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Adaptive synchronization of chaotic systems and its

application to secure communications

Teh-Lu Liao *, Shin-Hwa Tsai

Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan, ROC

Accepted 15 March 1999

Abstract

This paper addresses the adaptive synchronization problem of the drivedriven type chaotic systems via a scalar transmitted signal.

Given certain structural conditions of chaotic systems, an adaptive observer-based driven system is constructed to synchronize the drive

system whose dynamics are subjected to the systems disturbances and/or some unknown parameters. By appropriately selecting the

observer gains, the synchronization and stability of the overall systems can be guaranteed by the Lyapunov approach. Two well-known

chaotic systems: Rossler-like and Chua9s circuit are considered as illustrative examples to demonstrate the eectiveness of the proposed

scheme. Moreover, as an application, the proposed scheme is then applied to a secure communication system whose process consists of

two phases: the adaptation phase in which the chaotic transmitters disturbances are estimated; and the communication phase in which

the information signal is transmitted and then recovered on the basis of the estimated parameters. Simulation results verify the

proposed schemes success in the communication application. 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Synchronization in chaotic systems has received increasing attention [16] with several studies on the

basis of theoretical analysis and even realization in laboratory having demonstrated the pivotal role of this

phenomenon in secure communications [710]. The preliminary type of chaos synchronization consists of

drivedriven systems: a drive system and a custom designed driven system. The drive system drives the

driven system via the transmitted signals. More recently, synchronization of hyperchaotic systems was

investigated [1113] and the generalized synchronization was proposed [14,15], which makes communica-

tion more practicable and improves the degree of security. However, to our best knowledge, most of the

discussion on drivedriven type synchronization and its applications are under the following hypotheses:

(1) All parameters of the drive system are precisely known, and the driven system can be constructed with

those known parameters. (2) The dynamics of drive system are disturbance-free [16]. However, the system s

disturbances are always unavoidable and some systems parameters cannot be exactly known in priori. The

eects of these uncertainties will destroy the synchronization and even break it. Therefore, adaptive syn-

chronization of the drivedriven systems in the presence of systems disturbances and unknown parameters

is essential [6,17,18].

Recently, several investigations have linked observer-based concepts to chaos synchronization, which

construct all of the state information from only the transmitted signal [1923]. A systematic method

employing a nonlinear state observer is proposed to resolve the chaotic synchronization of a class of

www.elsevier.nl/locate/chaos

Chaos, Solitons and Fractals 11 (2000) 13871396

*Corresponding author. Fax: +886-6276-6549.

E-mail address: [email protected] (T.-L. Liao).

0960-0779/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 0 5 1 - X

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hyperchaotic systems via a scalar transmitted signal [22]. Although chaotic synchronization can be ensured

due to the eect of transmitting the nonlinear terms, the implementation of nonlinear functions is more

complicated in practice. Moreover, all parameters of chaotic system must be known in advance.

As synchronization-based communication schemes, chaotic signal masking and chaotic modulation [7,8]

have been successfully developed for analog communication systems. The idea of chaotic masking is that

the information signal is masked by directly adding a chaotic signal at the transmitter. Later the infor-

mation-bearing signal is received at the receiving end of the communication and recovered after some signal

processing operations [9,10]. The idea behind chaotic modulation is that the information signal is injectedinto a chaotic system or is modulated by means of an invertible transformation so that spread spectrum

transmission is achieved [7,8].

In the light of the above developments, the purpose of this work is to derive an adaptive observer-

based driven system via a scalar transmitted signal which can attain not only chaos synchronization

but also can be applied to secure communications of chaotic systems in the presence of systems

disturbances and unknown parameters. The class of chaotic systems considered in this work and the

problem formulation are presented in Section 2. Section 3 develops an adaptive observer-based driven

system to synchronize the drive system with systems disturbances and unknown parameters. By ap-

propriately selecting the observer gains, the synchronization and stability of the overall systems are

guaranteed by the Lyapunov stability theory with certain structural conditions. In Section 4, Rossler-

like and Chua9s systems are given as illustrative examples to demonstrate the eectiveness of the

proposed approach. In Section 5, the proposed scheme incorporated with the demodulation scheme

reported in [7] is then applied to a secure communication system and its numerical simulations are also

given to verify the proposed schemes success in communication application. Section 6 summarizes the

concluding remarks.

