Minimal state space realization, static output feedback and matrix completion of families of systems

TL;DR

This paper characterizes minimal realizations of linear systems using spectral intersection and static output feedback, introduces a controllability parameterization, and relates realization matrices of different dimensions for system families.

Contribution

It provides new spectral and feedback-based criteria for minimality, a controllability parameterization, and links between realization matrices of varying dimensions.

Findings

Minimality characterized by spectral intersection and pole placement.

Parameterization of controllable B matrices based on eigenvalue multiplicities.

Equivalence between realization matrices of different dimensions for system families.

Abstract

We here characterize the minimality of realization of arbitrary linear time-invariant dynamical systems through (i) intersection of the spectra of the realization matrix and of the corresponding state submatrix and (ii) moving the poles by applying static output feedback. In passing, we introduce, for a given square matrix A, a parameterization of all matrices B for which the pairs (A, B) are controllable. In particular, the minimal rank of such B turns to be equal to the smallest geometric multiplicity among the eigenvalues of A. Finally, we show that the use of a (not necessarily square) realization matrix L to examine minimality of realization, is equivalent to the study of a smaller dimensions, square realization matrix L_sq, which in turn is linked to realization matrices obtained as polynomials in L_sq. Namely a whole family of systems.