Chaos, Solitons and 11 (2000) 1387±1396 www.elsevier.nl/locate/chaos

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 drive±driven 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 systemÕs 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 Chua's circuit are considered as illustrative examples to demonstrate the e€ectiveness 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 transmitterÕs 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 schemeÕs success in the communication application. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Synchronization in chaotic systems has received increasing attention [1±6] 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 [7±10]. The preliminary type of chaos synchronization consists of drive±driven 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 [11±13] 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 drive±driven 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 systemÕs disturbances are always unavoidable and some systemÕs parameters cannot be exactly known in priori. The e€ects of these uncertainties will destroy the synchronization and even break it. Therefore, adaptive syn- chronization of the drive±driven systems in the presence of systemÕs 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 [19±23]. A systematic method employing a nonlinear state observer is proposed to resolve the chaotic synchronization of a class of

* Corresponding author. Fax: +886-6276-6549. E-mail address: [email protected] (T.-L. Liao).

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 051-X 1388 T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396 hyperchaotic systems via a scalar transmitted signal [22]. Although chaotic synchronization can be ensured due to the e€ect 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 injected into 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 systemÕs 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 systemÕs disturbances and unknown parameters. By ap- propriately selecting the observer gains, the synchronization and stability of the overall systems are guaranteed by the Lyapunov with certain structural conditions. In Section 4, Rossler- like and Chua's systems are given as illustrative examples to demonstrate the e€ectiveness 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 schemeÕs 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 di€erential 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 system output. Therefore, we will be particularly concerned with nonlinear continuous-time systems of the fol- lowing form: Á x_ ˆ Ax ‡ f y†‡Bd‡ hTg y† ; y ˆ CTx; 1† where y 2 R denotes the system output, x 2 Rn represents the state vector, d 2 R is a bounded disturbance, and A; B and C denote known matrices with appropriate dimensions, †T denotes the vector transpose. We assume that h 2 Rp is a constant parameter vector which may be unknown, and that f 2 Rn and g 2 Rp are real analytic vectors with f 0†ˆ0 and g 0†ˆ0, respectively. Furthermore, we assume that system (1) has a unique solution x t† passing through the initial state x 0†ˆx0 and this solution is well de®ned over an interval ‰ 0 1†. Many chaotic systems have a special structure on the matrices A; B and C; hence we make the following assumption: A1: The pair A; B† is controllable and the pair CT; A† is observable.

Remark 1. The class of nonlinear dynamical systems includes an extensive variety of chaotic systems such as Rossler system and Chua's 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. T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396 1389

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 drive±driven con®guration of chaos synchronization corresponding to (1) is given as follows: x^_ ˆ Ax^ ‡ f y†‡L y y^†‡Bu; y^ ˆ CTx^; 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 systemÕs disturbance and/or unknown parameters. Moreover, the constant vector L 2 Rn is chosen such that A LCT† is an exponentially stable matrix, which is possible since the pair CT; A† is observable.

3.1. Nonadaptive driven system design

By allowing the state error e ˆ x x^ and the output error e1 ˆ y y^, the error dynamics can be written as follows: Á _ T T T e_ ˆ x_ x^ ˆ A LC †e ‡ Bd‡ h g y†u ; e1 ˆ C e: 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 ˆ d ‡ hTg y†: 4† Applying the control law (4) to (3) yields the resulting error dynamics as follows: _ T T e_ ˆ x_ x^ ˆ A LC †e; e1 ˆ C e: 5† Since the matrix A LCT is exponentially stable, it can be easily veri®ed that error dynamics exponentially converge to zero for any initial condition e 0†ˆx 0†x^ 0†. Consequently, the dynamics of the chaotic drive system (1) and the driven system (2) are synchronized at the rate of convergence: T exp kmin A LC †, where kmin D† 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 systemÕs disturbance d and the parameter vector h. However, in many real applications it can be dicult to exactly determine the values of the systemÕs disturbance d and 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 e€ort 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 e€ort, thereby achieving the adaptive driven system. Owing to chaos, systems have a property of strange ; 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 the controlled 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 2 Rn such that H s†CT †sI A LCT† B is a strictly positive real transfer function.

