The module structure of a controllable system

Abstract

It is proved that the system matrix P(s) of an l-input, m-output controllable system  can be brought by strict system equivalence transformations into a special form [Ptilde](s) such that the system  is described by the module μ generated by the first L column vectors of the matrix [Ptilde](s). Invariants of this module are determined and related to invariants of the system  under similarity transformations, such as the controllability indices and the controllability subspaces.