2. Problem formulation

In general, dynamics of chaotic systems are described by a set of nonlinear dierential equations with

respect to state variables. Moreover, in many cases, the dynamical equations can be decomposed into parts:

a linear dynamics with respect to state variables; and a nonlinear feedback part with respect to the systemoutput. Therefore, we will be particularly concerned with nonlinear continuous-time systems of the fol-

lowing form:

x Ax fy B d

hTgyY y CTxY 1

where y P R denotes the system output, x P Rn represents the state vector, d P R is a bounded disturbance,and AYB and C denote known matrices with appropriate dimensions,

Tdenotes the vector transpose. We

assume that h P Rp is a constant parameter vector which may be unknown, and that f P Rn and gP Rp arereal analytic vectors with f0 0 and g0 0, respectively. Furthermore, we assume that system (1) has aunique solution xt passing through the initial state x0 x0 and this solution is well dened over aninterval 0 I .

Many chaotic systems have a special structure on the matrices AY B and C; hence we make the following

assumption:

A1: The pair AYB is controllable and the pair CTYA is observable.

Remark 1. The class of nonlinear dynamical systems includes an extensive variety of chaotic systems such as

ossler system and ghu9s circuit, which will be discussed in detail in the illustrative examples section.

This work largely focuses on the following objective: given the drive system modeled by (1), we want to

design an adaptive observer-based driven system on the basis of a scalar available transmitted signal so that

these drive and driven systems are to be synchronized. The synchronization system based on the observer

design method is shown in Fig. 1.

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3. Adaptive synchronization via an observer-based design

According to the control theory, when all state variables of system (1) are unavailable, a Luenberger-like

observer based on the available signal can be derived to estimate the state variables provided that the linear

part of system (1) is observable. On the basis of state observer design, a driven system in the drivedriven

conguration of chaos synchronization corresponding to (1) is given as follows:

x Ax fy L y y BuY y CTxY 2

where x denotes the dynamic estimate of the state x, and u is the control input which will be designed to

compensate for the systems disturbance and/or unknown parameters. Moreover, the constant vector

L P Rn is chosen such that A LCT is an exponentially stable matrix, which is possible since the pairCTYA is observable.

3.1. Nonadaptive driven system design

By allowing the state error e x x and the output error e1 y y, the error dynamics can be writtenas follows:

e x x A LCTe B d hTgy uY e1 C

TeX 3

In the case when the scalar disturbance d and the parameter vector h are known and available, the control

input is derived as follows:

u d hTgyX 4

Applying the control law (4) to (3) yields the resulting error dynamics as follows:

e x x A LCTeY e1 CTeX 5

Since the matrix A LCT is exponentially stable, it can be easily veried that error dynamics exponentiallyconverge to zero for any initial condition e0 x0 x0. Consequently, the dynamics of the chaoticdrive system (1) and the driven system (2) are synchronized at the rate of convergence:

exp k

minA LCT

, wherek

minD denotes the minimum eigenvalue of the matrix D.

3.2. Adaptive driven system design

The control law derived thus far requires that the knowledge of the systems disturbance d and the

parameter vector h. However, in many real applications it can be dicult to exactly determine the values of

the systems disturbance dand the parameter vector h. Consequently, the control law u shown as (4) cannot

be appropriately designed such that the driven system synchronizes the drive system. If u is overdesigned,

then an expensive and too conservative control eort is introduced. Moreover, the uniform stability of the

error dynamics cannot be ensured. To overcome these drawbacks, an adaptive control law is derived to

appropriately adjust the control eort, thereby achieving the adaptive driven system.

Owing to chaos, systems have a property of strange attractor; the matrix A is generally unstable. It is

also well known that the stability of the adaptive control system requires a strictly positive realness of thecontrolled system for only an available signal e1. Hence, to ensure the stability of the overall adaptive drive

driven chaos synchronization, the following assumption is made:

A2. There is a constant vector L P Rn such that Hs CT sI A LCT B is a strictly positive realtransfer function.

Remark 2. According to the Kalman and Yakubovich Lemma [24], the strict positive realness of Hsassures the existence of a symmetric and positive matrix P PT b 0, which satises

A LCTTP PA LCT QY PB CY 6

where Q QT b 0.