Remark 2. According to the Kalman and Yakubovich Lemma [24], the strict positive realness of H s† assures the existence of a symmetric and positive matrix P ˆ P T > 0, which satis®es A LCT†TP ‡ P A LCT†ˆQ; PB ˆ C; 6† where Q ˆ QT > 0. 1390 T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396

Now, control law (4) with the estimates of d and h is rewritten as follows: u ˆ d^ ‡ h^Tg y†: 7†

The estimates d^ and h^ are updated according to the following algorithm: _ d^ ˆ y y^†; h^ ˆ g y† y y^†: 8† Under control law (7) and the adaptation algorithm (8), the resulting error dynamics can be characterized as follows:   _ T ^ ^ T ^_ ^_ T e_ ˆ x_ x^ ˆ A LC †e ‡ Bd d ‡ h h† g y† ; d ˆ e1; h ˆ e1g y†; e1 ˆ C e: 9†

Now, the main theorem is stated as follows:

Main Theorem. Consider the drive chaotic system (1) satisfying the assumptions A1 and A2. The observer- based driven system (2) associated the proposed control law (7) and the adaptation algorithm (8) globally asymptotically synchronizes the drive system, i.e. ke t†k ˆ kx t†x^ t†k ! 0 as t !1for all initial condi- tions. Furthermore, all signals inside the closed-loop remain boundedness.

Proof. Consider a Lyapunov function as   V ˆ eTPe ‡ d d^†2 ‡ h h^†T h h^† : 10†

Taking the time derivative of V along the trajectories of the resulting error (9) leads to   V_ ˆ eT A LCT†TP ‡ P A LCT† e     _ _ ‡ 2eTPB d d^†‡ h h^†Tg y† ‡ 2 d d^† d^†‡ h h^†T h^†     ˆ eT A LCT†TP ‡ P A LCT† e ‡ 2eTPB d d^†‡ h h^†Tg y†   ^ ^ T ‡ 2 d d† e1†‡ h h† e1g y†† : 11†

Using the assumption A2 and the K±Y lemma in Remark 2 yield V_ ˆeTQe 6 0: 12† Since V is a positive and decrescent function and V_ is negative semide®nite, it follows that the equilibrium ^ ^ ^ ^ points e ˆ 0, d ˆ d and h ˆ h of system (9) are uniformly stable, i.e. e t†2L1; d t†2L1 and h t†2L1. From (12), we can easily show that the square of e t† is integrable with respect to time, i.e. e t†2L2. Next by BarbalatÕs lemma [24], for initial conditions, (9) implies that i.e. e_ t†2L1, which in turn implies e t†!0 as t !1. This completes the proof. Ã

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 Runge±Kutta method with ®xed step-size 0.0001 is used in all simulations.

Example 1. Rossler-like system. A chaotic system with six terms and one nonlinearity adopted from the case S of 19 distinct simple examples of chaotic ¯ows in [25] is described as follows:

_ _ 2 _ X1 ˆX1 4X2; X2 ˆ X1 ‡ X3 ; X3 ˆ 1 ‡ X1: 13† T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396 1391

System (13) topologically resembles the Rossler attractor and has one positive of 0.188. Moreover, its dimension is 2.151 [25]. Taking the consideration of systemÕs disturbances, the per- turbed chaotic system can be expressed by _ _ 2 _ X1 ˆX1 4X2 ‡ dX1; X2 ˆ X1 ‡ X3 ‡ dX2; X3 ˆ 1 ‡ X1 ‡ dX3; 14† where dX1; dX2 and dX3 are time-independent unknown disturbances. By applying state variable trans- formation to (14) dX dX x ˆ X ‡ dX ; x ˆ X 1 2 ; x ˆ X 15† 1 1 2 2 2 4 4 3 3 leads to

2 x_ 1 ˆx1 4x2; x_ 2 ˆ x1 ‡ x3; x_ 3 ˆ d ‡ x1 16† and d ˆ 1 dX2 ‡ dX3. Moreover, by introducing y ˆ x3, Eq. (16) can be rewritten in a compact form as follows: 2 3 2 3 2 3 1 40 0 0 6 7 6 7 6 7 x_ ˆ 4 10 05x ‡ 4 0 5 ‡ 4 0 5 d†Ax‡ f y†‡Bd; 17† 10 0 y2 1 T y ˆ x3 ˆ‰001Šx  C x:

