vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2023-59-2-110-120vestifm-710Research ArticleМАТЕМАТИКАMATHEMATICSРобастная стабилизируемость и стабилизация трехтемповых линейных стационарных сингулярно возмущенных систем с запаздываниемRobust stabilizability and stabilization of three-time-scale linear time-invariant singularly perturbed systems with delayНалигамаЧ. А.NaligamaCh. A.

Налигама Чамила Анурадха – аспирант, факультет математики и информатики; инженер-консультант

ул. Ожешко, 22, 230023, Гродно; 550/01/10, Михиндумаватха, Малватта, Ниттамбува, Шри-Ланка, 11880

Chamila A. Naligama – Postgraduate Student, Department of Mathematics and Informatics; Consulting Engineer

22, Ozheshko Str., 230023, Grodno; 550/01/10, Mihindumawatha, Malwatta, Nittambuwa, Sri Lanka, 11880

chammme@gmail.comhttps://orcid.org/0000-0001-7636-4589ЦеханО. Б.TsekhanO. B.

Цехан Ольга Борисовна – кандидат физико-математических наук, доцент, доцент кафедры математического и информационного обеспечения экономических систем

ул. Ожешко, 22, 230023, Гродно

Olga B. Tsekhan – Ph. D. (Physics and Mathematics),Associate Professor, Associate Professor of the Departmentof Mathematical and Information Support of EconomicSystems

22, Ozheshko Str., 230023, Grodno

tsekhan@grsu.byГродненский государственный университет имени Янки Купалы; Сенекс ИнжинирингБеларусьYanka Kupala State University of Grodno; Cenex EngineeringBelarusГродненский государственный университет имени Янки КупалыБеларусьYanka Kupala State University of GrodnoBelarus202306072023592110120Copyright © Налигама Ч.А., Цехан О.Б., 20232023Налигама Ч.А., Цехан О.Б.Naligama C.A., Tsekhan O.B.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/710

Целью работы является получение условий стабилизируемости и построение композитной стабилизирующей обратной связи по состоянию для трехтемповых линейных стационарных сингулярно возмущенных систем с кратными соизмеримыми запаздываниями в медленных переменных состояния и с двумя малыми параметрами при части старших производных (ТСВЛССЗ). Условия стабилизируемости и стабилизирующая обратная связь не зависят от малых параметров и действительны для всех их достаточно малых значений. Применяемый в работе подход использует невырожденное преобразование, которое полностью расщепляет зависящую от двух малых параметров сингулярно возмущенную систему на три регулярно зависящие от параметров подсистемы меньших размерностей, чем исходная система, которые аппроксимируются подсистемами, не зависящими от малых параметров. Доказано, что стабилизируемость аппроксимирующих подсистем гарантирует робастную (по малым параметрам) стабилизируемость исходной ТСВЛССЗ. Получено представление не зависящего от параметра композитного управления с обратной связью для ТСВЛССЗ, стабилизирующего ее при всех достаточно малых значениях параметров. Приведен численный пример. 

The objective of this study is to obtain the stabilizability conditions and a stabilizing composite state feedback control for the exponential stabilization of three-time-scale singularly perturbed linear time-invariant systems with multiple commensurate delays in the slow state variables and with two small parameters of perturbation (TSPLTISD). The stabilizability conditions and the stabilizing feedback do not depend on the small parameters and are valid for all of their sufficiently small values. The approach used in this work is the nondegenerate decoupling transformation that splits the TSPLTISD into three regularly dependent on the small parameters subsystems, which are lower in dimensions than the TSPLTISD. Further, the decoupled subsystems are approximated by three subsystems that do not depend on the small parameters. It is proven that the stabilizability of the approximating subsystems guarantees the robust (with respect to small parameters) stabilizability of the original TSPLTISD. Finally, we obtain a representation of a parameter free composite feedback control for the TSPLTISD, stabilizing it for all sufficiently small values of the parameters. A numerical example is given. 

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The authors declare that there are no conflicts of interest present.