{"title":"解决线性代数方程的控制论方法:连续时间视角","authors":"Yuxin Wu, Deyuan Meng","doi":"10.1016/j.sysconle.2024.105822","DOIUrl":null,"url":null,"abstract":"<div><p>In the applications of control theory, how to improve the solution methods of mathematical problems with some control methods is an attractive problem. This paper aims at exploiting a class of continuous-time solvers for linear algebraic equations (LAEs) via the incorporation of the idea of “control design”. The equivalence is established between the solving problem of LAEs and two basic control problems of continuous-time systems, namely, the output tracking problem and the state observation problem, in a unified manner. Then, the equivalence between the properties of LAEs and the controllability and observability properties of continuous-time systems are disclosed from the perspective of the output tracking and the state observation, respectively. Moreover, a continuous-time solver is developed based on the feedback control design such that all (least squares) solutions to any (un)solvable LAE can be determined by the selection of different initial conditions. Particularly, the design conditions to derive the best approximate solutions to LAEs are presented for the continuous-time solver.</p></div>","PeriodicalId":49450,"journal":{"name":"Systems & Control Letters","volume":"189 ","pages":"Article 105822"},"PeriodicalIF":2.1000,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Control-theoretic methods for solving linear algebraic equations: A continuous-time perspective\",\"authors\":\"Yuxin Wu, Deyuan Meng\",\"doi\":\"10.1016/j.sysconle.2024.105822\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In the applications of control theory, how to improve the solution methods of mathematical problems with some control methods is an attractive problem. This paper aims at exploiting a class of continuous-time solvers for linear algebraic equations (LAEs) via the incorporation of the idea of “control design”. The equivalence is established between the solving problem of LAEs and two basic control problems of continuous-time systems, namely, the output tracking problem and the state observation problem, in a unified manner. Then, the equivalence between the properties of LAEs and the controllability and observability properties of continuous-time systems are disclosed from the perspective of the output tracking and the state observation, respectively. Moreover, a continuous-time solver is developed based on the feedback control design such that all (least squares) solutions to any (un)solvable LAE can be determined by the selection of different initial conditions. Particularly, the design conditions to derive the best approximate solutions to LAEs are presented for the continuous-time solver.</p></div>\",\"PeriodicalId\":49450,\"journal\":{\"name\":\"Systems & Control Letters\",\"volume\":\"189 \",\"pages\":\"Article 105822\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-05-25\",\"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/S0167691124001105\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Systems & Control Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167691124001105","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}
Control-theoretic methods for solving linear algebraic equations: A continuous-time perspective
In the applications of control theory, how to improve the solution methods of mathematical problems with some control methods is an attractive problem. This paper aims at exploiting a class of continuous-time solvers for linear algebraic equations (LAEs) via the incorporation of the idea of “control design”. The equivalence is established between the solving problem of LAEs and two basic control problems of continuous-time systems, namely, the output tracking problem and the state observation problem, in a unified manner. Then, the equivalence between the properties of LAEs and the controllability and observability properties of continuous-time systems are disclosed from the perspective of the output tracking and the state observation, respectively. Moreover, a continuous-time solver is developed based on the feedback control design such that all (least squares) solutions to any (un)solvable LAE can be determined by the selection of different initial conditions. Particularly, the design conditions to derive the best approximate solutions to LAEs are presented for the continuous-time solver.
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.