A Refinement of the Generalized Chordal Distance

Abstract

For single input single output systems, we give a refinement dc,r of the generalized chordal metric dc introduced in [A. J. Sasane, Complex Anal. Oper. Theory, 7 (2013), pp. 1345-1356. Our metric dc,r is given in terms of coprime factorizations, but it coincides with the extension of Vinnicombe's ν-metric given in Ball and Sasane [Complex Anal. Oper. Theory, 6 (2012), pp. 65-89] if the coprime factorizations happen to be normalized. The advantage of the metric dc,r introduced in this article is its easy computability (since it relies only on coprime factorizations and does not require normalized coprime factorizations). We also give concrete formulations of our abstract metric for standard classes of stable transfer functions. Introduction. The Vinnicombe ν-metric introduced in [16]and its abstract version given in [1] both rely on finding normalized coprime factorizations. This is a troublesome aspect of the theory, because of the following points:1. Although merely coprime factorizations may exist, a normalized coprime factorization may fail to exist over the original algebra. For example, in the article [14] by Treil, it was shown that the set of plants in the field of fractions of the disk algebra possessing normalized coprime factorizations is strictly contained in the set of plants possessing coprime factorizations. 2. Even if they exist, normalized coprime factorizations might be impossible to find using a constructive procedure: for example, in the paper [8] by Partington and Sankaran, it is shown that in the case of delay systems, in general the relevant spectral factorizations for finding normalized coprime factorizations cannot be determined by solving any finite system of polynomial equations over the field R(s, e s ). The goal in this paper is to show that this problematic feature of the ν-metric can be eliminated, at least in the case of single input single output systems, by redefining the ν-metric, which relies only on coprime factorizations, rather than normalized coprime factorizations. The starting point is the generalized chordal distance d c introduced in [13] given in terms of coprime factorizations; then a refinement d c,r of this chordal distance akin to the ν-metric d ν is considered. It turns out that the metric d c,r coincides with the ν-metric if one has normalized coprime factorizations at hand. But since our metric is in fact only defined using coprime factorizations, the burden of working with normalized coprime factorizations is completely eliminated. Our main results are then that d c,r defines a metric on the set of all elements admitting a coprime factorization, and stabilizability is a robust property of the plant. The precise statements of the results are given in Theorems 4.3 and 4.5, after the notation has been introduced in the next section. *