Extension of the $\nu$-metric: the $H^\infty$ case

TL;DR

This paper extends the abstract $ u$-metric to the $H^ty$ case, connecting it with Vinnicombe's $ u$-metric for nonrational plants, thus broadening its applicability in infinite-dimensional control systems.

Contribution

It provides a concrete example verifying the assumptions of the abstract $ u$-metric in the $H^ty$ setting, linking it to Vinnicombe's metric for nonrational transfer functions.

Findings

Verified assumptions for the abstract $ u$-metric in the $H^ty$ case

Connected the abstract $ u$-metric with Vinnicombe's nonrational plant metric

Extended the $ u$-metric framework to infinite-dimensional systems

Abstract

An abtract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the classical $\nu$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. In this short note, we give an additional concrete special instance of the abstract $\nu$-metric, by verifying all the assumptions demanded in the abstract set-up. This example links the abstract $\nu$-metric with the one proposed by Vinnicombe as a candidate for the $\nu$-metric for nonrational plants.