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Article: Optimal representation matrices for solving polynomial systems via LMI

TitleOptimal representation matrices for solving polynomial systems via LMI
Authors
KeywordsSystem of polynomial equations and inequalities
Linear matrix inequality (LMI)
Square matricial representation (SMR)
Control systems
Issue Date2008
PublisherAcademic Publications. The Journal's web site is located at http://www.ijpam.eu/index.html
Citation
International Journal of Pure and Applied Mathematics, 2008, v. 45 n. 3, p. 397-412 How to Cite?
AbstractIt is known that LMI can be useful for solving systems of polynomial equations and inequalities provided that the dimensions of the null spaces of some matrices representing the systems are smaller than certain thresholds. The first contribution of this paper is to show that, unfortunately, there always exist representation matrices obtainable in the LMI optimization for which these dimensions are larger than the allowed thresholds and, consequently, the extraction mechanism of the sought solutions cannot be performed. Moreover, it is also shown that, if there exist representation matrices for which these dimensions are smaller than the allowed thresholds, then these matrices can be arbitrarily ill-conditioned since the smallest non-zero eigenvalue can be arbitrarily close to zero, hence affecting the computation of the null spaces. Another contribution is to show that an upper bound to the dimension of these null spaces can be imposed in a non-conservative way by adding suitable LMIs. This allows one to obtain the null spaces with the smallest dimension via a finite sequence of feasibility tests. Moreover, the introduced LMIs also allow to avoid ill-conditioned representation matrices, if possible, by simply turning the feasible tests into suitable convex maximizations.
Persistent Identifierhttp://hdl.handle.net/10722/73540
ISSN
2015 SCImago Journal Rankings: 0.299

 

DC FieldValueLanguage
dc.contributor.authorChesi, Gen_HK
dc.date.accessioned2010-09-06T06:52:20Z-
dc.date.available2010-09-06T06:52:20Z-
dc.date.issued2008en_HK
dc.identifier.citationInternational Journal of Pure and Applied Mathematics, 2008, v. 45 n. 3, p. 397-412en_HK
dc.identifier.issn1311-8080en_HK
dc.identifier.urihttp://hdl.handle.net/10722/73540-
dc.description.abstractIt is known that LMI can be useful for solving systems of polynomial equations and inequalities provided that the dimensions of the null spaces of some matrices representing the systems are smaller than certain thresholds. The first contribution of this paper is to show that, unfortunately, there always exist representation matrices obtainable in the LMI optimization for which these dimensions are larger than the allowed thresholds and, consequently, the extraction mechanism of the sought solutions cannot be performed. Moreover, it is also shown that, if there exist representation matrices for which these dimensions are smaller than the allowed thresholds, then these matrices can be arbitrarily ill-conditioned since the smallest non-zero eigenvalue can be arbitrarily close to zero, hence affecting the computation of the null spaces. Another contribution is to show that an upper bound to the dimension of these null spaces can be imposed in a non-conservative way by adding suitable LMIs. This allows one to obtain the null spaces with the smallest dimension via a finite sequence of feasibility tests. Moreover, the introduced LMIs also allow to avoid ill-conditioned representation matrices, if possible, by simply turning the feasible tests into suitable convex maximizations.-
dc.languageengen_HK
dc.publisherAcademic Publications. The Journal's web site is located at http://www.ijpam.eu/index.htmlen_HK
dc.relation.ispartofInternational Journal of Pure and Applied Mathematicsen_HK
dc.subjectSystem of polynomial equations and inequalities-
dc.subjectLinear matrix inequality (LMI)-
dc.subjectSquare matricial representation (SMR)-
dc.subjectControl systems-
dc.titleOptimal representation matrices for solving polynomial systems via LMIen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=1311-8080&volume=&spage=397&epage=412&date=2008&atitle=Optimal+representation+matrices+for+solving+polynomial+systems+via+LMIen_HK
dc.identifier.emailChesi, G: chesi@eee.hku.hken_HK
dc.identifier.authorityChesi, G=rp00100en_HK
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.hkuros149165en_HK
dc.identifier.volume45-
dc.identifier.issue3-
dc.identifier.spage397-
dc.identifier.epage412-
dc.publisher.placeBulgaria-

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