116 International Journal of Fuzzy Systems, Vol. 9, No. 2, June 2007 Chaotic Synchronization Using Fuzzy Control Approach H.K. Lam and Mahbub Gani Abstract 1 This paper presents the synchronization of chaotic systems subject to parameter uncertainties. Based on the fuzzy models of chaotic systems, a fuzzy con- troller is designed to realize chaotic synchronization. A design criterion of the membership functions of fuzzy controller is proposed to facilitate the stability design towards chaotic synchronization when pa- rameter uncertainties are under consideration. LMI-based stability conditions are derived to guar- antee the system stability using Lyapunov-based ap- proach. Simulation examples are given to illustrate the merits of the proposed fuzzy-model-based control approach. 1. Introduction Chaotic synchronization has drawn the researchers’ attention for many years due to its practical applications such as secure communication. The highly nonlinear nature of the chaotic systems and its sensitivity to initial conditions make the system analysis and controller de- sign for chaotic synchronization challenging. The situation becomes further complicated when chaotic systems are subject to parameter uncertainties, which is inevitable in most practical applications. Fuzzy-model-based control approach has been shown to be beneficial in dealing with ill-defined and nonlinear systems. Recently, various fuzzy-model-control approaches have been proposed to realize chaotic synchronization and promising stability analysis results have been achieved. In general, under the fuzzy-model-based control approach, a TS-fuzzy model [1], [2], which exhibits favourable properties to facilitate the stability analysis and controller design, is employed to provide a general and systematical frame- work to represent the dynamics of the chaotic systems. It was shown in [3]-[5] that some common chaotic sys- tems can be represented by fuzzy models with simple rules. Based on the fuzzy models, a fuzzy controller is then designed to realize chaotic synchronization. In [3], Corresponding Author: H.K. Lam, Division of Engineering, King's College London, Strand, London, WC2R 2LS, United Kingdom E-mail: hak-keung.lam@kcl.ac.uk Manuscript accepted 11 th June. 2007. [6], by taking advantage of the identical structure of the chaotic systems and the favourable property given by sharing the same premises between fuzzy model and controller, an exact-linearization fuzzy control approach was proposed and stability conditions in terms of linear matrix inequalities (LMIs) were derived. By employ- ing some convex programming techniques, the solution, which includes the feedback gains of the fuzzy controller, to the LMI-based conditions can be solved numerically and efficiently. This idea was extended to H ∞ approach of which the synchronization performance is guaranteed by an H ∞ performance index [4], [5]. In [3]-[6], only uncertainty-free chaotic systems were considered. When the chaotic systems are subject to parameter uncertainties, the stability conditions in [3]-[6] are not applicable to reach a stable design of fuzzy controller to realize chaotic synchronization. To deal with the parameter uncertainties, adaptation ability [7] was endowed to the fuzzy controller. By taking advantage of the superior approximation ability of the fuzzy system, the values of parameter uncertainties can be estimated in an online manner for the fuzzy controller to realize synchronization. Consequently, compared with the fuzzy-model-based control approach in [3]-[6], the adaptive fuzzy controller offers an outstanding ro- bustness property to handle parameter uncertainties at the cost of high structural complexity and computational demand. Various adaptive fuzzy control approaches were reported in [8], [9]. In this paper, a fuzzy controller is employed to syn- chronize chaotic systems subject to parameter uncertain- ties. As parameter uncertainties are considered, the favourable property given by sharing the same premises between the fuzzy model and controller [3]-[6] cannot facilitate the stability analysis and design. Instead, by designing properly the membership functions of the fuzzy controller, some arbitrary matrices can be intro- duced to ease the stability analysis. LMI-based stabil- ity conditions are derived using Lyapunov-based ap- proach to aid the design of fuzzy controllers. © 2007 TFSA