Tutorial-Modelred

Tutorial-Modelred - WORKING PAPER A TUTORIAL ON HANKELNORM...

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W O R K I N G PAPER A TUTORIAL ON HANKELNORM APPROXIMATIONS K. Glovcr October 1989 WP-89079 - I n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis
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A TUTORIAL ON HANKELNORM APPROXIMATIONS K. Glover October 1989 WP-84079 Cambridge University Engineering Department, Control and Management Systems Division, England. Working Papere are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria
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FOREWORD This is a contribution to the activity on the topic From Data to Model initiated at the Systems and Decision Sciences Program of IIASA by Professor J. C. Willems. A. Kurzhanski Program Leader System and Decision Sciences Program.
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A TUTORIAL ON HANKEL-NORM APPROXlMATlON KEITH GLOVER Abstract A self-contained derivation is presented of the characterization of all optimal Hankel-norm approximations to a given matrix-valued transfer function. The approach involves a state-space characterization of dl-pass systems as in the author's previous work, but has been greatly simplified. A section of preliminary results is included giving general results on linear fractional transformations, Hankel operators and all-pass systems. These results then can be applied to give the characterization of all optimal Hankel-norm approximations of a given stable transfer function. Frequency response bounds for these approximations are then derived from finite rank perturbation results. Keywords Hankel norm, Hardy spaces, H,, model order reduction, rational approximation.
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1 INTRODUCTION An important question when modelling dynamic systems is whether a model can be simplified without undue loss of accuracy. A measure of the complexity of a linear state- space model, is the dimension, n, of its state vector z(t). In (1.1)-(1.2), u(t) E Cm, z(t) E Cn, y(t) E CP for all t, and A, B, C are complex matrices of compatible dimensions. Low order models will give more efficient simulations and, for example, control system design calculations. Approximating (1.1)-(1.2) by a reduced order system, where i(t) E ck, k < n, is termed a model reduction problem. Substantial progress has been made on problems of this type in recent years by the use of truncated balanced realizations as introduced by Moore (1981) and optimal Hankel-norm approximations as given in Adamjan, Arov and Krein (1971). The first method truncates states from a particular realization but has not been shown to be optimal in any sense; whereas the second method minimizes a specific norm of the error between (1.1)-(1.2) and (1.3)-(1.4). Both methods have been shown to give excellent results in many application areas.
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