Once Again: The Adamjan-Arov-Krein Approximation Theory

Abstract

This paper is largely devoted to a tutorial presentation of an extensive background material related to a Glover-like solution of the AAK problem of optimal Hankel-norm approximation for discrete-time multivariable systems of finite degree. A special emphasis is laid on the rationale for the Hankel-matrix approach and on the singular value decomposition of bounded infinite Hankel matrices of finite rank (with some original matrix-theoretic complements). Among the main AAK results, which are briefly summarized here in a form suitable for system-theoretic applications, one of the most remarkable states that the number of zeros (inside the unit circle) of the rational functions obtained by z-transforming the Schmidt pair belonging to any singular value of such a Hankel matrix is related simply to its serial number. Unlike the classical proofs, which are notoriously sophisticated, the pure matrix proof we outline here is both reasonably simple and transparent.