Variable Structure Control of Second Order Systems With Nonlinear Uncertain Discontinuous Input Functions Using Approximate Inverses

We consider second order nonlinear systems in control canonical form, where the functional relation between the control variable and the system acceleration is nonlinear, uncertain and discontinuous. We approximate the nonlinear uncertain and discontinuous function with an invertible one, and use variable structure control theory to suppress the error of the approximate inverse. We rewrite the finite time reaching law or the terminal reaching law into the form of inequalities and use these inequalities to design a control law that drives the system state to the sliding surface in finite time, and one that ensures the convergence of the system speed and position errors to a narrow, adjustable width band around zero. The stability of the control system is also analyzed and proven. We demonstrate how to ensure the sufficiency of the available control effort, by the proper design of control parameters. The method is then validated on a numerical model of a nonlinear system with nonlinear, uncertain and discontinuous input function.