Synchronization of Two Fractional-Order Chaotic Systems via Nonsingular Terminal Fuzzy Sliding Mode Control

Xiaona Song,Ines Tejado Balsera,Lei Zhang,Leipo Liu,Shuai Song

doi:10.1155/2017/9562818

Abstract

The synchronization of two fractional-order complex chaotic systems is discussed in this paper. The parameter uncertainty and external disturbance are included in the system model, and the synchronization of the considered chaotic systems is implemented based on the finite-time concept. First, a novel fractional-order nonsingular terminal sliding surface which is suitable for the considered fractional-order systems is proposed. It is proven that once the state trajectories of the system reach the proposed sliding surface they will converge to the origin within a given finite time. Second, in terms of the established nonsingular terminal sliding surface, combining the fuzzy control and the sliding mode control schemes, a novel robust single fuzzy sliding mode control law is introduced, which can force the closed-loop dynamic error system trajectories to reach the sliding surface over a finite time. Finally, using the fractional Lyapunov stability theorem, the stability of the proposed method is proven. The proposed method is implemented for synchronization of two fractional-order Genesio-Tesi chaotic systems with uncertain parameters and external disturbances to verify the effectiveness of the proposed fractional-order nonsingular terminal fuzzy sliding mode controller.

Highlights

Fractional calculus has received an enormous amount of research effort in recent years, because it allows one to describe the behavior of a real system more accurately and more adequately compared to the standard calculus [1,2,3]
Due to the ease of fractional chaotic systems’ electronics implementation and the rapid development of the stability of fractional differential equations, fractional chaotic systems have attracted a great deal of attention [9]
Many researchers have made great contributions; for example, for integer-order chaotic systems with uncertain parameters or disturbances, the finite-time nonsingular terminal sliding mode controller was designed in [34, 35], and, for fractional-order chaotic systems, a novel fractional-order terminal sliding mode controller was proposed in [27] based on finite-time scheme

Summary

Introduction

Fractional calculus has received an enormous amount of research effort in recent years, because it allows one to describe the behavior of a real system more accurately and more adequately compared to the standard calculus [1,2,3]. Many researchers have made great contributions; for example, for integer-order chaotic systems with uncertain parameters or disturbances, the finite-time nonsingular terminal sliding mode controller was designed in [34, 35], and, for fractional-order chaotic systems, a novel fractional-order terminal sliding mode controller was proposed in [27] based on finite-time scheme. To the best of our knowledge, there is little work in the literature on finite-time control for fractional-order chaotic systems which combines the nonsingular terminal sliding mode control and the fuzzy control theory, which has remained as an open and challenging problem to be solved in this paper. Motivated by the above discussions, this paper proposes a new fractional-order nonsingular fuzzy sliding mode controller for robust synchronization of two fractional-order chaotic systems with uncertain parameters and external disturbances.

Preliminaries of Fractional Calculus

Problem Statement

Fractional-Order Nonsingular Terminal Fuzzy Sliding Mode Controller Design

Numerical Example

Conclusions