One-step extension approach to optimal Hankel-norm approximation and H∞-optimization problems

A methodology for Hankel approximation and H^∞ -optimization problems is presented, based on a formulation of the one-step extension problem of V.M. Adamjan, D.Z. Arov and M.G. Krein (1971). The problem is solved by the interpolation theorem of D. Sarason (1976). The parametrization of all optimal solutions is given in terms of the eigenvalue decomposition of a Hermitian matrix composed directly from the coefficients of the given transfer function matrix. The proposed method does not require an initial balanced realization; nevertheless, the method itself provides a very simple, natural way to achieve this. In the method, if the minimal balanced realization of a given transfer function matrix is obtained, then the Hankel approximants of that transfer function are obtained at the same time