¼ �k � signðxÞjxja ð2Þ

with k > 0, a 2 ð0; 1Þ.

The solution trajectory of (1) and (2) is

xðtÞ1�a ¼ xð0Þ1�a � ð1� aÞkt; xð0ÞP 0

½�xðtÞ1�a ¼ ½�xð0Þ1�a � ð1� aÞkt; xð0Þ < 0

�ð3Þ

For any initial value of state xðtÞ at t ¼ 0, i.e., xð0Þ, it is easily computed that the solution trajectory of (1) and (2) will

reach x ¼ 0 in finite time ts determined by ts ¼ jxð0Þj1�a=ð1� aÞk.

Lemma 2 ([9]). The system

dxðtÞdt

¼ vðtÞ;

dvðtÞdt

¼ uðtÞð4Þ

can be globally stabilized in finite time under the feedback control law

u ¼ �k1 signðxÞjxja1 � k2 signðvÞjvja2 ð5Þ

with k1, k2 > 0, a1 2 ð0; 1Þ, a2 ¼ 2a1=ð1þ a1Þ.

Now, let us first consider the synchronization problem of second order chaotic systems which can be written in the

following form [3]

dx1ðtÞdt

¼ v1ðtÞ;

dv1ðtÞdt

¼ f1 x1ðtÞ; v1ðtÞ; tð Þð6Þ

where f1ðx1ðtÞ; v1ðtÞ; tÞ is a nonlinear function. This system is called the master system. The equations that describe the

slave system are

dx2ðtÞdt

¼ v2ðtÞ þ u1ðtÞ;

dv2ðtÞdt

¼ f2 x2ðtÞ; v2ðtÞ; tð Þ þ u2ðtÞð7Þ

where f2ðx2ðtÞ; v2ðtÞ; tÞ is a nonlinear function and u1ðtÞ, u2ðtÞ are the control signals to be designed.

The difference between the two distances and the two velocities is described by

x3 ¼ x2 � x1;

v3 ¼ v2 � v1:ð8Þ

Thus, we get

dx3ðtÞdt

¼ v3ðtÞ þ u1ðtÞ;

dv3ðtÞdt

¼ f2 x2ðtÞ; v2ðtÞ; tð Þ � f1 x1ðtÞ; v1ðtÞ; tð Þ þ u2ðtÞð9Þ

Several possible choices for the control signals u1ðtÞ, u2ðtÞ can be taken to synchronize the slave system to the master

system.

304 S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310

Page 3: Finite time synchronization of chaotic systems

2.1. Control strategy 1

One possible choice is given by

u1ðtÞ ¼ �v3ðtÞ � k1 signðx3Þjx3ja;

u2ðtÞ ¼ �f2 x2ðtÞ; v2ðtÞ; tð Þ þ f1 x1ðtÞ; v1ðtÞ; tð Þ

� k2 signðv3Þjv3jb; k1; k2 > 0; a; b;2 ð0; 1Þ:

ð10Þ

Both signals u1ðtÞ, u2ðtÞ are available in the synchronization procedure. With (10), (9) can be rewritten in the following

notation:

dx3ðtÞdt

¼ �k1 signðx3Þjx3ja;

dv3ðtÞdt

¼ �k2 signðv3Þjv3jb; k1; k2 > 0; a; b 2 ð0; 1Þ:ð11Þ

Due to Lemma 1, (11) implies that the two chaotic systems are synchronized with continuous state feedbacks in finite

time.

2.2. Control strategy 2

Another possible choice is given by

u1ðtÞ ¼ 0;

u2ðtÞ ¼ �f2 x2ðtÞ; v2ðtÞ; tð Þ þ f1ðx1ðtÞ; v1ðtÞ; tÞ � k1 signðx3Þjx3ja1

� k2 signðv3Þjv3ja2 ; k1; k2 > 0; a1 2 ð0; 1Þ; a2 ¼2a1

1þ a1

:

ð12Þ

Here, only control signal u2ðtÞ is made available while u1ðtÞ � 0.

