Model reduction in large scale MIMO dynamical systems via the block Lanczos method
Autor(a) principal: | |
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Data de Publicação: | 2008 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Computational & Applied Mathematics |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022008000200006 |
Resumo: | In the present paper, we propose a numerical method for solving the coupled Lyapunov matrix equations A P + P A T + B B T = 0 and A T Q + Q A + C T C = 0 where A is an n ×n real matrix and B, C T are n × s real matrices with rank(B) = rank(C) = s and s << n . Such equations appear in control problems. The proposed method is a Krylov subspace method based on the nonsymmetric block Lanczos process. We use this process to produce low rank approximate solutions to the coupled Lyapunov matrix equations. We give some theoretical results such as an upper bound for the residual norms and perturbation results. By approximating the matrix transfer function F(z) = C (z In - A)-1 B of a Linear Time Invariant (LTI) system of order n by another one Fm(z) = Cm (z Im - Am)-1 Bm of order m, where m is much smaller than n , we will construct a reduced order model of the original LTI system. We conclude this work by reporting some numerical experiments to show the numerical behavior of the proposed method. |
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Computational & Applied Mathematics |
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Model reduction in large scale MIMO dynamical systems via the block Lanczos methodcoupled Lyapunov matrix equationsKrylov subspace methodsnonsymmetric block Lanczos processreduced order modeltransfer functionsIn the present paper, we propose a numerical method for solving the coupled Lyapunov matrix equations A P + P A T + B B T = 0 and A T Q + Q A + C T C = 0 where A is an n ×n real matrix and B, C T are n × s real matrices with rank(B) = rank(C) = s and s << n . Such equations appear in control problems. The proposed method is a Krylov subspace method based on the nonsymmetric block Lanczos process. We use this process to produce low rank approximate solutions to the coupled Lyapunov matrix equations. We give some theoretical results such as an upper bound for the residual norms and perturbation results. By approximating the matrix transfer function F(z) = C (z In - A)-1 B of a Linear Time Invariant (LTI) system of order n by another one Fm(z) = Cm (z Im - Am)-1 Bm of order m, where m is much smaller than n , we will construct a reduced order model of the original LTI system. We conclude this work by reporting some numerical experiments to show the numerical behavior of the proposed method.Sociedade Brasileira de Matemática Aplicada e Computacional2008-01-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022008000200006Computational & Applied Mathematics v.27 n.2 2008reponame:Computational & Applied Mathematicsinstname:Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)instacron:SBMAC10.1590/S0101-82052008000200006info:eu-repo/semantics/openAccessHeyouni,M.Jbilou,K.Messaoudi,A.Tabaa,K.eng2008-07-21T00:00:00Zoai:scielo:S1807-03022008000200006Revistahttps://www.scielo.br/j/cam/ONGhttps://old.scielo.br/oai/scielo-oai.php||sbmac@sbmac.org.br1807-03022238-3603opendoar:2008-07-21T00:00Computational & Applied Mathematics - Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)false |
dc.title.none.fl_str_mv |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method |
title |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method |
spellingShingle |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method Heyouni,M. coupled Lyapunov matrix equations Krylov subspace methods nonsymmetric block Lanczos process reduced order model transfer functions |
title_short |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method |
title_full |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method |
title_fullStr |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method |
title_full_unstemmed |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method |
title_sort |
Model reduction in large scale MIMO dynamical systems via the block Lanczos method |
author |
Heyouni,M. |
author_facet |
Heyouni,M. Jbilou,K. Messaoudi,A. Tabaa,K. |
author_role |
author |
author2 |
Jbilou,K. Messaoudi,A. Tabaa,K. |
author2_role |
author author author |
dc.contributor.author.fl_str_mv |
Heyouni,M. Jbilou,K. Messaoudi,A. Tabaa,K. |
dc.subject.por.fl_str_mv |
coupled Lyapunov matrix equations Krylov subspace methods nonsymmetric block Lanczos process reduced order model transfer functions |
topic |
coupled Lyapunov matrix equations Krylov subspace methods nonsymmetric block Lanczos process reduced order model transfer functions |
description |
In the present paper, we propose a numerical method for solving the coupled Lyapunov matrix equations A P + P A T + B B T = 0 and A T Q + Q A + C T C = 0 where A is an n ×n real matrix and B, C T are n × s real matrices with rank(B) = rank(C) = s and s << n . Such equations appear in control problems. The proposed method is a Krylov subspace method based on the nonsymmetric block Lanczos process. We use this process to produce low rank approximate solutions to the coupled Lyapunov matrix equations. We give some theoretical results such as an upper bound for the residual norms and perturbation results. By approximating the matrix transfer function F(z) = C (z In - A)-1 B of a Linear Time Invariant (LTI) system of order n by another one Fm(z) = Cm (z Im - Am)-1 Bm of order m, where m is much smaller than n , we will construct a reduced order model of the original LTI system. We conclude this work by reporting some numerical experiments to show the numerical behavior of the proposed method. |
publishDate |
2008 |
dc.date.none.fl_str_mv |
2008-01-01 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022008000200006 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1807-03022008000200006 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.1590/S0101-82052008000200006 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
text/html |
dc.publisher.none.fl_str_mv |
Sociedade Brasileira de Matemática Aplicada e Computacional |
publisher.none.fl_str_mv |
Sociedade Brasileira de Matemática Aplicada e Computacional |
dc.source.none.fl_str_mv |
Computational & Applied Mathematics v.27 n.2 2008 reponame:Computational & Applied Mathematics instname:Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC) instacron:SBMAC |
instname_str |
Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC) |
instacron_str |
SBMAC |
institution |
SBMAC |
reponame_str |
Computational & Applied Mathematics |
collection |
Computational & Applied Mathematics |
repository.name.fl_str_mv |
Computational & Applied Mathematics - Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC) |
repository.mail.fl_str_mv |
||sbmac@sbmac.org.br |
_version_ |
1754734890165731328 |