State-space algorithm for the computation of superoptimal matrix interpolating functions

A state-space algorithm is studied which generates the (unique) superoptimal Nehari extension of a general rational matrix $. The procedure is to use a set of all-pass transformations to sequentially minimize each frequency-dependent singular value (of the interpolating function) in a dimension peeling algorithm. These all-pass transformations are determined by the maximal Schmidt pairs of a sequence of Hankel operators. The process terminates when the original problem is reduced to one of rank one; at this stage all the available degrees of freedom have been exhausted. The work is an extension of that by Young (1986) and gives a ‘concrete’ state-space implementation of his operator-theoretic arguments. In addition, bounds are given on the minimum achievable values for s₁ ^∞ (E) = sup_ωεRSi (E(jω)), i = 1, 2,..., rank (G₀), and also the McMillan degree of the final superoptimal extension. Here S_i.(.) denotes the ith singular value of a (frequency-dependent) matrix, and the numbering is taken to be in decreasing order of magnitude. The algorithm has the property that it may be stopped after minimizing s_i ^∞ (.),i = 1. 2,...,l< rank (G₀) if it continues further it is deemed ‘not worth it’ in some sense. A premature termination of the algorithm carries with it an expected saving in computational effort and a predictable reduction in the degree of the extension. A shortened version of the present work has already appeared in work by Limebeeref al, (1987).