Matricial decomposition of systems over rings

Abstract

This paper extends to non-controllable linear systems over rings the property FC^s (s > 0), which means “feedback cyclization with s inputs”: given a controllable system (A, B), there exist a matrix K and a matrix U with s columns such that (A + BK, BU) is controllable. Clearly, FC¹ is the usual FC property. The main technique used in this work is the obtention of block decompositions for systems, with controllable subsystems of a certain size. Each of the studied decompositions is associated to a class of commutative rings for which all systems can be decomposed accordingly. Finally, examples are shown of FC^s rings (for s > 1) which are not FC rings.