On Assessing the Stability of Linear Systems through Modifying the Classical Frequency Stability Criterion
DOI:
https://doi.org/10.24160/1993-6982-2026-1-128-134Keywords:
stability, Mikhailov criterion, frequency methodsAbstract
One of the key issues of control theory is stability. A special group of methods has been developed for stability analysis: frequency stability criteria. They are based on the principle of an argument, which links the order of the initial system’s characteristic equation with variations in the complex function phase obtained by transforming this characteristic equation into the frequency domain. The Mikhailov criterion is a universal one among frequency criteria, which makes it possible to determine the complex function stability through constructing its locus. However, being literally a graphical implementation of the argument principle, the Mikhailov criterion is used only for analyzing the phase change; therefore, it seems interesting to reduce the calculations necessary for stability analysis. The article considers an alternative to the Mikhailov criterion in the form of an imaginary-real characteristic (IRC). By using this characteristic, one can perform exactly the same analysis of phase change, but now through constructing a graph representing the initial complex function’s imaginary to real component ratio rather than through constructing a full-fledged Mikhailov locus. The article presents evidence that IRC fully represents each requirement of the Mikhailov criterion, which means that IRC is also suitable to perform stability assessment. The IRC applicability limits are given along with the requirements according to which the system can be identified as a stable one. The IRC applicability areas are the same as those of the classical Mikhailov criterion, with the only exception that IRC, while having no less information content, is more compact and convenient to implement.
References
1. Ляпунов А.М. Общая задача об устойчивости движения. Изд-во технико-теоретической лит-ры, 1950.
2. Гантмахер Ф.Р. Теория матриц. М.: Физматлит, 2004.
3. Постников М.М. Устойчивые многочлены. М.: Едиториал УРСС, 2004.
4. Ляшенко А.Л. Применение критерия Михайлова для оценки устойчивости нелинейных систем с распределёнными параметрами // Актуальные проблемы инфотелекоммуникаций в науке и образовании: Материалы IV Междунар. науч.-техн. и научно-методической конф. СПб.: Санкт-Петербургский гос. ун-т телекоммуникаций им. проф. М.А. Бонч-Бруевича, 2015. С. 534—538.
5. Авсиевич А.В. Модификация критерия устойчивости Михайлова для анализа моделей автоматического управления с нецелыми показателями // Мехатроника, автоматизация и управление на транспорте: Материалы II Всерос. науч.-практ. конф. Самара: Самарский гос. ун-т путей сообщения, 2020. С. 7—12.
6. Melchor-Aguilar D., Mendiola-Fuentes J. Mikhailov Stability Criterion for Fractional Commensurate Order Systems with Delays // J. Franklin Institute. 2022. V. 359(1). Pp. 8395—8408.
7. Thai Ha Duc, Hoang The Tuan. Modified Mikhailov Stability Criterion for Non-commensurate Fractional-order Neutral Differential Systems with Delays // J. Franklin Institute. 2024. V. 362(1). P. 107384.
8. Шелуденко А.С., Филер З.Е. Методы финитизации критерия Михайлова // Актуальные направления научных исследований XXI века: теория и практика. 2015. Т. 3. № 7—3(18—3). С. 215—219.
9. Keel L., Bhattacharyya S.P. A Generalization of Mikhailov's Criterion with Applications // Proc. American Control Conf. 2000. V. 6. P. 4311—4315.
10. Михайлов А.В. Метод гармонического анализа в теории регулирования // Автоматика и телемеханика. 1938. № 3. С. 27—81.
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Для цитирования: Вершинин Д.В., Кулешов В.В., Се Ю., Юрасов А.О. Об оценке устойчивости линейных систем через модификацию классического частотного критерия устойчивости // Вестник МЭИ. 2026. № 1. С. 128—134. DOI: 10.24160/1993-6982-2026-1-128-134.
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1. Lyapunov A.M. Obshchaya Zadacha ob Ustoychivosti Dvizheniya. Izd-vo Tekhniko-teoreticheskoy Lit-ry, 1950. (in Russian).
2. Gantmakher F.R. Teoriya Matrits. M.: Fizmatlit, 2004. (in Russian).
3. Postnikov M.M. Ustoychivye Mnogochleny. M.: Editorial URSS, 2004. (in Russian).
4. Lyashenko A.L. Primenenie Kriteriya Mikhaylova dlya Otsenki Ustoychivosti Nelineynykh Sistem s Raspredelennymi Parametrami. Aktual'nye Problemy Infotelekommunikatsiy v Nauke i Obrazovanii: Materialy IV Mezhdunar. Nauch.-tekhn. i Nauchno-metodicheskoy Konf. SPb.: Sankt-Peterburgskiy Gos. Un-t Telekommunikatsiy im. Prof. M.A. Bonch-Bruevicha, 2015:534—538. (in Russian).
5. Avsievich A.V. Modifikatsiya Kriteriya Ustoychivosti Mikhaylova dlya Analiza Modeley Avtomaticheskogo Upravleniya s Netselymi Pokazatelyami. Mekhatronika, Avtomatizatsiya i Upravlenie na Transporte: Materialy II Vseros. Nauch.-prakt. Konf. Samara: Samarskiy Gos. Un-t Putey Soobshcheniya, 2020:7—12. (in Russian).
6. Melchor-Aguilar D., Mendiola-Fuentes J. Mikhailov Stability Criterion for Fractional Commensurate Order Systems with Delays. J. Franklin Institute. 2022;359(1):8395—8408.
7. Thai Ha Duc, Hoang The Tuan. Modified Mikhailov Stability Criterion for Non-commensurate Fractional-order Neutral Differential Systems with Delays. J. Franklin Institute. 2024;362(1):107384.
8. Sheludenko A.S., Filer Z.E. Metody Finitizatsii Kriteriya Mikhaylova. Aktual'nye Napravleniya Nauchnykh Issledovaniy XXI Veka: Teoriya i Praktika. 2015;3;7—3(18—3):215—219. (in Russian).
9. Keel L., Bhattacharyya S.P. A Generalization of Mikhailov's Criterion with Applications. Proc. American Control Conf. 2000;6:4311—4315.
10. Mikhaylov A.V. Metod Garmonicheskogo Analiza v Teorii Regulirovaniya. Avtomatika i Telemekhanika. 1938;3:27—81. (in Russian)
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For citation: Vershinin D.V., Kuleshov V.V., Xie Yu., Yurasov A.O. On Assessing the Stability of Linear Systems through Modifying the Classical Frequency Stability Criterion. Bulletin of MPEI. 2026;1:128—134. (in Russian). DOI: 10.24160/1993-6982-2026-1-128-134.
