msiСовременная наука и инновацииModern Science and Innovations2307-910XNorth-Caucasus Federal University10.37493/2307-910X.2022.3.1msi-1357Research ArticleТЕХНИЧЕСКИЕ НАУКИ. ИНФОРМАТИКА, ВЫЧИСЛИТЕЛЬНАЯ ТЕХНИКА И УПРАВЛЕНИЕTECHNICAL SCIENCES. INFORMATION, COMPUTING AND MANAGEMENTПостроение области притяжения на основе функций Ляпунова для нелинейных систем общего видаConstruction of the domain of attraction based on Lyapunov functions for general nonlinear systemshttps://orcid.org/0000-0002-8647-9096РомановС. А.RomanovS. A.

Романов Сергей Алексеевич - научный сотрудник кафедры автоматики и процессов управления.

Санкт-Петербург, ул. Профессора Попова, дом 5

Sergey A. Romanov - Researcher at the Department of Automation and Control Processes of the St. Petersburg State Electrotechnical University "LETI" named after V.I. Ulyanov (Lenin).

St. Petersburg, Professora Popova str., house 5

sarmnv@gmail.comДушинС. Е.DushinS. E.

Душин Сергей Евгеньевич - доктор технических наук, профессор кафедры автоматики и процессов управления.

Санкт-Петербург, ул. Профессора Попова, дом 5

Тел.: +7 (921) 970-46-31

Sergey E. Dushin - Doctor of Technical Sciences, Professor of the Department of Automation and Control Processes of the St. Petersburg State Electrotechnical University "LETI" named after V.I. Ulyanov (Lenin).

St. Petersburg, Professora Popova str., house 5

Tel.: +7 (921) 970-46-31

dushins@yandex.ruhttps://orcid.org/0000-0002-5851-6160ШпаковскаяИ. И.ShpakovshayaI. I.

Шпаковская Ирина Игоревна - ассистент кафедры автоматики и процессов управления.

Санкт-Петербург, ул. Профессора Попова, дом 5

Тел.: +7 (950) 049-71-40

Irina I. Shpakovskaya - Assistant of the Department of Automation and Control Processes of the St. Petersburg State Electrotechnical University "LETI" named after V.I. Ulyanov (Lenin).

St. Petersburg, Professora Popova str., 5

Tel.: +7 (950) 049-71-40

iishpakovskaia@etu.ruСанкт-Петербургский государственный электротехнический университет «ЛЭТИ» им. В.И. Ульянова (Ленина)РоссияSaint Petersburg Electrotechnical University "LETI”Russian Federation202213012023031019Copyright © Романов С.А., Душин С.Е., Шпаковская И.И., 20232023Романов С.А., Душин С.Е., Шпаковская И.И.Romanov S.A., Dushin S.E., Shpakovshaya I.I.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://msi.elpub.ru/jour/article/view/1357

В работе предложен алгоритм для построения нейросетевых функций-кандидатов Ляпунова с целью максимизации оценки области притяжения положения равновесия. Для этого необходимо, чтобы инвариантное подмножество, задаваемое множеством уровня, занимало как можно большую долю полученной симуляцией эмпирической оценки области притяжения. Реализация данной цели осуществляется путём введения дополнительного слагаемого в функцию потерь. Алгоритм позволяет строить функции-кандидаты для систем с нелинейностями достаточно общего вида. Работа алгоритма проиллюстрирована на примере.

The paper proposes an algorithm for constructing Lyapunov candidate neural network functions in order to maximize the estimate of the area of attraction of the equilibrium position. To do this, it is necessary that the invariant subset, given by the level set, occupy the largest possible share of the empirical estimate of the attraction region obtained by the simulation. The implementation of this goal is carried out by introducing an additional term in the loss function. The algorithm makes it possible to construct candidate functions for systems with non-linearities of a rather general form. The operation of the algorithm is illustrated by an example.

