Computing Invariant Zeros of a Linear System Using State-Space Realization

TL;DR

This paper introduces a state-space realization method that simplifies the computation of invariant zeros in linear systems by reducing it to an eigenvalue problem, applicable to MIMO systems regardless of controllability or observability.

Contribution

The paper proposes a new realization technique for linear systems that enables invariant zeros to be computed via eigenvalues, extending to wide MIMO systems and clarifying their relation to zero dynamics.

Findings

Invariant zeros can be computed from eigenvalues of a constructed realization.

Method applies to square MIMO systems regardless of controllability or observability.

Zeros are shown to be the poles of the system's zero dynamics.

Abstract

It is well known that zeros and poles of a single-input, single-output system in the transfer function form are the roots of the transfer function's numerator and the denominator polynomial, respectively. However, in the state-space form, where the poles are a subset of the eigenvalue of the dynamics matrix and thus can be computed by solving an eigenvalue problem, the computation of zeros is a non-trivial problem. This paper presents a realization of a linear system that allows the computation of invariant zeros by solving a simple eigenvalue problem. The result is valid for square multi-input, multi-output (MIMO) systems, is unaffected by lack of observability or controllability, and is easily extended to wide MIMO systems. Finally, the paper illuminates the connection between the zero-subspace form and the normal form to conclude that zeros are the poles of the system's zero dynamics