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

TL;DR

This paper presents a realization of a linear system that allows the computation of invariant zeros by solving a simple eigenvalue problem, 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.

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

Theorems & Definitions (15)

• Proposition III.1
• Proposition III.2
• Proposition III.3
• Lemma IV.1
• Proposition IV.1
• Proposition IV.2
• Theorem IV.1
• Remark IV.1
• Remark IV.2
• Theorem IV.2