Reformulation of the extension of the $\nu$-metric for $H^{\infty}$

TL;DR

This paper demonstrates that the extension of the nu-metric for H-infinity transfer functions aligns with the abstract framework previously established, unifying different approaches in robust control theory.

Contribution

It shows that the nu-metric extension for H-infinity fits into the existing abstract framework, clarifying its theoretical foundation.

Findings

The nu-metric for H-infinity is consistent with the abstract framework.

The paper introduces a Banach algebra as the inductive limit of C*-algebras.

The extended nu-metric matches the one previously defined by Sasane.

Abstract

The classical nu-metric introduced by Vinnicombe in robust control theory for rational plants was extended to classes of nonrational transfer functions in Ball and Sasane [Complex Analysis and Operator Theory; 2012]. In Sasane [Mathematics of Control and Related Fields; 2012], an extension of the classical nu-metric was given when the underlying ring of stable transfer functions is the Hardy algebra, H^infty. However, this particular extension to H^infty did not directly fit in the abstract framework given in Ball and Sasane [CAOT 2012]. In this paper we show that the case of H^infty also fits into the general abstract framework in [BS CAOT 2012] and that the nu-metric defined in this setting is identical to the extension of the nu-metric defined in Sasane [MCRF 2012]. This is done by introducing a particular Banach algebra, which is the inductive limit of certain C*-algebras.