A new adaptive variable structure control for chaotic synchronization and secure communication

Chaos, Solitons and Fractals 20 (2004) 967–977 www.elsevier.com/locate/chaos

A new adaptive variable structure control for chaotic synchronization and secure communication Chun-Chieh Wang a

c

a,b

, Juhng-Perng Su

c,*

Graduate School of Engineering Science & Technology (Doctoral Program), National Yunlin University of Science & Technology, No. 123, Section 3, University Road, Douliu, Yunlin 640, Taiwan, ROC b Department of Electrical Engineering C.K.I.T., No. 1, Chieh-Sou N. Rd., Changhua City, Taiwan, ROC Department of Electrical Engineering, National Yunlin University of Science and Technology, No. 123, Section 3, University Road, Yunlin 640, Taiwan, ROC Accepted 13 October 2003

Abstract A novel adaptive complementary variable structure control is proposed in this paper for chaotic synchronization. The bounded parameters of the model approximation error and the external disturbance are all regarded as unknown constants in this paper. Based on Lyapunov’s stability theory and the Babalat’s lemma the proposed controller has been shown to render the synchronous error to zero. The Duffing–Holmes oscillator was used as an illustrative example. Simulation results validated that the proposed scheme in the application of secure communication.
2003 Elsevier Ltd. All rights reserved.

1. Introduction Researchers from different areas, such as mathematicians, physicists, chemist, as well as control engineers have devoted themselves to examine the issue of synchronization over the past decade [1,2]. Chaotic systems, in particular, have been applied to the development of secure communication systems [3–9]. The system which received the most attention among chaotic communication systems perhaps is the Chua oscillator [2]. This system belongs to general class of Lur’e systems [6]. In publications regarding synchronization of chaotic systems, the controller design is often based upon the assumptions that the master chaotic model is precisely known, and the slave system can be easily constructed with those well known parameters. However, in real chaotic synchronization, unknown parameters do exist and the external disturbances are always unavoidable. These unknown parameters and disturbances may cause chaotic perturbations to originally regular behavior, or induce additional chaos in originally chaotic but known behavior. Lots of efforts have been dedicated to synchronization of chaotic systems which contain uncertainties [10–13]. However, they often incorporate the control input to every state equation, i.e. there usually exists more than one control input. Such methods would significantly increase the implementation cost. The authors of this article have successfully dealt with uncertain chaotic systems via a novel VSC scheme [14]. In that paper, we defined a new sliding variable as a complement to the conventional sliding variable to form a useful error transformation by which an nth-order problem can be transformed into an equivalent first-order problem such that an efficient continuous sliding control can be devised to achieve a better performance of the system. In addition, we incorporated only one control input to one of the state equations. By this method, the implementation cost decreased significantly.

*

Corresponding author. Tel.: +886-5-5342061x4246; fax: +886-5-5312065. E-mail address: [email protected] (J.-P. Su).

0960-0779/$ - see front matter
2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.10.026

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In this paper, we will propose an adaptive complementary variable structure control scheme to deal with synchronization of chaotic systems with unknown parameters and external disturbances. Based on Lyapunov synthesis method and Babalat’s lemma, the proposed control law is shown to guarantee that all the closed-loop signals are uniformly bounded and the synchronous error converges to zero. The effectiveness of the newly developed control scheme will be demonstrated through the synchronization of Duffing–Holmes oscillator. A visible improvement of system performance, under this control, can be observed. Finally, the presented controller is applied successfully to a secure communication system. 2. Problem statement In order to observe the synchronization behavior in the chaotic systems which are subjected to unknown parameters, we consider a single-input single-output (SISO) nth-order nonlinear master (drive) system described by the following differential equation ymðnÞ ¼ fm ðt; ym ; y_ m ; . . . ; ymðn1Þ Þ

ð1Þ

zm ¼ ym

ð2Þ

and an SISO nth-order nonlinear slave (response) system described by the following differential equation ysðnÞ ¼ fs ðt; ys ; y_ s ; . . . ; ysðn1Þ Þ þ u þ d

