Tomomichi Hagiwara , Taichi Yuyama , Jung Hoon Kim
{"title":"L∞/L2 Hankel norm analysis of linear periodically time-varying systems and detection of critical instants","authors":"Tomomichi Hagiwara , Taichi Yuyama , Jung Hoon Kim","doi":"10.1016/j.sysconle.2025.106235","DOIUrl":null,"url":null,"abstract":"<div><div>This paper deals with the Hankel norm analysis of linear periodically time-varying (LPTV) systems, where <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is taken as the (past) input space and the (future) output is regarded as an element in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span>. This norm is known to be important because of its coincidence with the well-known <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> norm for the special case of single-output linear time-invariant systems. For the period <span><math><mi>h</mi></math></span> of LPTV systems, an arbitrary instant <span><math><mrow><mi>Θ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>h</mi><mo>)</mo></mrow></mrow></math></span> is first taken to separate past and future, and then the quasi <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> Hankel norm at <span><math><mi>Θ</mi></math></span> is defined. A computation method for the quasi <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> Hankel norm for each <span><math><mi>Θ</mi></math></span> is further derived, and for <em>the</em> <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> Hankel norm defined as the supremum of the quasi <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> Hankel norms over <span><math><mrow><mi>Θ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>h</mi><mo>)</mo></mrow></mrow></math></span>, its alternative and direct computation method is also provided, which is actually completely free from dealing with any quasi <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> Hankel norms. A relevant question of whether the supremum is attained as the maximum is also studied, in which case each maximum-attaining <span><math><mi>Θ</mi></math></span> is called a critical instant. In particular, it is discussed when and how the existence/absence of a critical instant can be determined and some or all critical instants can be detected without directly computing all (or any of) the quasi <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> Hankel norms over <span><math><mrow><mi>Θ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>h</mi><mo>)</mo></mrow></mrow></math></span>. Finally, numerical examples are provided to illustrate the arguments of this paper.</div></div>","PeriodicalId":49450,"journal":{"name":"Systems & Control Letters","volume":"205 ","pages":"Article 106235"},"PeriodicalIF":2.5000,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Systems & Control Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167691125002178","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper deals with the Hankel norm analysis of linear periodically time-varying (LPTV) systems, where is taken as the (past) input space and the (future) output is regarded as an element in . This norm is known to be important because of its coincidence with the well-known norm for the special case of single-output linear time-invariant systems. For the period of LPTV systems, an arbitrary instant is first taken to separate past and future, and then the quasi Hankel norm at is defined. A computation method for the quasi Hankel norm for each is further derived, and for the Hankel norm defined as the supremum of the quasi Hankel norms over , its alternative and direct computation method is also provided, which is actually completely free from dealing with any quasi Hankel norms. A relevant question of whether the supremum is attained as the maximum is also studied, in which case each maximum-attaining is called a critical instant. In particular, it is discussed when and how the existence/absence of a critical instant can be determined and some or all critical instants can be detected without directly computing all (or any of) the quasi Hankel norms over . Finally, numerical examples are provided to illustrate the arguments of this paper.
期刊介绍:
Founded in 1981 by two of the pre-eminent control theorists, Roger Brockett and Jan Willems, Systems & Control Letters is one of the leading journals in the field of control theory. The aim of the journal is to allow dissemination of relatively concise but highly original contributions whose high initial quality enables a relatively rapid review process. All aspects of the fields of systems and control are covered, especially mathematically-oriented and theoretical papers that have a clear relevance to engineering, physical and biological sciences, and even economics. Application-oriented papers with sophisticated and rigorous mathematical elements are also welcome.