Linear Fractional Transformations and Balanced Realization of Discrete-Time Stable All-Pass Systems

Abstract

The tangential Schur algorithm provides a means of constructing the class of multivariable discrete-time stable all-pass transfer functions of a fixed finite McMillan degree. In each iteration step a linear fractional transformation is employed which is associated with a J-inner rational matrix of McMillan degree 1. In this set-up, the emphasis is exclusively on transfer functions. In the present contribution we present a unified framework in which linear fractional transformations on transfer functions are represented by corresponding linear fractional transformations on state-space realization matrices. When applied to the case of the tangential Schur algorithm, minimal balanced realizations of stable all-pass systems in terms of the parameters used are obtained. The balanced state-space approach of (Hanzon and Peeters, 2000) is incorporated as a special case.