Vector Spaces of Linearizations for Multivariable State-Space Systems

TL;DR

This paper explores vector spaces of matrix pencils related to multivariable state-space systems, demonstrating that most are linearizations of the transfer function and constructing symmetric/Hermitian linearizations for specific cases.

Contribution

It introduces and analyzes two vector spaces of matrix pencils associated with multivariable systems, showing most are linearizations and providing symmetric/Hermitian linearizations for regular symmetric systems.

Findings

Most pencils in the defined vector spaces are linearizations of G(λ)

Constructs symmetric/Hermitian linearizations for regular symmetric G(λ)

Provides a framework for analyzing matrix pencils in multivariable systems

Abstract

Consider a multivariable state space system and associated transfer function G({\lambda}). The aim of this paper is to define and analyze two vector spaces of matrix pencils associated with the matrix G({\lambda}) and show that almost all of these pencils are linearizations of G({\lambda}). We also construct symmetric/Hermitian linearizations of G({\lambda}) when G({\lambda}) is regular and symmetric/Hermitian.