Fiedler Linearizations of Multivariable State-Space System and its Associated System Matrix

TL;DR

This paper introduces Fiedler pencils for multivariable state-space systems, providing a systematic way to linearize the associated system matrix for eigenvalue analysis.

Contribution

It extends Fiedler linearizations to multivariable systems and presents an algorithm for their construction, enhancing numerical methods for system analysis.

Findings

Fiedler pencils are proven to be linearizations of the system matrix.

An explicit algorithm for constructing Fiedler pencils is provided.

The approach generalizes existing linearization techniques to multivariable systems.

Abstract

Linearization is a standard method in the computation of eigenvalues and eigenvectors of matrix polynomials. In the last decade a variety of linearization methods have been developed in order to deal with algebraic structures and in order to construct efficient numerical methods. An important source of linearizations for matrix polynomials are the so called Fiedler pencils, which are generalizations of the Frobenius companion form and these linearizations have been extended to regular rational matrix function which is the transfer function of LTI State-space system in [1, 6]. We consider a multivariable state-space system and its associated system matrix S({\lambda}). We introduce Fiedler pencils of S({\lambda}) and describe an algorithm for their construction. We show that Fiedler pencils are linearizations of the system matrix S({\lambda}).