Sliding Mode Controller Design for the Global Stabilization of Chaotic Systems and Its Application to Vaidyanathan Jerk System

Abstract

Chaos in nonlinear dynamics occurs widely in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Control of chaotic systems is an important research problem in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. In this work, we derive a novel sliding mode control method for the global stabilization of chaotic systems. The general control result derived using novel sliding mode control method is proved using Lyapunov stability theory. As an application of the general result, the problem of global stabilization of the Vaidyanathan jerk chaotic system (2015) is studied and a new sliding mode controller is derived. The Lyapunov exponents of the Vaidyanathan jerk system are obtained as \(L_1 = 0.12476\), \(L_2 = 0\) and \(L_3 = -1.12451\). Since the Vaidyanathan jerk system has a positive Lyapunov exponent, it is chaotic. The Maximal Lyapunov Exponent (MLE) of the Vaidyanathan jerk system is given by \(L_1 = 0.12476\). Also, the Kaplan–Yorke dimension of the Vaidyanathan jerk system is obtained as \(D_{KY} = 2.11095\). Numerical simulations using MATLAB have been shown to depict the phase portraits of the Vaidyanathan jerk system and the sliding mode controller design for the global stabilization of the Vaidyanathan jerk system.