ð3Þ

zs ¼ y s

ð4Þ n

where fm ; fs : R  R ! R, are unknown continuous functions, d is the bounded external disturbance, u 2 R is the control; and zm ; zs 2 R, are the outputs of the master and slave systems, respectively. Set xm ¼ ðxm1 ; xm2 ; . . . ; xmn ÞT ¼ ðym ; y_ m ; . . . ; ymðn1Þ ÞT 2 Rn and xs ¼ ðxs1 ; xs2 ; . . . ; xsn ÞT ¼ ðys ; y_ s ; . . . ; ysðn1Þ ÞT 2 Rn Then (1)–(4) can be expressed as the state and output equations as follows: x_ m ¼ Axm þ Bfm

ð5Þ

zm ¼ Cxm

ð6Þ

x_ s ¼ Axs þ Bðfs þ u þ dÞ

ð7Þ

zs ¼ Cxs

ð8Þ

and

where 2

0 1 0 60 0 1 6. . . .. . . . A¼6 . 6. . . 40 0 0 0 0 0

3 0 07 .. 7 .7 7; 15 0

2 3 0 607 6.7 .7 B¼6 6 . 7; 405 1

C ¼ ½1

0 0

0

and the initial values on the drive system are different from that of the response system. Define the state error: eðtÞ ¼ xs  xm ¼ ½ e1 e2 en T and the output error: ye ¼ zs  zm . Then, the error equation can be obtained as follows: e_ ¼ Ae þ Bðfs  fm þ u þ dÞ ye ¼ Ce

ð9Þ ð10Þ

The object is to design an adaptive complementary variable structure control (ACVSC) law such that output zs of the slave system can asymptotically track the output zm of the master system, i.e. lim ye ¼ 0

t!1

ð11Þ

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3. Adaptive complementary variable structure control In this paper, we devise a scheme of the communication transmission via an ACVSC, shown in Fig. 1, for the synchronization of chaotic circuits for the secure communications. For any k; a > 0, define the following transformations:

 n1   d d n sm ðtÞ ¼ ð12Þ þk þ a ne ðtÞ ¼ D1 e e dt dt

 n1   d d n ð13Þ se ðtÞ ¼ þk  a ne ðtÞ ¼ D2 e e dt dt Rt where ne ðtÞ ¼ ye ðsÞ ds, D1 ¼ ½ akn1 1 , D2 ¼ ½ akn1 1 . The following significant relationship can be easily established: s_ e þ aðse þ sm Þ ¼ s_ m

ð14Þ

Differentiation of sm with respect to t yield s_ m þ asm ¼ fs  fm þ w þ u þ d

ð15Þ

" #    n1  n1  n2  X X X n  1 k ðnkÞ n  1 k ðn1kÞ n  1 k ðn2kÞ n1 2 þ k ne þ 2a þa w¼ k ye k ye k ye k k k k¼1 k¼0 k¼0

ð16Þ

where

Since fm and fs are unknown and ym can be measured in the response system, a proper approximator, f^s , should be employed and the control law would be designed as follows: u ¼ ^u þ vr

ð17Þ

where ^u ¼ f^s þ ymðnÞ  w

ð18Þ 



Km ðsm þ se Þ ð1  k1 Þlm h ih iT f^s ¼ ^h1fs ; ^h2fs ; . . . ; ^hnfs u1f ; u2f ; . . . ; unf ¼ ^hTfs uf

ð20Þ

h ih iT fs ¼ h1fs ; h2fs ; . . . ; hnfs u1f ; u2f ; . . . ; unf ¼ hTfs uf

ð21Þ

vr ¼ 

Km tanh 1  k1

ð19Þ

Thus, we have Km tanh s_ m þ asm ¼ fs  f^s  1  k1



Km ðsm þ se Þ ð1  k1 Þlm

 þd

ð22Þ

External disturbance

d (Master)

zm Controller

Chaotic Transmitter

(Slave)

zs

Chaotic Receiver Controller parameters

I (t ) Information signal

Parameter adjustment

Fig. 1. Block diagram of an adaptive synchronization of chaotic systems.

