Extension of the $\nu$-metric for stabilizable plants over $H^\infty$

TL;DR

This paper extends the $ u$-metric to infinite-dimensional control systems by verifying its applicability over the Hardy algebra $H^infty$, addressing an open question in the field.

Contribution

It provides a concrete example of the abstract $ u$-metric for $H^infty$, confirming its validity for stabilizable plants in this setting.

Findings

Confirmed the assumptions of the abstract $ u$-metric for $H^infty$

Extended the $ u$-metric to nonrational transfer functions

Resolved an open question in the $ u$-metric theory

Abstract

An abstract $\nu$-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 important concrete special instance of the abstract $\nu$-metric, by verifying that all the assumptions demanded in the abstract set-up are satisfied when the ring of stable transfer functions is the Hardy algebra $H^\infty$. This settles the open question implicit in \cite{BalSas2}.