Chaos, Solitons and Fractals 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 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 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 eects 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 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 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 stability theory with certain structural conditions. In Section 4, Rossler- like and Chua's 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 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 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 system output. Therefore, we will be particularly concerned with nonlinear continuous-time systems of the fol- lowing form: Á x_ Ax f yBd hTg y ; y 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 00 and g 00, respectively. Furthermore, we assume that system (1) has a unique solution x t passing through the initial state x 0x0 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^_ Ax^ f yL y y^Bu; y^ 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 x x^ and the output error e1 y y^, the error dynamics can be written as follows: Á _ T T T e_ x_ x^ A LC e Bd h g yu ; e1 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 d hTg y: 4 Applying the control law (4) to (3) yields the resulting error dynamics as follows: _ T T e_ x_ x^ A LC e; e1 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 0x 0x^ 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 dicult 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 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.