I r (t ) Recovered signal

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We need the following lemma from [15] for subsequent analysis. Lemma 1. For any lm > 0 and any s 2 R, s tanhðlsm Þ P 0, and the following inequality holds   s 6 clm 0 < jsj  s tanh lm

ð23Þ

where c is a constant that satisfies c ¼ eðcþ1Þ ; i.e., c ¼ 0:2785. Assumption 1. jdj < D, 8t, where D is a given constant or time function. Let ~hf ¼ ^hfs  hfs and choose the following Lyapunov function candidate 1 V ¼ ðs2m þ s2e þ ~hTf ~hf Þ 2 The derivative of V is V_ ¼ sm s_ m þ se s_ e  ~hTf h_ fs

ð24Þ

ð25Þ

By Lemma 1 and Assumption 1, we have

 Km  D  ðse þ sm Þ~ hTf uf 1  k1   Km  D  ðse þ sm Þ~ hTf uf  ~ hTf h_ fs . Then V_ 6  aðse þ sm Þ2 þ clm  jse þ sm j 1k 1 se s_ e þ sm s_ m 6  aðse þ sm Þ2 þ clm  jse þ sm j



ð26Þ

The adaptive law with r-modification [15,16] for parameter hfs is designed as follows: h_ fs ¼ ½ðse þ sm Þuf  rhfs 

ð27Þ

where r > 0. It is worth noting that the r-modification will prevent parameter from drifting. It follows   Km V_ 6  aðse þ sm Þ2 þ clm  jse þ sm j  D þ r~ hTf hfs 1  k1

ð28Þ

Remark 2. The following identity can be easily shown by completing the square: ~hT hfs ¼  1 k~hf k2  1 khfs k2 þ 1 k^hfs k2 f 2 2 2 2 2 2

ð29Þ

where k k2 denotes Euclidean norm. Consequently, V_ 6  aðse þ sm Þ2 þ clm  jse þ sm j



 Km r hfs k22  D þ k^ 2 1  k1

Let b :¼ r2 k^hfs k22 , we have, from Eq. (30),   Km V_ 6  aðse þ sm Þ2 þ clm  jse þ sm j D þb 1  k1

ð30Þ

ð31Þ

For g > 0, if Km verifies Km P ð1  k1 Þðg þ DÞ

ð32Þ

then V_ 6  aðse þ sm Þ2 þ clm  gjse þ sm j þ b. For any 0 < # < 1, set ^g :¼ ð1  #Þg and Um ¼

clm  b #g

the following condition V_ 6  aðse þ sm Þ2  ^gjse þ sm j is satisfied for jse þ sm j P Um .

ð33Þ

ð34Þ

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In analysis, we need the following lemmas to prove limt!1 kek ¼ 0. Lemma 3 (Babalat’s lemma). If f ðtÞ is a uniformly continuous function and limt!1 limt!1 f ðtÞ ¼ 0.

Rt 0

f ðsÞ ds exists, then

Lemma 4. If f ; f_ 2 L1 and f 2 Lp , 8p 2 ½1; 1Þ, then limt!1 f ðtÞ ¼ 0. 2 _ gjse þ sm j 6 0, we have V 2 L1 , which implies that Because V is a positive definite function and V 6  aðs e þ sm Þ  ^ n n e e and se ¼ D2 , we also have that ne ; e 2 L1 . sm , se and h~f 2 L1 . Because sm ¼ D1 e e 2 3 ye  n1  n1 6 . 7 d þk se þ sm ¼ 2 ye ¼ 2k ð35Þ 2 4 .. 5 ¼ Ke dt ðn1Þ y e 3 2 ye 6 . 7 where K ¼ ½ 2kn1 2 , e ¼ 4 .. 5. We can obtain

yeðn1Þ

2

V_ 6  aðse þ sm Þ  ^gjse þ sm j ¼ ajKej2  ^gjKej 6  ajKej2 Integrating both sides of (36), we have Z t V ðtÞ  V ð0Þ 6  a jKej2 ds

ð36Þ

ð37Þ

0

Then Z

t

1 1 ½V ð0Þ  V ðtÞ 6 ½jV ð0Þj þ jV ðtÞj a a   1 Define p ¼ a jV ð0Þj þ supt P 0 jV ðtÞj , then we have Z t jKej2 ds 6 p < 1; t P 0 jKej2 ds 6

