Minimal dimension realization versus minimal realization and a correction to "Minimal dimension realization and identifiability of input-output sequences"

Abstract

A minimal dimension realization is a realization, with minimal dimension, from a given input-output sequence; while a minimal realization is one from a given transfer function. It is known that minimal realizations are reachable and observable, and are unique up to an isomorphism. On the other hand, minimal dimension realizations are unique up to an isomorphism if and only if the I/O sequence is identifiable. Every minimal dimension realization is observable. However, not every minimal dimension realization is reachable, even when the I/O sequence is identifiable.