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Now, control law (4) with the estimates of d and h is rewritten as follows:

u d hTgyX 7

The estimates d and h are updated according to the following algorithm:

d y yY h gyy yX 8

Under control law (7) and the adaptation algorithm (8), the resulting error dynamics can be characterizedas follows:

e x x A LCTe B d

d h hTgy

Y

d e1Yh e1gyY e1 C

TeX 9

Now, the main theorem is stated as follows:

Main Theorem. gonsider the drive hoti system (1) stisfyin the ssumtions A1 and A2. he oserverE

sed driven system (2) ssoited the roosed ontrol lw (7) nd the dttion lorithm (8) lolly

symtotilly synhronizes the drive system, iFeF ketk kxt xtk 3 0 s t 3 I for ll initil ondiEtions. purthermoreD ll sinls inside the losedEloo remin oundedness .

Proof. Consider a Lyapunov function as

V eTPe d

d2 h hTh h

X 10

Taking the time derivative of along the trajectories of the resulting error dynamical system (9) leads to

V eT A

LCTTP PA LCT

e

2eTPB d

d h hTgy

2 d

d

d h hT

h

eT A

LCTTP PA LCT

e 2eTPB d

d h h

Tgy

2 d

d e1 h hT e1gy

X 11

Using the assumption A2 and the KY lemma in Remark 2 yield

V eTQeT 0X 12

Since is a positive and decrescent function and V is negative semidenite, it follows that the equilibrium

points e 0, d d and h h of system (9) are uniformly stable, i.e. et P LIY dt P LI and ht P LI.From (12), we can easily show that the square of et is integrable with respect to time, i.e. et P L2. Nextby Barbalats lemma [24], for initial conditions, (9) implies that i.e. et P LI, which in turn implies et 3 0as t 3 I. This completes the proof.

4. Illustrative examples

To illustrate the adaptive synchronization scheme proposed herein, two examples of well-known chaotic

systems are considered and their numerical simulations are performed. A fourth-order RungeKutta

method with xed step-size 0.0001 is used in all simulations.

Example 1. osslerElike system. A chaotic system with six terms and one nonlinearity adopted from the case

of 19 distinct simple examples of chaotic ows in [25] is described as follows:

X1 X1 4X2Y X2 X1 X2

3

Y X3 1 X1X 13


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System (13) topologically resembles the Rossler attractor and has one positive Lyapunov exponent of 0.188.

Moreover, its fractal dimension is 2.151 [25]. Taking the consideration of systems disturbances, the per-

turbed chaotic system can be expressed by

X1 X1 4X2 dX1Y X2 X1 X2

3 dX2Y X3 1 X1 dX3Y 14

where dX1Y dX2 and dX3 are time-independent unknown disturbances. By applying state variable trans-

formation to (14)

x1 X1 dX2Y x2 X2 dX1

4

dX2

4Y x3 X3 15

leads to

x1 x1 4x2Y x2 x1 x23Y x3 d x1 16

and d 1 dX2 dX3. Moreover, by introducing y x3, Eq. (16) can be rewritten in a compact form asfollows:

x

1 4 0

1 0 0

1 0 0

PTR

QUS

x

0

0

y

2

PTR

QUS

0

0

1

PTR

QUS

d A x fy B dY

y x3 0 0 1x CTxX

17

It can be easily veried that AYB is a controllable pair and CTYA is an observable pair. Also, the vectorLT 21

26126

1

can be found so that the eigenvalues of matrix A LCT are 0X8606 and0X5697 j2X1219, and so that the transfer function

Hs CTsI A LCT1B

s2 s 4

s3 2s2 5X8077s 4X1538

is strictly positive real. Hence, the assumptions A1 and A2 are satised. Moreover, the following symmetric

and positive-denite matrices

P 1X25 0X254 0

0X254 5X25 0

0 0 1

PR

QSY Q 2 0 00 2 0

0 0 2

PR

QS 18

satisfy Eq. (6). As derived earlier, an observer-based driven system is designed as follows:

x

1 4 0

1 0 0

1 0 0

PTR

QUSx

0

0

y2

PTR

QUS

0

0

1

PTR

QUS d L y y

y 0 0 1 x

19

and the estimated is updated according to the following algorithm:

d y yX 20

In numerical simulations, the time variable is re-scaled by a factor of 20 for observing the chaotic behavior

in a short interval; and systems disturbances are chosen as dX1 1YdX2 2X1 and dX3 2X0, therebyimplying d 0X9. Figs. 2(a)(d) show the trajectories of the state variables x1tY x1t, x2tY x2t, x3tY x3tand dt, respectively, for the initial conditions: x10 0X1Y x20 0X1Y x30 0X1Y x10 1Y x20 2Y x30 1 and d0 0X1.