ð38Þ

0

ð39Þ

0

R1

kek2 ds < 1. Moreover,     Km Km ðsm þ se Þ  f^s þ ymðnÞ  w þ d tanh e_ ¼ Ae þ Bðfs  fm þ u þ dÞ ¼ Ae þ B fs  fm  ð1  k1 Þlm 1  k1     Km Km ðsm þ se Þ wþd tanh ¼ Ae þ B ~hf uf  ð1  k1 Þlm 1  k1

which implies that e 2 L2 , i.e.,

0

ð40Þ

Since all the variables on the right-hand side of (40) are bounded, we have e_ 2 L1 . According to the Barbalat’s Lemma, if e 2 L2 \ L1 and e_ 2 L1 , then limt!1 kek ¼ 0.

4. Illustrative example In this section, A Duffing–Holmes oscillator is used to demonstrate the design for communication systems. 4.1. Duffing–Holmes oscillator We consider two Duffing–Holmes oscillators with different unknown parameters in which the master system is described as (41) and the slave system as (42) €ym ¼ rm1 ym  rm2 y_ m  rm3 ym3 þ rm4 cos xt

ð41Þ

€ys ¼ rs1 ys  rs2 y_ s  rs3 ys3 þ rs4 cos xt þ u þ d

ð42Þ

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where ym , ys are state variables, x is a constant frequency parameter; rm1 , rm4 , rs1 , rs4 are unknown parameters; rm2 , rm3 , rs2 , rs3 are known parameters; u is the controller and d is the external disturbance. Let xm ¼ ðxm1 ; xm2 ÞT ¼ ðym ; y_ m ÞT 2 R2 and xs ¼ ðxs1 ; xs2 ÞT ¼ ðys ; y_ s ÞT 2 R2 . zm and zs are the outputs of the master and slave systems respectively. The Eqs. (41) and (42) can be rewritten as x_ m ¼ Axm þ Bðrm1 ym  rm2 y_ m  rm3 ym3 þ rm4 cos xtÞ ¼ Axm þ Bfm zm ¼ Cxm and x_ s ¼ Axm þ Bðrs1 ys  rs2 y_ s  rs3 ys3 þ rs4 cos xt þ u þ dÞ ¼ Axs þ Bðfs þ u þ dÞ zs ¼ Cxs Thus the error equations can be obtained as (9) and (10). The bounds for unknown parameters in Eqs. (41) and (42) are taken as follows: 1:2 6 rm1 6  0:8 1:7 6 rm4 6 2:1 We take ðrm2 ; rm3 ; rs2 ; rs3 Þ ¼ ð0:4; 1; 0:4; 1Þ and ð^rs1 ; ^rs4 Þ ¼ ð1; 1:9Þ. According to the design given in Section 3, we take f^s ¼ ½^rs1 ; rs2 ; rs3 ; ^rs4 ½zs ; z_ s ; z3s ; cos xtT ¼ ^hTfs uf , jdðtÞj 6 D, and the initial conditions ðym ð0Þ; ys ð0ÞÞ ¼ ð0; 1Þ. By setting r ¼ 1, k1 ¼ 0:9, lm ¼ 5, g ¼ 1, k ¼ 5, and a ¼ 15, the control law can, therefore, be obtained from (17)–(19), where Km ¼ ð1  k1 Þðg þ DÞ. The above systems were simulated using SIMULINK with a fourth order Runge–Kutta algorithm at a fixed-step integration time of 0.01 s. 4.1.1. Case 1. Chaotic synchronization without disturbances In this case, the information signal IðtÞ and the disturbance dðtÞ are assumed to be zero. Then the trajectories of the output of the master and slave systems are depicted in Fig. 2. Fig. 3 shows that the time-response of the error, ye ðtÞ, where the magnitude of the error seems to fall into the bound: jye ðtÞj 6 5  103 very rapidly (within 1.45 s).

