A dynamic graph characterisation of the fixed part of the controllable subspace of a linear structured system

In this paper we study linear structured systems described by means of system matrices of which only the zero/non-zero structure is known and where the non-zeros are supposed to have independent values. The structure of linear structured systems can be represented by means of various types of graphs, like directed graphs or dynamic graphs. Here we use both type of graphs because they enable us to formulate and study certain controllability properties in a uniform and straightforward way. In this paper we extend the results of a previous paper containing a partial characterisation of the fixed part of the controllable subspace of linear structured systems. This fixed part is defined as the part of the controllable subspace that is independent of the values to the non-zeros, and therefore can be seen as the robust part of the controllable subspace. It turns out that, by considering the generic dimension of the controllable subspace, a characterisation of the fixed part can be obtained. The latter dimension equals the size of the minimal set of nodes in the dynamic graph that separates between the set of input nodes and the set of final state nodes. Computing the supremal of such minimal separating sets, we are capable of characterising the fixed part. In the paper we indicate how this supremal minimal separating set can be obtained insightfully and efficiently using the recursive nature of the dynamic graph. Our results are illustrated by some meaningful examples.