Generic dimensions of output-nulling invariant and controllability subspaces of structured linear systems

In this paper we consider linear structured systems which represent a large class of parameter dependent linear systems. We study the generic dimension of the maximal output-nulling invariant and controllability subspaces of such systems. We give characterizations of these generic dimensions in terms of properties of the directed graphs that can be naturally associated with linear structured systems. The systems are not a priori assumed to satisfy some sort of invertibility or minimality requirements.

1 Introduction

Many aspects of linear systems can be phrased in terms of their invariant properties. These invariant properties contain essential information concerning a system, stated in a simple form, for instance in terms of some integer numbers. Examples of invariants are the dimensions of the maximal output-nulling invariant and controllability subspaces of a linear system.