Matrix rational H/sup 2/ approximation: a state-space approach using Schur parameters

This paper deals with the problem of computing a best stable rational L/sup 2/ approximation of specified order to a given multivariable transfer function. The problem is equivalently formulated as a minimization problem over the manifold of stable all-pass (or lossless) transfer functions of fixed order. Some special Schur parameters are used to describe this manifold. Such a description presents numerous advantages: it takes into account the stability constraint, possesses a good numerical behavior and provides a model in state-space form, which is very useful in practice. A rigorous and convergent algorithm is proposed to compute local minima which has been implemented using standard MATLAB subroutines. The effectiveness of our approach to solve model reduction problems as well as identification problems in frequency domain is demonstrated through several examples, including real-data simulations.

[1]  V. Potapov.  The multiplicative structure of J-contractive matrix functions , 1960 .

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

[3]  M. Olivi,et al.  Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm , 1999, 1999 European Control Conference (ECC).

[4]  Y. Hung,et al.  Multivariable Feedback: A Quasi-Classical Approach , 1982 .

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

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

[7]  K. Hoffman.  Banach Spaces of Analytic Functions , 1962 .

[8]  Martine Olivi,et al.  Identification and rational L2 approximation A gradient algorithm , 1991, Autom..

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

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

[11]  Laurent Baratchart,et al.  Existence and Generic Properties of L2 Approximants for Linear Systems , 1986 .

[12]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[13]  James Lam,et al.  An approximate approach to H2 optimal model reduction , 1999, IEEE Trans. Autom. Control..

[14]  D. Bernstein,et al.  The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore , 1985 .