Global Stabilization of Nonlinear Systems via Novel Second Order Sliding Mode Control with an Application to a Novel Highly Chaotic System

Abstract

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 second order sliding mode control method for the global stabilization of any nonlinear system. The global stabilization result is derived using novel second order sliding mode control method and established using Lyapunov stability theory. Chaos in nonlinear dynamics occurs widely in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Synchronization of chaotic systems is an important research problem in chaos theory. As an application of the general result, the problem of global chaos control of a novel highly chaotic system is studied and a new sliding mode controller is derived. The Lyapunov exponents of the novel chaotic system are obtained as \(L_1 = 12.8393\), \(L_2 = 0\) and \(L_3 = -33.1207\). The large value of the maximal Lyapunov exponent (MLE) shows that the novel chaotic system is highly chaotic. The Kaplan-Yorke dimension of the novel chaotic system is obtained as \(D_{KY} = 2.3877\). We show that the novel highly chaotic system has three unstable equilibrium points. Numerical simulations using MATLAB have been shown to depict the phase portraits of the novel highly chaotic system and the global chaos control of the state trajectories of the novel highly chaotic system.