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.
Categories: math.OC math.CV math.FA