Representation of rational positive real functions of several variables by means of positive long resolvent

TL;DR

This paper presents a novel representation of rational positive real functions of multiple variables using positive long resolvent, linking their structure to Schur complements and sums of squares of polynomials.

Contribution

It introduces a new method to represent multivariable rational positive real functions via positive long resolvent and Schur complements, expanding theoretical understanding.

Findings

Representation as Schur complement of a linear pencil with positive semidefinite matrices

Partial derivatives expressed as sums of squares of polynomials

Provides a new framework for analyzing multivariable positive real functions

Abstract

A rational homogeneous (of degree one) positive real matrix-valued function is presented as the Schur complement of a block of the linear pencil with positive semidefinite matrix coefficients. The partial derivative numerators of a rational positive real function are the sums of squares of polynomials.