A Unifying Characterization of Robust Sliding Mode Control: A Lyapunov Approach

Abstract

This paper sets forth general conditions on the existence, boundedness, and proper gains of a control for stabilizing a nonlinear plant state trajectory to a sliding manifold denoted by S contained in the state space as characterized by a smooth quadratic Lyapunov function, V. To state such conditions we define a time-varying (possibly discontinuous in time) state-dependent decision manifold by considering the time-derivative of the quadratic Lyapunov function. The decision manifold disconnects the control space. At each instant of time, stability is achieved by choosing a control in an appropriate half space defined by the decision manifold so that the derivative of the Lyapunov function is negative definite. If the decision manifold moves continuously, then there is no need for a discontinuous (classical VSC) controller unless robustness in the presence of matched disturbances is desired. If the decision manifold is discontinuous, then the need for a discontinuous control is clear. The formulation unifies the various VSC control strategies found in the literature under a single umbrella and suggests new structures. The formulation also provides a simple geometric understanding of the effect of norm bounded but not necessarily matched disturbances and parameter variations on the system. Two examples illustrate the design aspects of the formulation. [S0022-0434(00)02904-X]