Impact of Estimation Errors of a Matrix of Transfer Functions onto Its Analytic Singular Values and Their Potential Algorithmic Extraction

TL;DR

This paper investigates how estimation errors in transfer function matrices affect their analytic singular values, revealing that perturbations eliminate multiplicities and enforce non-negativity, impacting algorithms for their extraction.

Contribution

It demonstrates that stochastic perturbations remove algebraic multiplicities of analytic singular values and ensure their non-negativity, affecting analytic SVD extraction methods.

Findings

Perturbations cause singular values to lose multiplicities.

Analytic singular values become strictly non-negative with probability one.

Impacts on algorithms for extracting analytic SVD are significant.

Abstract

A matrix of analytic functions A(z), such as the matrix of transfer functions in a multiple-input multiple-output (MIMO) system, generally admits an analytic singular value decomposition (SVD), where the singular values themselves are functions. When evaluated on the unit circle, for the sake of analyticity, these singular values must be permitted of become negative. In this paper, we address how the estimation of such a matrix, causing a stochastic perturbation of A}(z), results in fundamental changes to the analytic singular values: for the perturbed system, we show that their analytic singular values lose any algebraic multiplicities and are strictly non-negative with probability one. We present examples and highlight the impact that this has on algorithmic solutions to extracting an analytic or approximate analytic SVD.