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

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.

[1]  Raimund J. Ober,et al.  Asymptotically Stable All-pass Transfer Functions: Canonical Form, Parametrization and Realization , 1987 .

[2]  Bernard Hanzon,et al.  Balanced Parametrizations of Stable SISO All-Pass Systems in Discrete Time , 2000, Math. Control. Signals Syst..

[3]  Thomas Kailath,et al.  Linear Systems , 1980 .

[4]  Bernard Hanzon,et al.  Overlapping Block-Balanced Canonical Forms and Parametrizations: The Stable SISO Case , 1997 .

[5]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .

[6]  Michel Verhaegen,et al.  Linear and Non-linear System Identification Using Separable Least-Squares , 1997, Eur. J. Control.

[7]  Paul Van Dooren,et al.  On Sigma-lossless transfer functions and related questions , 1983 .

[8]  Martine Olivi,et al.  Inner-unstable factorization of stable rational transfer functions , 1991 .

[9]  J. Leblond,et al.  Weighted $H^2$ Approximation of Transfer Functions , 1998 .

[10]  Nicholas Young,et al.  Linear fractional transformations in rings and modules , 1984 .

[11]  B. Hanzon,et al.  OVERLAPPING BLOCK-BALANCED CANONICAL FORMS FOR VARIOUS CLASSES OF LINEAR SYSTEMS , 1998 .

[12]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[13]  L. Rodman,et al.  Interpolation of Rational Matrix Functions , 1990 .

[14]  Harry Dym,et al.  On Reproducing Kernel Spaces, J Unitary Matrix Functions, Interpolation and Displacement Rank , 1989 .

[15]  T. Constantinescu,et al.  Schur's Algorithm and Several Applications , 1992 .

[16]  A. Shields,et al.  Cyclic vectors and invariant subspaces for the backward shift operator , 1970 .

[17]  Joseph A. Ball,et al.  Review: Harry Dym, $J$ Contractive matrix functions, reproducing kernel Hilbert spaces and interpolation , 1990 .

[18]  Raimund J. Ober,et al.  Balanced realizations: canonical form, parametrization, model reduction , 1987 .

[19]  H. Kimura.  Chain-Scattering Approach to H∞Control , 1996 .

[20]  Bernard Hanzon,et al.  Constructive algebra methods for the L2-problem for stable linear systems , 1996, Autom..

[21]  Martine Olivi,et al.  Matrix Rational H 2 Approximation: A Gradient Algorithm Based on Schur Analysis , 1995 .

[22]  Laurent Baratchart,et al.  On the differential structure of matrix-valued rational inner functions , 1992 .