Example 2. ghu9s iruit. A typical Chuas circuit and its physical meaning can be found in Ref. [26]. After

appropriate variable and parameter transformations, the set of nondimensional dierential equations is

given by


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x1 10 x2 x1 fx1Y

x2 x1 x2 x3Y

x3 15x2 0X0385x3

21

and

fx1 bx1 0X5a b x1j 1j x1j 1jY 22

wherefx1

denotes a three-segment piecewise linear function, where and are two negative real con-

stants and a ` 1Y 1 ` b ` 0. Herein, the parameters and are assumed to be unknown. By intro-ducing y x1, Eq. (22) can be rewritten in a compact form as follows:

x

10 10 0

1 1 1

0 15 0X385

PTR

QUSx

1

0

0

PTR

QUS 10 by 5a b yj 1j yj 1jY A x B hTgyY

y x1 1 0 0x CTxY

23

where hT h1 h2 10b 5a b and gyT

g1y g2y y y 1j j y 1j j . It can be easily

veried that AYB is a controllable pair and CTYA is an observable pair. Also, the vectorLT 9 1X6883 0X6044 can be found so that the eigenvalues of matrix A LCT are 0X9737 and0X5324 j4X6518, and that the transfer function

Hs CTsI A LCT1

B s2 1X0385s 15X0385

s3 2X0385s2 22X9599s 21X3474

is strictly positive real. Hence, the assumptions A1 and A2 are satised. Moreover, the following symmetric

and positive-denite matrices

P 1 0 0

0 15X37 0X9580 0X958 1X091

PR

QSY Q

2 0 0

0 2 0

0 0 2

PR

QS 24

Fig. 1. Chaotic synchronization system based on adaptive observer design.

Fig. 2. Trajectories ofRossler-like chaos synchronization (a) x1t and

x1t (b) x2t and

x2t (c) x3t and

x3t (d) the

estimated parameter dt.


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performed in which the information signal communicates and is then recovered using the estimated pa-

rameters.

The transmitter is a chaotic system described by (1) with a slight modication and represented as follows:

x A x fyH B d

hTgyH

LsY yH CTx s y sY 27

where s P R is the information signal and masked by the systems output, and yH P R is the chaoticallytransmitted signal, which drives the receiver [6]. Employing the observer-based driven system design, the

receiver is constructed as follows:

x Ax fyH B d

hTgyH

L yH yY y CTxX 28

If the systems disturbances and parameters can be exactly estimated by the proposed adaptive scheme fromthe adaptation phase in which the information signal is set to be zero, i.e. st 0, then d d and h h.Similarly, in the communication phase, dening the synchronization error e x x yields

e x x A LCTeX 29

Furthermore, according to Fig. 4, the recovered signal is achieved by

sRt yHt yt 30

and, by using (27), we have

limt3I

sRt limt3I

CTxt

xt st

limt3I

CTet

st

stX 31

Fig. 4. Secure communication system based on adaptive observer design.

Fig. 5. Trajectories of Rossler-like chaos synchronization-based communication (a) x3t and x3t (b) the estimated parameterdt (c) the information s(t) (d) the error between s(t) and sRt.


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Consequently, the information signal can be asymptotically recovered at the receiving end of communi-

cation.

In the following communication application illustration, we will use the perturbed Rossler-like system

shown in Example 1 as the chaotic transmitter, which is described as follows:

X1 X1 4X2 dX1 Y X2 X1 X3 s2 dX2 Y X3 1 X1 dX3

yH X3 sX32

By the adaptive observer-based synchronization proposed earlier, the receiver can be easily designed. In the

numerical simulations, the time variable of system (32) is re-scaled by a factor of 20 for observing the

chaotic behaviors in a short time interval. The information signal is st 0X1sin 20t. Figs. 5(a) and (b)show the trajectories of both x3t and x3t and the estimated parameter d in the adaptation phase0T t` 0X5, respectively. After the adaptation phase, the information signal is then injected into the

transmitter. Figs. 5(c)(d) show the time-response of the transmitted information signal s(t) and the error

between and information single and the recovery signal, respectively, in the communication phase tP 0X5.

The above simulations conrm that the proposed scheme operates satisfactorily.

6. Conclusions

In this work, an adaptive observer-based approach has been developed to resolve the chaos synchro-

nization of a class of nonlinear systems in the presence of systems disturbances and unknown parameters.

