Topological aspects of controllability and observability on the manifold of singular and regular systems

Abstract

In a previous article (J. D. Cobb, J. Math. Anal Appl., Nov. 1986) we considered the class of all singular and regular linear time-invariant systems and proved some basic topological properties of that set. In this paper we examine specific implications of those results to control theory and demonstrate, among other things, that controllability and observability are generic properties even when singular systems are included in the construction. We also derive related results for other important subclasses of systems, proving that only some of the remaining fundamental system properties are generic. Finally, we extend existing results on connectedness and show that the number of connected components of the controllable and observable sets is diminished whenever singular systems are brought into the picture.