Interlacing properties of system-poles, system-zeros and spectral-zeros in MIMO systems

TL;DR

This paper extends interlacing properties of poles and zeros from SISO to MIMO systems, introduces conditions for spectral zeros to be real, and explores new algebraic Riccati equation results for system analysis.

Contribution

It formulates conditions for interlaced poles, zeros, and spectral zeros in MIMO systems, and develops new algebraic Riccati equation techniques for system analysis.

Findings

MIMO systems can exhibit interlaced poles and zeros under certain conditions.

Spectral zeros in MIMO systems can be real with specific criteria.

New algebraic Riccati equation results facilitate system property analysis.

Abstract

SISO passive systems with just one type of memory/storage element (either only inductive or only capacitative) are known to have real poles and zeros, and further, with the zeros interlacing poles (ZIP). Due to a variety of definitions of the notion of a system zero, and due to other reasons described in the paper, results involving ZIP have not been extended to MIMO systems. This paper formulates conditions under which MIMO systems too have interlaced poles and zeros. This paper next focusses on the notion of a `spectral zero' of a system, which has been well-studied in various contexts: for example, spectral factorization, optimal charging/discharging of a dissipative system, and even model order reduction. We formulate conditions under which the spectral zeros of a MIMO system are real, and further, conditions that guarantee that the system-zeros, spectral zeros, and the poles are all interlaced. The techniques used in the proofs involve new results in Algebraic Riccati equations (ARE) and Hamiltonian matrices, and these results help in formulating new notions of positive-real balancing, and inter-relations with the existing notion of positive-real balancing; we also relate the positive-real singular values with the eigenvalues of the extremal ARE solutions in the proposed `quasi-balanced' forms.