{"title":"A method based on linear feasibility tests for full-rank characterization of convex combinations of matrices","authors":"","doi":"10.1016/j.automatica.2024.111842","DOIUrl":null,"url":null,"abstract":"<div><p>Given a set of full-rank matrices <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>p</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></math></span>, this brief paper proposes a method based on linear feasibility tests to determine whether a convex combination <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math></span>, with <span><math><mrow><mi>α</mi><mo>=</mo><msup><mrow><mrow><mo>[</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mspace></mspace><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mo>⋯</mo><mspace></mspace><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>]</mo></mrow></mrow><mrow><mi>T</mi></mrow></msup></mrow></math></span> in the unit simplex <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>, may result in a rank-deficient matrix. The method is based on a sequence of linear programs with increasingly tightened constraints, and is guaranteed to reach an outcome after a finite number of iterations. Given a tolerance <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span> arbitrarily chosen by the user, the method will either (i) certify that <span><math><mrow><mo>∄</mo><mspace></mspace><mi>α</mi><mo>∈</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math></span> such that <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> is rank-deficient or (ii) yield <span><math><mrow><mi>α</mi><mo>∈</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math></span>, <span><math><mrow><mi>v</mi><mo>≠</mo><mn>0</mn></mrow></math></span> such that <span><math><mrow><mo>‖</mo><mi>A</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mi>v</mi><mo>‖</mo><mo>/</mo><mo>‖</mo><mi>v</mi><mo>‖</mo><mo><</mo><mi>ɛ</mi></mrow></math></span>, which certifies that the smallest singular value of <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> is less than <span><math><mi>ɛ</mi></math></span>. This method bridges a gap in the literature, as no other numerically verifiable test for generic <span><math><mi>p</mi></math></span>, <span><math><mi>n</mi></math></span>, <span><math><mi>r</mi></math></span> has been proposed to reach the conclusion (ii). Three numerical examples are provided to showcase the advantages of the proposed method with respect to other tests reported in previous papers. The code employed in this work is available at <span><span>https://github.com/rubensjma/full-rank-characterization</span><svg><path></path></svg></span>.</p></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":null,"pages":null},"PeriodicalIF":4.8000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Automatica","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0005109824003364","RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
Given a set of full-rank matrices , this brief paper proposes a method based on linear feasibility tests to determine whether a convex combination , with in the unit simplex , may result in a rank-deficient matrix. The method is based on a sequence of linear programs with increasingly tightened constraints, and is guaranteed to reach an outcome after a finite number of iterations. Given a tolerance arbitrarily chosen by the user, the method will either (i) certify that such that is rank-deficient or (ii) yield , such that , which certifies that the smallest singular value of is less than . This method bridges a gap in the literature, as no other numerically verifiable test for generic , , has been proposed to reach the conclusion (ii). Three numerical examples are provided to showcase the advantages of the proposed method with respect to other tests reported in previous papers. The code employed in this work is available at https://github.com/rubensjma/full-rank-characterization.
期刊介绍:
Automatica is a leading archival publication in the field of systems and control. The field encompasses today a broad set of areas and topics, and is thriving not only within itself but also in terms of its impact on other fields, such as communications, computers, biology, energy and economics. Since its inception in 1963, Automatica has kept abreast with the evolution of the field over the years, and has emerged as a leading publication driving the trends in the field.
After being founded in 1963, Automatica became a journal of the International Federation of Automatic Control (IFAC) in 1969. It features a characteristic blend of theoretical and applied papers of archival, lasting value, reporting cutting edge research results by authors across the globe. It features articles in distinct categories, including regular, brief and survey papers, technical communiqués, correspondence items, as well as reviews on published books of interest to the readership. It occasionally publishes special issues on emerging new topics or established mature topics of interest to a broad audience.
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