With (12), the system (9) can be rewritten in the following notation:

dx3ðtÞdt

¼ v3;

dv3ðtÞdt

¼ �k1 signðx3Þjx3ja1 � k2 signðv3Þjv3ja2 ; k1; k2 > 0; a1 2 ð0; 1Þ; a2 ¼2a1

1þ a1

:

ð13Þ

According to Lemma 2, the differences will converge to zero in finite time.

Example 1 (Duffing systems). As an example, let us consider the synchronization of two Duffing systems [3]. The

master Duffing system is given by

dx1ðtÞdt

¼ v1;

dv1ðtÞdt

¼ 1:8x1ðtÞ � 0:1v1ðtÞ � ½x1ðtÞ3 þ 1:1 cosðtÞ:ð14Þ

The slave Duffing system is given as follows:

dx2ðtÞdt

¼ v2 þ u1ðtÞ;

dv2ðtÞdt

¼ 1:8x2ðtÞ � 0:1v2ðtÞ � ½x2ðtÞ3 þ 1:1 cosð0:4tÞ þ u2ðtÞ:ð15Þ

S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310 305

Page 4: Finite time synchronization of chaotic systems

Following the same procedure as outlined above, we write

x3 ¼ x2 � x1;

v3 ¼ v2 � v1:ð16Þ

The differences equations of x3, v3 satisfy

dx3ðtÞdt

¼ v3ðtÞ þ u1ðtÞ;

dv3ðtÞdt

¼ 1:8x3ðtÞ � 0:1v3ðtÞ þ ½x1ðtÞ3 � ½x2ðtÞ3 þ 1:1½cosð0:4tÞ � cosðtÞ þ u2ðtÞ:ð17Þ

According to Eq. (10), the control signals u1ðtÞ, u2ðtÞ are given by

u1ðtÞ ¼ �v3ðtÞ � k1 signðx3Þjx3ja;u2ðtÞ ¼ �1:8x3ðtÞ þ 0:1v3ðtÞ � ½x1ðtÞ3 þ ½x2ðtÞ3 � 1:1½cosð0:4tÞ � cosðtÞ

� k2 signðv3Þjv3jb; k1; k2 > 0; a; b 2 ð0; 1Þ:ð18Þ

We choose values for the constants k1 ¼ k2 ¼ 4, a ¼ b ¼ 13. After a time t ¼ 100, the slave Duffing system (15) is to be

controlled with the master Duffing system (14). The simulation results are shown in Fig. 1(a) and (b).

According to the second control strategy, the controllers can be described by

u1ðtÞ ¼ 0;

u2ðtÞ ¼ �1:8x3ðtÞ þ 0:1v3ðtÞ � ½x1ðtÞ3 þ ½x2ðtÞ3 � 1:1½cosð0:4tÞ � cosðtÞ

� k1 signðx3Þjx3ja1 � k2 signðv3Þjv3ja2 ; k1; k2 > 0; a1 2 ð0; 1Þ; a2 ¼2a1

1þ a1

:

ð19Þ

Fig. 1. Synchronization of a slave system to a master system after t ¼ 100.

306 S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310

Page 5: Finite time synchronization of chaotic systems

In numerical simulations, we select values for the constants k1 ¼ k2 ¼ 4, a1 ¼ 13. Then we get a2 ¼ 2a1=ð1þ a1Þ ¼ 1

2. The

controller was activated at t ¼ 100 (i.e., u2 ¼ 0 for all t < 100). Note that the slave Duffing system (15) is to be con-

trolled with the master Duffing system (14). The simulated response is shown in Fig. 2(a) and (b).

3. Finite time synchronization of Lorenz systems

Now, let us consider the synchronization of two Lorenz systems [2]. The master Lorenz system is given by

dx1ðtÞdt

¼ rðy1 � x1Þ;

dy1ðtÞdt

¼ rx1 � y1 � x1z1;

dz1ðtÞdt

¼ x1y1 � bz1:

ð20Þ

The slave Lorenz system is given as follows:

dx2ðtÞdt

¼ rðy2 � x2Þ þ u1;

dy2ðtÞdt

¼ rx2 � y2 � x2z2 þ u2;

dz2ðtÞdt

¼ x2y2 � bz2 þ u3:

ð21Þ

Letting the error state be

x3 ¼ x2 � x1;

y3 ¼ y2 � y1;

z3 ¼ z2 � z1;

ð22Þ

Fig. 2. Synchronization of a slave system to a master system after t ¼ 100.

