Synchronization of two different chaotic systems using Legendre polynomials with applications in secure communications

Abstract

In this study, a new controller for chaos synchronization is proposed. It consists of a state feedback controller and a robust control term using Legendre polynomials to compensate for uncertainties. The truncation error is also considered. Due to the orthogonal functions theorem, Legendre polynomials can approximate nonlinear functions with arbitrarily small approximation errors. As a result, they can replace fuzzy systems and neural networks to estimate and compensate for uncertainties in control systems. Legendre polynomials have fewer tuning parameters than fuzzy systems and neural networks. Thus, their tuning process is simpler. Similar to the parameters of fuzzy systems, Legendre coefficients are estimated online using the adaptation rule obtained from the stability analysis. It is assumed that the master and slave systems are the Lorenz and Chen chaotic systems, respectively. In secure communication systems, observer-based synchronization is required since only one state variable of the master system is sent through the channel. The use of observer-based synchronization to obtain other state variables is discussed. Simulation results reveal the effectiveness of the proposed approach. A comparison with a fuzzy sliding mode controller shows that the proposed controller provides a superior transient response. The problem of secure communications is explained and the controller performance in secure communications is examined.

Keywords

Observer-based synchronization / Chaotic systems / Legendre polynomials / Secure communications

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