On the Controllability Matrix Realization Problem

The problem of characterizing the sequences of matrices that can be realized as the block-matrices of the controllability matrices of controllable systems is considered. Necessary and sufficient conditions for the existence of realizations are provided. It is then shown that when realizations exist they are unique if and only if the rank of the input matrix in the realization is greater than 1. The non-uniqueness of the Single Input realizations is associated with the assignability property of these type of systems. This realization problem is shown to be useful to provide a characterization of the closure of the orbit under similarity when restricted to controllable systems. Finally, the sequences which are realizable as the controllability matrix of controllable systems with prescribed controllability indices is shown to be a differentiable manifold and its dimension is computed. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.