Introduction

Abstract

This book is intended to develop analysis tools and the control synthesis of discontinuous systems operating under uncertainty conditions. Throughout this book, discontinuous systems are regarded as the ones consisting of a family of continuoustime subsystems, equipped with the rule of switching between them. Conceptually, these systems are described as follows. Let the continuous state space, say an Euclidean space or a Hilbert space, be partitioned into an infinite number of operating domains by means of a family of switching surfaces. Inside an operating domain, such a system is governed by an ordinary differential equation with a differentiable right-hand side. Whenever the system trajectory hits a switching surface, the continuous state makes a jump, specified by a restitution rule. Such a rule is a map whose domain and range are the union of the switching surfaces and the entire state space, respectively. The state jumps are typically referred to as impulse effects. A special case is when impulse effects are absent. Just in case, the state trajectory is always continuous, but in general it is not differentiable when it hits a switching surface. After hitting a switching surface, the trajectory can either cross it or evolve along the surface on a finite time interval. In the latter case, sliding motions occur in the system. To summarize, a discontinuous system is specified by i) the family of switching surfaces determining the operating domains, ii) the family of continuous time subsystems, well-posed for each domain, iii) the restitution rule, determining potential jumps of the system, and iv) the sliding mode dynamics, governing potential state motions on the switching surfaces. We introduce here some basic models of discontinuous systems to preview issues that will arise in our study.