функция Ляпуноваобласть притяженияискусственная нейронная сетьLyapunov functionattraction domainartificial neural networkReferencesPablo Parrilo. ―Structured Semidenite Programs and Semialgebraic Geometry Methods in Robustness and Optimization‖. В: PhD thesis, 2000Pablo Parrilo. ―Structured Semidenite Programs and Semialgebraic Geometry Methods in Robustness and Optimization‖. V: PhD thesis, 2000Daniel Richardson. ―Some Undecidable Problems Involving Elementary Functions of a Real Variable‖. В: The Journal of Symbolic Logic 33.4 (1968), с. 514— 520. issn: 00224812. url: http://www.jstor.org/stable/2271358 (дата обр. 08.04.2022)Daniel Richardson. ―Some Undecidable Problems Involving Elementary Functions of a Real Variable‖. V: The Journal of Symbolic Logic 33.4 (1968), s. 514— 520. issn: 00224812. url: http://www.jstor.org/stable/2271358 (data obr. 08.04.2022)Hongkai Dai и др. Lyapunov-stable neural-network control. 2021. arXiv: 2109.14152Hongkai Dai i dr. Lyapunov-stable neural-network control. 2021. arXiv: 2109.14152Lars Gr¨une. ―Computing Lyapunov functions using deep neural networks‖. В: Journal of Computational Dynamics 8.2 (2021), с. 131—152.Lars Gr¨une. ―Computing Lyapunov functions using deep neural networks‖. V: Journal of Computational Dynamics 8.2 (2021), s. 131—152.Spencer M. Richards, Felix Berkenkamp и Andreas Krause. The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems. 2018. arXiv: 1808.00924Spencer M. Richards, Felix Berkenkamp i Andreas Krause. The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems. 2018. arXiv: 1808.00924Guy Katz и др. Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks. 2017. doi: 10.48550/ARXIV.1702.01135. url: https://arxiv.org/abs/1702.01135Guy Katz i dr. Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks. 2017. doi: 10.48550/ARXIV.1702.01135. url: https://arxiv.org/abs/1702.01135Ляпунов А.М. Общая задача об устойчивости движения. Москва: Гостехиздат; 1950. 472 сLyapunov A.M. Obshchaya zadacha ob ustoichivosti dvizheniya. Moskva: Gostekhizdat; 1950. 472 sY. Long и M.M. Bayoumi. ―Feedback stabilization: Control Lyapunov functions modelled by neural networks‖. В: Proceedings of 32nd IEEE Conference on Decision and Control. IEEE. 1993, с. 2812—2814Y. Long i M.M. Bayoumi. ―Feedback stabilization: Control Lyapunov functions modelled by neural networks‖. V: Proceedings of 32nd IEEE Conference on Decision and Control. IEEE. 1993, s. 2812—2814D.V. Prokhorov. ―A Lyapunov machine for stability analysis of nonlinear systems‖. В: Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN‘94). Т. 2. 1994, 1028—1031 vol.2. doi: 10.1109/ICNN.1994.374324D.V. Prokhorov. ―A Lyapunov machine for stability analysis of nonlinear systems‖. V: Pro-ceedings of 1994 IEEE International Conference on Neural Networks (ICNN‘94). T. 2. 1994, 1028—1031 vol.2. doi: 10.1109/ICNN.1994.374324Gursel Serpen. ―Empirical approximation for Lyapunov functions with artificial neural nets‖. В: Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2005. Т. 2. IEEE. 2005, с. 735—740Gursel Serpen. ―Empirical approximation for Lyapunov functions with artificial neural nets‖. V: Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2005. T. 2. IEEE. 2005, s. 735—740Alessandro Abate и др. ―Formal synthesis of lyapunov neural networks‖. В: IEEE Control Systems Letters 5.3 (2020), с. 773—778Alessandro Abate i dr. ―Formal synthesis of lyapunov neural networks‖. V: IEEE Con-trol Systems Letters 5.3 (2020), s. 773—778Ya-Chien Chang, Nima Roohi и Sicun Gao. Neural Lyapunov Control. 2020. arXiv: 2005.00611Ya-Chien Chang, Nima Roohi i Sicun Gao. Neural Lyapunov Control. 2020. arXiv: 2005.00611Nathan Gaby, Fumin Zhang и Xiaojing Ye. ―Lyapunov-net: A deep neural network architecture for Lyapunov function approximation‖. В: arXiv preprint arXiv:2109.13359, 2021Nathan Gaby, Fumin Zhang i Xiaojing Ye. ―Lyapunov-net: A deep neural network archi-tecture for Lyapunov function approximation‖. V: arXiv preprint arXiv:2109.13359, 2021H. Edelsbrunner, D. Kirkpatrick и R. Seidel. ―On the shape of a set of points in the plane‖. В: IEEE Transactions on Information Theory 29.4 (1983), с. 551—559. doi: 10.1109/TIT.1983.1056714H. Edelsbrunner, D. