{"title":"Solution of the Spectrum Allocation Problem for a Linear Control System with Closed Feedback","authors":"S. P. Zubova, E. V. Raetskaya","doi":"10.1134/s0012266124060065","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A method for constructing a feedback matrix to solve the spectrum allocation (spectrum\ncontrol; pole assignment) problem for a linear dynamical system is given. A new proof of the\nwell-known theorem about the connection between the complete controllability of a dynamical\nsystem and the existence of a feedback matrix is formed in the process of constructing the cascade\ndecomposition method. The entire set of arbitrary elements affecting the nonuniqueness of the\nmatrix is identified. Examples of constructing a feedback matrix in the case of a real spectrum\nand in the presence of complex conjugate eigenvalues as well as for the case of multiple eigenvalues\nare given. The stability of the specified spectrum under small perturbations of system parameters\nwith a fixed feedback matrix is studied.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"33 7 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124060065","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A method for constructing a feedback matrix to solve the spectrum allocation (spectrum
control; pole assignment) problem for a linear dynamical system is given. A new proof of the
well-known theorem about the connection between the complete controllability of a dynamical
system and the existence of a feedback matrix is formed in the process of constructing the cascade
decomposition method. The entire set of arbitrary elements affecting the nonuniqueness of the
matrix is identified. Examples of constructing a feedback matrix in the case of a real spectrum
and in the presence of complex conjugate eigenvalues as well as for the case of multiple eigenvalues
are given. The stability of the specified spectrum under small perturbations of system parameters
with a fixed feedback matrix is studied.
期刊介绍：
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.
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