On two natural extensions of Vinnicombe's metric: their noncoincidence yet equivalence on stabilizable plants over A_+

TL;DR

This paper compares two extensions of Vinnicombe's metric for stabilizable plants over A_+, showing they differ but induce the same topology, which aligns with the gap metric topology.

Contribution

It introduces and compares two natural extensions of Vinnicombe's metric for plants over A_+, demonstrating their non-coincidence yet topological equivalence.

Findings

The two metrics do not coincide on stabilizable plants over A_+.

Both metrics induce the same topology on the set of stabilizable plants.

This topology coincides with the gap metric topology.

Abstract

Let A_+ be the ring of Laplace transforms of complex Borel measures on R with support in [0,+\infty) which do not have a singular nonatomic part. We compare the nu-metric d_{A_+} for stabilizable plants over A_+ given in the article by Ball and Sasane [2010], with yet another metric d_{H^\infty}|_{A_+}, namely the one induced by the metric d_{H^\infty} for the set of stabilizable plants over H^\infty given in teh article by Sasane in 2011. Both d_{A_+} and d_{H^\infty} coincide with the classical Vinnicombe metric defined for rational transfer functions, but we show here by means of an example that these two possible extensions of the classical nu-metric for plants over A_+ do not coincide on the set of stabilizable plants over A_+. We also prove that they nevertheless give rise to the same topology on stabilizable plants over A_+, which in turn coincides with the gap metric topology.