T It canÂà be easily veri®ed that A; B† is a controllable pair and C ; A† is an observable pair. Also, the vector T 21 1 1 T L ˆ 26 26 can be found so that the eigenvalues of matrix A LC are 0:8606 and 0:5697 Æ j2:1219, and so that the transfer function s2 ‡ s ‡ 4 H s†ˆCT sI A LCT††1B ˆ s3 ‡ 2s2 ‡ 5:8077s ‡ 4:1538 is strictly positive real. Hence, the assumptions A1 and A2 are satis®ed. Moreover, the following symmetric and positive-de®nite matrices 2 3 2 3 1:25 0:254 0 200 P ˆ 4 0:254 5:25 0 5; Q ˆ 4 0205 18† 001 002 satisfy Eq. (6). As derived earlier, an observer-based driven system is designed as follows: 2 3 2 3 2 3 1 40 0 0 6 7 6 7 6 7 x^_ ˆ 4 10 05x^ ‡ 4 0 5 ‡ 4 0 5 d^†‡L y y^† 19† 10 0 y2 1 y^ ˆ‰001Š x^ and the estimate d^ is updated according to the following algorithm: d^ ˆ y y^†: 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 systemÕs disturbances are chosen as dX1 ˆ1; dX2 ˆ 2:1anddX3 ˆ 2:0, thereby implying d ˆ 0:9. Figs. 2(a)±(d) show the trajectories of the state variables x1 t†; x^1 t†, x2 t†; x^2 t†, x3 t†; x^3 t† ^ and d t†, respectively, for the initial conditions: x1 0†ˆ0:1; x2 0†ˆ0:1; x3 0†ˆ0:1; x^1 0†ˆ ^ 1; x^2 0†ˆ2; x^3 0†ˆ1 and d 0†ˆ0:1.

Example 2. Chua's circuit. A typical ChuaÕs circuit and its physical meaning can be found in Ref. [26]. After appropriate variable and parameter transformations, the set of nondimensional di€erential equations is given by 1392 T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396

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

Fig. 2. Trajectories of Rossler-like chaos synchronization (a) x1 t† ÁÁÁ†and x^1 t† (b) x2 t† ÁÁÁ†and x^2 t† (c) x3 t† ÁÁÁ†and x^3 t† (d) the estimated parameter d^ t†.

x_ 1 ˆ 10 x2 x1 f x1††;

x_ 2 ˆ x1 x2 ‡ x3; 21†

x_ 3 ˆ15x2 0:0385x3 and

f x1†ˆbx1 ‡ 0:5 a b† ‡jx1 1j jx1 1j†; 22† where f x1† denotes a three-segment piecewise linear function, where a and b are two negative real con- stants and a < 1; 1 < b < 0. Herein, the parameters a and b are assumed to be unknown. By intro- ducing y ˆ x1, Eq. (22) can be rewritten in a compact form as follows: 2 3 2 3 10 10 0 1 6 7 6 7 x_ ˆ 4 1 11 5x ‡ 4 0 5 10by 5 a b† ‡jy 1j jy 1j††;  Ax‡ B hTg y††; 0 15 0:385 0

T y ˆ x1 ˆ‰100Š x  C x; 23†

T T where h ˆ ‰Šˆh1 h2 ‰Š10b 5 a b† and g y† ˆ ‰Šˆg1 y† g2 y† ‰Šyyjj‡ 1 jjy ‡ 1 . It can be easily veri®ed that A; B† is a controllable pair and CT; A† is an observable pair. Also, the vector LT ˆ‰Š91:6883 0:6044 can be found so that the eigenvalues of matrix A LCT are 0:9737 and 0:5324 Æ j4:6518, and that the transfer function s2 ‡ 1:0385s ‡ 15:0385 H s†ˆCT sI A LCT††1B ˆ s3 ‡ 2:0385s2 ‡ 22:9599s ‡ 21:3474 is strictly positive real. Hence, the assumptions A1 and A2 are satis®ed. Moreover, the following symmetric and positive-de®nite matrices 2 3 2 3 10 0 200 P ˆ 4 015:37 0:958 5; Q ˆ 4 0205 24† 0 0:958 1:091 002 T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396 1393

Fig. 3. Trajectories of Chao's circuit chaos synchronization (a) x1 t† ÁÁÁ†and x^1 t† (b) x2 t† ÁÁÁ†and x^2 t† (c) x3 t† ÁÁÁ†and x^3 t† ^ ^ (d) the estimated parameters h1 t† ÁÁÁ†and h2 t†. satisfy Eq. (6). As derived earlier, an observer-based driven system is designed as follows: 2 3 2 3 10 10 0 1   _ 6 7 6 7 ^ ^ ^T x^ˆ 4 1 11 5x^‡ 405 h1y ‡ h2 ‡jy 1j jy 1j† ‡ L y y^†Ax^‡ B h g y†† ‡ L y y^†; 0 15 0:385 0 y^ˆ‰100Šx^ CTx^ 25† ^ ^ and the estimates h1 and h2 are updated according to the following algorithm: _ h^ ˆ y y^†y; 1 26† ^_ h2 ˆ y y^† ‡jy 1j jy 1j†:

