Design of a sliding mode controller to eliminate the limit cycle in chaotic Van der Pol system in comparison with state feedback controllers

In this paper sliding mode controller is designed in comparison with the linear state space feedback controllers namely the pole placement controller and linear quadratic regulator to eliminate the chaotic limit cycle oscillations in Van der Pol system induced by an external sinusoidal excitation in order to avoid failure in physical structures. The controller design is based on the mathematical model of the Van der Pol oscillator presented in this paper. The sliding mode controller is robust in chaotic motion as it achieves global asymptotic stability about an unstable equilibrium point which is the origin in contrast to the state feedback controllers which fail to achieve asymptotic stability about the origin. The lyapunov stability of the system is proved to ensure that the system hits the sliding surface in a global asymptotic sense. The sliding dynamics of the sliding surface is proved to exhibit finite time convergence to the origin. The dynamics of the sliding surface is examined for three different conditions of sliding surface coefficients. The validity of the design is verified through MATLAB /Simulink based simulations.