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

A novel methodology for Hankel approximation and H^∞-optimization problems is presented, based on a new formulation of the Adamjan-Arov-Krein one-step extension problem. The problem is solved by the Sarason interpolation theorem. 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 wayt to achieve this. In the present method, if one obtains the minimal balanced realization of a given transfer function matrix, then one also obtains the Hankel approximants of that transfer function at the same time.