A unifying characterization of robust sliding mode control: A Lyapunov approach

This paper sets forth general conditions on the existence, boundedness, and proper gains of a control for stabilizing a nonlinear plant state trajecto ry to a sliding manifold denoted by S contained in the state space as chara cterized by a smooth quadratic Lyapunov function, V. To state such conditio ns we define a time-varying (possibly discontinuous in time) state-dependen t decision manifold by considering the time-derivative of the quadratic Lya punov function. The decision manifold disconnects the control space. At eac h instant of time, stability is achieved by choosing a control in an approp riate 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) co ntroller unless robustness in the presence of matched disturbances is desir ed. If the decision manifold is discontinuous, then the need for a disconti nuous control is clear. The formulation unifies the various VSC control str ategies found in the literature under a single umbrella and suggests new st ructures. The formulation also provides a simple geometric understanding of the effect of norm bounded but not necessarily matched disturbances and pa rameter variations on the system. Two examples illustrate the design aspect s of the formulation. [S0022-0434(00)02904-X].