Discrete-time/discrete-event sliding mode design via Lyapunov approach

This paper considers the development of a sliding mode control strategy for a discrete-time dynamic system in an abstract metric space. The control objective is to stabilize the state trajectory of the system to an equilibrium surface in the state space without assuming any particular control structure. Stability to the surface is characterized by a possibly time varying quadratic Lyapunov function V. Conditions on the control input which will guarantee that the Lyapunov function has negative (semi) definite increment are developed. This is accomplished by defining a decision manifold in the control space which defines an elliptic time varying area. It is shown that if the value of the control input vector is chosen inside the elliptic area at each instant of time, the increment of V is negative (semi) definite. Such a formulation allows one to unify different variable structure control strategies in discrete-time systems in Euclidean state space X and in discrete event systems, where X is finite