Nested Bases of Invariants for Minimal Realizations of Finite Matrix Sequences

Abstract

The problem of finding minimal realizations of linear constant systems from finite order input-output Markov matrix sequences is considered. The paper identifies from the sequences sets of independent structural and numerical quantities which are invariants of equivalent state space representations and completely characterize any minimal realization of the sequence. These sets, termed bases of invariants, acquire a “nesting” property by which a subsequent basis of a higher order finite sequence is obtained from the previous basis by addition of some new invariants. Two canonical state space representations of special forms that reflect the input and output structural properties of the underlying systems are presented and readily derived from these bases by a simple algorithm which is provided. Necessary and sufficient conditions for the existence of a unique minimal partial realization to a given finite Markov sequence are given in terms of the invariants of its nested basis. The set of all minimal partia...