Algebraic Construction of Normalized Coprime Factors for Delay Systems

Abstract

We introduce an algebraic approach to the problem of constructing normalized coprime factorizations for retarded delay systems. A parametrization is given of all the possible factorizations that can be obtained by solving algebraic equations over the field generated by s and e Às . This enables us to provide a means of determining when such factors can be calculated by solving such algebraic equations, and to show that in general they cannot. Some illustrative examples are given.

A Refinement of the Generalized Chordal Distance

“…It was shown in [8] that an algebraic spectral factorization leading to normalized coprime factors is not possible in this example. So calculating the ν-metric between this plant and a perturbed one, say p a := 1 s − ae −s , is problematic.…”

“…This is a troublesome aspect of the theory, since (1) Although merely coprime factorizations may exist, a normalized coprime factorization may fail to exist: for example in the article [14] by Sergei 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 Jonathan Partington and Gregory 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 ).…”

“…(2) Even if they exist, normalized coprime factorizations might be impossible to find using a constructive procedure: for example, in the paper [8] by Jonathan Partington and Gregory 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 ).…”

Reformulation of the extension of theν-metric forH∞

“…This factorization was found using the algorithm given in [7,Example 4.1]. Set s := ϕ(z) for z ∈ D, and consider the two plants…”