Given certain structural conditions of the drive chaotic system, an adaptive observer-based driven system is

constructed so that those drive systems and driven systems are to be synchronized. By appropriately

selecting observer gain vector such that the strictly positive real (SPR) condition is satised, the syn-

chronization and stability of the overall systems are guaranteed by the Lyapunov stability theory. Two

well-known chaotic systems: osslerElike and ghu9s iruit are considered as illustrative examples to

demonstrate the eectiveness of the proposed approach. Moreover, the proposed scheme is then success-

fully applied to an Rossler-like chaos synchronization-based communication system. Simulation results

also verify the proposed schemes success in the communication application.

Acknowledgements

This research was supported by the National Science Council of the Republic of China, under Grant

NSC88-2213-E-006-092.

References

[1] Carroll TL, Pecora LM. Synchronizing chaotic circuits. IEEE Trans Circuits Sys I 1991;38:4536.

[2] Chua LO, Kocarev LJ, Eckert K, Itoh M. Experimental chaos synchronization in Chuas circuit. Int J Bifurc Chaos 1992;2:7058.

[3] Ogorzalek MJ. Taming chaos-part I: synchronization. IEEE Trans Circuits Syst I 1993;40:6939.[4] Chen G. Control and Synchronization of Chaos, a Bibliography, Department of Electrical Engineering, University of Houston,

Houston, TX, available via ftp:uhoop.egr.uh.edu/pub/TeX/chaos.tex (login name and password both ``anonymous''), 1997.

[5] Bai EW, Lonngran EE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:518.

[6] Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons, Fractals 1998;9:155561.

[7] Halle KS, Wu CW, Itoh M, Chua LO. Spread spectrum communication through modulation of chaos. Int J of Bifurc Chaos

3:469477.

[8] Itoh M, Wu CW, Chua LO. Communication systems via chaotic signals from a reconstruction viewpoint. Int J Bifurc Chaos

1997;7:27586.

[9] Cuomo KM, Oppenheim AV, Strogatz SH. Synchronization of Lorenz-based chaotic circuits with applications to communi-

cations. IEEE Trans Circuits Syst II 1993;40:62633.

[10] Kocarev LJ, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communications via chaotic

synchronization. Int J Bifurc Chaos 1992;2:70913.

[11] Peng JH, Ding EJ, Yang W. Synchronizing hyperchaos with a scalar transmitted signal. Phys Rev Lett 1996;76:9047.


7/28/2019 Liao00

10/10

[12] Tamasevicius A, Cenys A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic oscillators with gyrators. Electron Lett

1997;33:5424.

[13] Grassi G, Mascolo S. Synchronization of hyperchaotic oscillators using a scalar signal. Electron Lett 1998;34:4234.

[14] Rulkov NF, Sushchik MM, Tsimring LS, Abarbanel HDI. Generalized synchronization of chaos in directly coupled chaotic

systems. Phys Rev E 1995;51:91480.

[15] Abarbanel HDI, Rulkov NF, Sushchik MM. Generalized synchronization of chaos: the auxiliary system approach. Phys Rev E

1995;53:452835.

[16] Duan CK, Yang SS, Min M, Xia S. Feedback-control to realize and stabilize chaos synchronization. Chaos, Solitons & Fractals

1998;9:9219.[17] Wu CW, Yang T, Chua LO. On adaptive synchronization and control of nonlinear dynamical systems. Int J Bifurc Chaos

1996;6:45571.

[18] Bernardo MD. An adaptive approach control and synchronization of continuous-time chaotic systems. Int J Bifurc Chaos

1996;6:55768.

[19] Morgul O, Solak E. Observer based synchronization of chaotic systems. Phys Rev E 1996;54:480311.

[20] Morgul O, Solak E. On the synchronization of chaotic systems by using state observations. Int J Bifurc and Chaos

1997;7:130722.

[21] Nijmeijer H, Mareels LMY. An observer looks at synchronization. IEEE Trans Circuits Syst I 1997;44:8829.

[22] Grassi G, Mascolo S. Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal. IEEE Trans Circuits Syst

I 1997;44:10134.

[23] Liao TL. Observer-based approach for controlling chaotic systems. Phys Rev E 1998;57:160410.

[24] Ioannou PA, Sun J. Robust adaptive control. Englewood Clis, NJ: Prentice-Hall, 1996.

[25] Sprott JC. Some simple chaotic ows. Phys Rev E 1994;50:R64750.

[26] Chua LO, Lin GN. Canonical realization of Chua s circuit family. IEEE Trans Circuits Syst 1990;37:885902.


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