A new quasi-finite-rank approximation of compression operators on L∞[0,H) with applications to sampled-data and time-delay systems: Piecewise linear kernel approximation approach
{"title":"A new quasi-finite-rank approximation of compression operators on L∞[0,H) with applications to sampled-data and time-delay systems: Piecewise linear kernel approximation approach","authors":"","doi":"10.1016/j.jfranklin.2024.107271","DOIUrl":null,"url":null,"abstract":"<div><p>This paper provides a new quasi-finite-rank approximation (QFRA) of infinite-rank compression operators defined on the Banach space <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>, which are associated with tractable representations of infinite-dimensional systems such as time-delay and sampled-data systems. We first formulate the QFRA by an optimization problem with a matrix-valued parameter <span><math><mi>X</mi></math></span> to minimize the associated error in terms of the <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>-induced norm. To facilitate solving the optimization problem, we next employ the piecewise linear kernel approximation (PLKA) technique, by which the optimization problem is then converted to a linear programming (LP) problem. The solution of the LP problem is shown to converge to the optimal solution of the original QFRA with the order of <span><math><mrow><mn>1</mn><mo>/</mo><mi>M</mi></mrow></math></span>, where <span><math><mi>M</mi></math></span> is the PLKA parameter. The PLKA-based QFRA is shown to lead to practical methods of the stability analysis for time-delay systems and the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> optimal controller synthesis for sampled-data systems. Finally, the overall arguments developed in this paper are demonstrated through some numerical and experimental studies.</p></div>","PeriodicalId":17283,"journal":{"name":"Journal of The Franklin Institute-engineering and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.7000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of The Franklin Institute-engineering and Applied Mathematics","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0016003224006926","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper provides a new quasi-finite-rank approximation (QFRA) of infinite-rank compression operators defined on the Banach space , which are associated with tractable representations of infinite-dimensional systems such as time-delay and sampled-data systems. We first formulate the QFRA by an optimization problem with a matrix-valued parameter to minimize the associated error in terms of the -induced norm. To facilitate solving the optimization problem, we next employ the piecewise linear kernel approximation (PLKA) technique, by which the optimization problem is then converted to a linear programming (LP) problem. The solution of the LP problem is shown to converge to the optimal solution of the original QFRA with the order of , where is the PLKA parameter. The PLKA-based QFRA is shown to lead to practical methods of the stability analysis for time-delay systems and the optimal controller synthesis for sampled-data systems. Finally, the overall arguments developed in this paper are demonstrated through some numerical and experimental studies.
期刊介绍:
The Journal of The Franklin Institute has an established reputation for publishing high-quality papers in the field of engineering and applied mathematics. Its current focus is on control systems, complex networks and dynamic systems, signal processing and communications and their applications. All submitted papers are peer-reviewed. The Journal will publish original research papers and research review papers of substance. Papers and special focus issues are judged upon possible lasting value, which has been and continues to be the strength of the Journal of The Franklin Institute.