A refinement of the generalized chordal distance

TL;DR

This paper introduces a new, easily computable refinement of the generalized chordal distance for SISO systems, based on coprime factorizations, improving on previous metrics by simplifying calculations.

Contribution

It presents a novel refinement of the generalized chordal metric that is easier to compute and applicable to standard classes of stable transfer functions.

Findings

The new metric coincides with Vinnicombe's nu-metric for normalized factorizations.

It simplifies the computation process by not requiring normalized coprime factorizations.

Concrete formulations are provided for standard classes of stable transfer functions.

Abstract

For single input single output systems, we give a refinement of the generalized chordal metric. Our metric is given in terms of coprime factorizations, but it coincides with the extension of Vinnicombe's nu-metric given in earlier work by Ball and Sasane if the coprime factorizations happens to be normalized. The advantage of the metric introduced in this article is its easy computability (since it relies only on coprime factorizations, and does not require normalized coprime factorizations). We also give concrete formulations of our abstract metric for standard classes of stable transfer functions.