Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm

TL;DR

This paper explores the relationship between two methods for parametrizing discrete-time stable all-pass systems, linking the tangential Schur algorithm with balanced state-space realizations, and generalizing scalar results to multivariable systems.

Contribution

It establishes a connection between the tangential Schur algorithm and balanced realizations for multivariable systems, extending scalar case results to higher dimensions.

Findings

Unified parametrization of multivariable systems

Recursive construction of balanced realizations

Direct link between Schur parameters and state-space forms

Abstract

In this paper, the connections are investigated between two different approaches towards the parametrization of multivariable stable all-pass systems in discrete-time. The first approach involves the tangential Schur algorithm, which employs linear fractional transformations. It stems from the theory of reproducing kernel Hilbert spaces and enables the direct construction of overlapping local parametrizations using Schur parameters and interpolation points. The second approach proceeds in terms of state-space realizations. In the scalar case, a balanced canonical form exists that can also be parametrized by Schur parameters. This canonical form can be constructed recursively, using unitary matrix operations. Here, this procedure is generalized to the multivariable case by establishing the connections with the first approach. It gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential Schur algorithm.