The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by ?(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions ?(s) + F(s) that minimize where ?(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.
%0 Journal Article
%1 Glover2007All
%A Glover, Keith
%D 2007
%I Taylor & Francis
%J International Journal of Control
%K 93a15-large-scale-systems 93a30-systems-theory-mathematical-modeling qep
%N 6
%P 1115--1193
%R 10.1080/00207178408933239
%T All optimal Hankel-norm approximations of linear multivariable systems and their
L
,
∞
-error bounds\dag
%U http://dx.doi.org/10.1080/00207178408933239
%V 39
%X The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by ?(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions ?(s) + F(s) that minimize where ?(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.
@article{Glover2007All,
abstract = {{The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by ?(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions ?(s) + F(s) that minimize where ?(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.}},
added-at = {2019-03-01T00:11:50.000+0100},
author = {Glover, Keith},
biburl = {https://www.bibsonomy.org/bibtex/2e3166f4bf2ffc862ed20e65de82b574c/gdmcbain},
citeulike-article-id = {5492304},
citeulike-linkout-0 = {http://dx.doi.org/10.1080/00207178408933239},
citeulike-linkout-1 = {http://www.tandfonline.com/doi/abs/10.1080/00207178408933239},
day = 27,
doi = {10.1080/00207178408933239},
interhash = {9f4f122909fb0be0a35acb7cc7fabb67},
intrahash = {e3166f4bf2ffc862ed20e65de82b574c},
issn = {0020-7179},
journal = {International Journal of Control},
keywords = {93a15-large-scale-systems 93a30-systems-theory-mathematical-modeling qep},
month = mar,
number = 6,
pages = {1115--1193},
posted-at = {2018-09-11 01:02:21},
priority = {2},
publisher = {Taylor \& Francis},
timestamp = {2019-03-01T00:11:50.000+0100},
title = {{All optimal Hankel-norm approximations of linear multivariable systems and their
L
,
∞
-error bounds{\dag}}},
url = {http://dx.doi.org/10.1080/00207178408933239},
volume = 39,
year = 2007
}