Linear Fractional Transformations and Balanced Realization of Discrete-Time Stable All-Pass Systems R.L.M. Peeters 1 , B. Hanzon 2 , M. Olivi 3 1 Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht, The Netherlands, Fax: +31-43-3884910, Email: ralf.peeters@math.unimaas.nl 2 Dept. Econometrics, FEWEC, Vrije Universiteit, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands, Fax: +31-20-4446020, Email: bhanzon@econ.vu.nl 3 INRIA Sophia-Antipolis, B.P. 93, 06902 Sophia-Antipolis Cedex, France, Fax: +33-4 92 38 78 58, Email: olivi@sophia.inria.fr Keywords : All-pass systems, Balanced realization, Linear fractional transformation, Schur parameters, Tangential Schur algorithm. Abstract The tangential Schur algorithm provides a means of constructing the class of p × p discrete- time stable all-pass transfer functions of a prescribed McMillan degree n. In each of its n iteration steps a linear fractional transformation is employed which is associated with a J - inner rational matrix of McMillan degree 1 involving certain parameters. In this set-up, the issue of generating corresponding state-space realizations in terms of these parameters is not addressed. In the present contribution we present a unified framework in which linear frac- tional transformations on transfer functions are represented by corresponding linear fractional transformations on state-space realization matrices. When applied to the case of the tan- gential Schur algorithm, minimal balanced realizations of stable all-pass systems in terms of the parameters used are obtained. The balanced state-space approach of [9] for SISO stable all-pass systems is incorporated as a special case. 1 Introduction Stable all-pass systems of finite order have several applications in linear systems theory. Within the fields of system identification, approximation and model reduction, they have been used in connection with the Douglas-Shapiro-Shields factorization, see e.g., [3, 2, 13, 7], to obtain effective algorithms for various purposes. The differential structure of the one-to-one related class of inner functions has been studied in [1]. There, a parametrization has been obtained in the multivariable case by means of the tangential Schur algorithm which involves Schur parameter vectors, interpo- lation points and normalized direction vectors. In the scalar case, a single coordinate chart suffices to entirely describe the manifold of stable all-pass (or inner) systems of a fixed finite order. In 1