In numerical simulations, the time variable is re-scaled by a factor of 120. All systemÕs parameters are chosen as a ˆ1:28 and b ˆ0:69, thereby implying h1 ˆ 6:9andh2 ˆ 2:95. Figs. 3(a)±(d) show the ^ ^ waveforms of the state variables x1 t†; x^1 t†, x2 t†; x^2 t†, x3 t†; x^3 t† and h1 t†; h2 t†, respectively, for the initial ^ ^ conditions: x1 0†ˆ0:1; x2 0†ˆ0:1; x3 0†ˆ0:1, x^1 0†ˆ1; x^2 0†ˆ1; x^3 0†ˆ1 and h1 0†ˆ3; h2 0†ˆ2. Simulation results of these chaotic systems demonstrate that: (i) the trajectories of the driven system asymptotically converge to those of the drive system; (ii) the value of d can be actually identi®ed.

5. Application to secure communication

In this section, the preceding adaptive observer-based synchronization scheme is applied to chaotic secure communications. Fig. 4 illustrates the proposed communication system consisting of a transmitter (or drive) and a receiver (or driven) at the receiving end of communication. The transmitted signal is a sum of the information and the output of the chaotic transmitter. In addition, the transmitted signal is also injected into the transmitter and, simultaneously, transmitted to the receiver. By the proposed adaptive observer design technique, a chaotic receiver is then derived to recover the information signal at the re- ceiving end of the communication. The process of communication consists of the following two phases: the adaptation phase is ®rst proceeded in which systemÕs disturbances and unknown parameters are estimated by the adaptive synchronization scheme proposed in the preceding section; the communication phase is then 1394 T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396

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

Fig. 5. Trajectories of Rossler-like chaos synchronization-based communication (a) x3 t† ÁÁÁ†and x^3 t† (b) the estimated parameter ^ d t† (c) the information s(t) (d) the error between s(t) and sR t†. 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 modi®cation and represented as follows: Á x_ ˆ Ax‡ f y0†‡Bd‡ hTg y0† ‡ Ls; y0 ˆ CTx ‡ s ˆ y ‡ s; 27† where s 2 R is the information signal and masked by the systemÕs output, and y0 2 R is the chaotically transmitted signal, which drives the receiver [6]. Employing the observer-based driven system design, the receiver is constructed as follows:   x^_ ˆ Ax^ ‡ f y0†‡B d^ ‡ h^Tg y0† ‡ L y0 y^†; y^ ˆ CTx^: 28† If the systemÕs disturbances and parameters can be exactly estimated by the proposed adaptive scheme from the adaptation phase in which the information signal is set to be zero, i.e. s t†ˆ0, then d^ ˆ d and h^ ˆ h. Similarly, in the communication phase, de®ning the synchronization error e ˆ x x^ yields e_ ˆ x_ x^ ˆ A LCT†e: 29† Furthermore, according to Fig. 4, the recovered signal is achieved by

0 sR t†ˆy t†y^ t† 30† and, by using (27), we have   Á T T lim sR t†† ˆ lim C x t† x^ t†† ‡ s t† ˆ lim C e t† ‡ s t† ˆ s t†: 31† t!1 t!1 t!1 T.-L. Liao, S.-H. Tsai / Chaos, Solitons and Fractals 11 (2000) 1387±1396 1395

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:

X_ ˆX 4X ‡ dX ; X_ ˆ X ‡ X ‡ s†2 ‡ dX ; X_ ˆ 1 ‡ X ‡ dX 1 1 2 1 2 1 3 2 3 1 3 32† 0 y ˆ X3 ‡ s: 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 s t†ˆ0:1sin 20t†. Figs. 5(a) and (b) ^ show the trajectories of both x3 t† and x^3 t† and the estimated parameter d in the adaptation phase 0 6 t < 0:5, 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 t P 0:5. The above simulations con®rm 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 systemÕs 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 satis®ed, the syn- chronization and stability of the overall systems are guaranteed by the theory. Two well-known chaotic systems: Rossler-like and Chua's circuit are considered as illustrative examples to demonstrate the e€ectiveness 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 schemeÕs 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.

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