{"title":"Structural Spectral Methods for Solving Continuous Lyapunov Equations","authors":"I. B. Yadykin, I. A. Galyaev","doi":"10.1134/s0005117923120081","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>For linear multivariable continuous stationary stable control systems with a simple spectrum, presented in the form of a canonical diagonal form, controllability and observability forms, a method was developed and analytical formulas for spectral decompositions of gramians in the form of various Xiao matrices were obtained. A method and algorithm for calculating generalized Xiao matrices in the form of the Hadamard product for multiply connected continuous linear systems with many inputs and many outputs have been developed. This allows us to calculate the elements of the corresponding controllability and observability gramians in the form of products of the corresponding elements of the multiplier matrices and a matrix that is the sum of all possible products of the numerator matrices of the matrix transfer function of the system. New results are obtained in the form of spectral and singular decompositions of the inverse gramians of controllability and observability. This makes it possible to obtain invariant decompositions of energy functionals and formulate new criteria for the stability of linear systems taking into account the nonlinear effects of mode interaction.</p>","PeriodicalId":55411,"journal":{"name":"Automation and Remote Control","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Automation and Remote Control","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1134/s0005117923120081","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
For linear multivariable continuous stationary stable control systems with a simple spectrum, presented in the form of a canonical diagonal form, controllability and observability forms, a method was developed and analytical formulas for spectral decompositions of gramians in the form of various Xiao matrices were obtained. A method and algorithm for calculating generalized Xiao matrices in the form of the Hadamard product for multiply connected continuous linear systems with many inputs and many outputs have been developed. This allows us to calculate the elements of the corresponding controllability and observability gramians in the form of products of the corresponding elements of the multiplier matrices and a matrix that is the sum of all possible products of the numerator matrices of the matrix transfer function of the system. New results are obtained in the form of spectral and singular decompositions of the inverse gramians of controllability and observability. This makes it possible to obtain invariant decompositions of energy functionals and formulate new criteria for the stability of linear systems taking into account the nonlinear effects of mode interaction.
期刊介绍：
Automation and Remote Control is one of the first journals on control theory. The scope of the journal is control theory problems and applications. The journal publishes reviews, original articles, and short communications (deterministic, stochastic, adaptive, and robust formulations) and its applications (computer control, components and instruments, process control, social and economy control, etc.).
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