State-space approach to zero-modules of proper transfer functions

TL;DR

This paper provides a state-space framework to analyze zero-modules of proper transfer functions, linking algebraic zero concepts with geometric control invariants, and introduces methods to simplify transfer functions via inner functions.

Contribution

It offers a novel state-space description of zero-modules for proper transfer functions, connecting algebraic and geometric perspectives, and proposes explicit equations for transfer function transformation.

Findings

Explicit state-space equations for zero-modules

Method to transform transfer functions into invertible form

Connection between zeros and invariant subspaces

Abstract

The poles and zeros of a transfer function can be studied by various means. The main motivation of the present paper is to give a state-space description of the module theoretic definition of zeros introduced and analyzed by Wyman et al. (1989, 1991). This analysis is carried out for proper transfer functions. The obtained explicit equations determined by the system matrices are used for defining two inner functions to transform the original transfer function into a square, invertible one via multiplication eliminating the "generic" zeros corresponding to the kernel and the image of the transfer function. As it is well-known the zeros are connected to various invariant subspaces arising in geometric control, see e.g. Aling and Schumacher (1984) for a complete description. The connections to these subspaces are also mentioned in the paper.