On the Stability of Sliding Mode Control for a Class of Underactuated Nonlinear Systems

A system is considered underactuated if the number of the actuator inputs is less than the number of degrees of freedom for the system. Sliding mode control for underactuated systems has been shown to be an effective way to achieve system stabilization. It involves exponentially stable sliding surfaces so that when the closed-loop system trajectory reaches the surface it moves along the surface while converging to the origin. In this paper, we present a general framework that provides sufficient conditions for asymptotic stabilization by a sliding mode controller for a class of underactuated nonlinear systems with two degrees of freedom. We show that, with the sliding mode controller presented, the closed-loop system trajectories reach the sliding surface in finite time. Furthermore, we develop a constructive methodology to determine exponential stability of the reduced-order closed-loop system while on the sliding surface thus ensuring asymptotic stability of the overall closed-loop system and provide a way to determine an estimate of the domain of attraction. Finally, we implement this framework on the example of an inverted pendulum.