Geometric tools and methods in the study of switching systems

The objective of this Chapter is that of reviewing and illustrating the basic concepts of the geometric approach to switching systems and to illustrate the results that follow from their application. Both structural aspects and aspects which are related to qualitative properties, like stability and stabilization, are considered and investigated. We begin with by discussing how the notions of invariance, controlled invariance and conditioned invariance are extended from the linear framework to the framework of switching systems and by characterizing the subspaces of the state space that enjoy those properties. Compliance, in a suitable sense, with switching is the key aspect of (controlled or conditioned) invariance that has to be considered. Feedback characterization of controlled invariant subspaces, as well as output injection characterization of conditioned invariant subspaces, for a switching system are illustrated and shown to be relevant in synthesizing controllers or observers which solve structural problems. Then, the properties of internal stabilizability and of external stabilizability for controlled invariant subspaces and for conditioned invariant subspaces are introduced and investigated. Geometric objects which enjoy stabilizability properties are instrumental in dealing with problems which are characterized, at the same time, by structural requirements and by qualitative requirements. Examples are given by disturbance decoupling problems with stability, model matching problems, asymptotic regulation problems and unknown input observation problems. In all those cases, the geometric approach deals separately with structural requirements, which are characterized by the possibility of constraining suitable components of the system dynamics inside specific invariant subspaces, and with qualitative requirements, which involve quadratic or asymptotic stabilizability of the same components of the dynamics. This simplifies the analysis and, at the same time, it provides algorithms and practical procedures that, constructing subspaces and checking simple geometric or algebraic conditions, makes it possible to investigate solvability of the problems and to synthesize solutions. Recent results are used to show how geometric methods are effective in tackling the above mentioned control and observation problems and in constructing switching feedback compensators or switching observers which, possibly, solve them.