On the parametrization of linear systems with given cyclic structure

Abstract

Let Sn be the manifold of linear constant systems over R, with m inputs, p outputs, and n-dimensional state space. In this paper, we are concerned with the subset Sn,l of Sn, consisting in systems whose cyclic structure is l. It is first stated to be a submanifold of Sn, and an atlas is given for it. When l ranges over all cyclic structures, (Sn,l) is a partition of Sn, one element of which is open (and dense), namely the submanifold of cyclic systems. We then introduce special factorizations for transfer functions, which allow us to give another parametrization for Sn,l, in particular, transfer functions of cyclic systems admit a rather simple description. As this paper is a shortened version of [2], most proofs are omitted.