S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310 307

Page 6: Finite time synchronization of chaotic systems

then the error state dynamic equations satisfy

dx3ðtÞdt

¼ rðy3 � x3Þ þ u1ðtÞ;

dy3ðtÞdt

¼ rx3 � y3 � x2z2 þ x1z1 þ u2ðtÞ;

dz3ðtÞdt

¼ x2y2 � x1y1 � bz3 þ u3ðtÞ:

ð23Þ

Similarly, one possible choice of control signals u1ðtÞ, u2ðtÞ, u3ðtÞ is

u1ðtÞ ¼ �rðy3 � x3Þ � k1 signðx3Þjx3ja;

u2ðtÞ ¼ �rx3 þ y3 þ x2z2 � x1z1 � k2 signðy3Þjy3jb;

u3ðtÞ ¼ �x2y2 þ x1y1 þ bz3 � k3 signðz3Þjz3jc; k1; k2; k3 > 0; a; b; c 2 ð0; 1Þ:

ð24Þ

which implies that the error state described by the following equations

dx3ðtÞdt

¼ �k1 signðx3Þjx3ja;

dy3ðtÞdt

¼ �k2 signðy3Þjy3jb;

dz3ðtÞdt

¼ �k3 signðz3Þjz3jc:

ð25Þ

Thus, the three error states will converge to zero in finite time.

For the two Lorenz systems, the parameters are r ¼ 10, b ¼ 83, r ¼ 28. We choose values for the constants

k1 ¼ k2 ¼ 20, k3 ¼ 4, a ¼ b ¼ c ¼ 12. After a time t ¼ 100, the slave Lorenz system (21) is to be controlled with the

master Duffing system (20). The simulation results are shown in Fig. 3(a)–(c).

Another choice is only taking u2ðtÞ, u3ðtÞ as available control signals while u1ðtÞ � 0. The control signals are designed

as follows:

u1ðtÞ ¼ 0;

u2ðtÞ ¼ �ðr þ 1Þx3 þ 2y3 þ x2z2 � x1z1 �k1rsignðx3Þjx3ja1

� k2rsignðy3 � x3Þ½rjy3 � x3ja2 ;

u3ðtÞ ¼ �x2y2 þ x1y1 þ bz3 � k3 signðz3Þjz3ja3 ; k1; k2; k3 > 0; a1; a3 2 ð0; 1Þ; a2 ¼2a1

1þ a1

:

ð26Þ

which implies that

d2x3ðtÞdt2

¼ �k1 signðx3Þjx3ja1 � k2 signdx3ðtÞdt

� �dx3ðtÞdt

��������a2

;

dz3ðtÞdt

¼ �k3 signðz3Þjz3ja3 k1; k2; k3 > 0; a1; a3 2 ð0; 1Þ; a2 ¼2a1

1þ a1

:

ð27Þ

Hence, the closed-loop system (25) is stabilized in finite time.

For the two Lorenz systems, we choose values for the constants k1 ¼ 40, k2 ¼ 20, k3 ¼ 4, a1 ¼ a3 ¼ 12. Thus we get

a2 ¼ 2a1=ð1þ a1Þ ¼ 23. After a time t ¼ 100, the slave Lorenz system (21) is to be controlled with the master Duffing

system (20). The simulation result is shown in Fig. 4(a)–(c).

308 S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310

Page 7: Finite time synchronization of chaotic systems

Fig. 4. Synchronization of a slave system to a master system after t ¼ 100.

Fig. 3. Synchronization of a slave system to a master system after t ¼ 100.

S. Li, Y.-P. Tian / Chaos, Solitons and Fractals 15 (2003) 303–310 309

Page 8: Finite time synchronization of chaotic systems

4. Conclusion

In this paper, the synchronization problem of chaotic systems is studied. Based on finite time control techniques,

continuous state feedback control laws are designed for the synchronization of chaotic systems such as Duffing systems,

Lorenz systems. The convergence of the synchronization procedure is in finite time.

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