2 zm zs

1.5

zm and zs

1 0.5 0 -0.5 -1 -1.5 -2

0

1

2

3

4

5

6

7

8

Time (sec) Fig. 2. The trajectories of zm ðtÞ and zs ðtÞ when IðtÞ ¼ dðtÞ ¼ 0.

9

10

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1.2

The error between zm and zs

1

0.8

0.6

0.4

0.2

0

-0.2

0

1

2

3

4

5

6

7

8

9

10

Time (sec) Fig. 3. Synchronization error in two Duffing–Holmes oscillator.

-3

1.5

x 10

1

Noise

0.5

0

-0.5

-1

-1.5

0

5

10

15

20

25

30

35

40

45

50

Time (sec) Fig. 4. The external disturbance dðtÞ.

4.1.2. Case 2. Chaotic synchronization with disturbances In the second case, we assume that the information signal IðtÞ is zero and the disturbance dðtÞ is a white Gaussian noise, as shown in Fig. 4. Choose D ¼ 1. Then the tracking response of zm ðtÞ of zs ðtÞ are depicted in Fig. 5. This good performance can be further justified from the observation of the error, ye ðtÞ ¼ zs ðtÞ  zm ðtÞ, shown in Fig. 6, where the magnitude of the error seems to fall into 6 · 103 within 1.6 s.

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The tracking responses of zm and zs

2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0

1

2

3

4

5

6

7

8

9

10

Time (sec) Fig. 5. The responses of zm ðtÞ and zs ðtÞ when IðtÞ ¼ 0 and jdðtÞj 6 0:001.

1.2

The error between zm and zs

1

0.8

0.6

0.4

0.2

0

-0.2

0

1

2

3

4

5

6

7

8

9

10

Time (sec) Fig. 6. Synchronization error in two Duffing–Holmes oscillator.

4.1.3. Case 3. Secure communication without disturbances The information signal is IðtÞ ¼ 0:001 sin 20t and the disturbance dðtÞ is assumed to be zero. Fig. 7 shows that the error, eðtÞ ¼ IðtÞ  Ir ðtÞ with respect to time, where the magnitude of the error seems to fall into the bound: jeðtÞj 6 5  103 (within 1.6 s). The control action uðtÞ is depicted in Fig. 8.

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1.2

The error between I(t) and Ir(t)

1

0.8

0.6

0.4

0.2

0

-0.2

0

1

2

3

4

5

6

7

8

9

10

Time (sec) Fig. 7. The error between IðtÞ and Ir ðtÞðdðtÞ ¼ 0Þ.

80 60

The controller

40 20 0 -20 -40 -60 -80

0

1

2

3

4

5

6

7

8

9

10

Time (sec) Fig. 8. The controller uðtÞ.

4.1.4. Case 4. Secure communication with disturbances The information signal is IðtÞ ¼ 0:001 sin 20t and the disturbance dðtÞ is assumed to be a white Gaussian noise, as shown in Fig. 9. Choose D ¼ 1. The magnitude of the controller is the same as the case 3. Then Fig. 10 shows that the magnitude of the error between the information single and the recovery signal falls into 5 · 103 within 1.6 s.

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x 10

1 0.8 0.6 0.4

Noise

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

5

10

15

20

25 30 Time (sec)

35

40

45

50

7

8

9

10

Fig. 9. The external noise dðtÞ.

1.2

The error between I(t) and Ir(t)

1

0.8

0.6

0.4

0.2

0

-0.2

0

1

2

3

4

5

6

Time (sec) Fig. 10. The error between IðtÞ and Ir ðtÞðjdðtÞj 6 0:0005Þ.

5. Conclusion An adaptive law with r-modification for parameters using complementary variable structure control scheme is proposed in this paper to synchronize a class of chaotic systems. Based on Lyapunov synthesis method and Babalat’s lemma, the proposed control scheme has been shown to result in a closed-loop system that all signals are uniformly

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bounded and the synchronous error converges to zero. To illustrate the effectiveness of the design, the synchronization of Duffing–Holmes oscillator was used as an illustrative example. Both theoretical and simulative results reveal that the proposed r-adaptive complementary variable structure control is promising for synchronizing chaotic dynamics with unknown parameters and external disturbances.

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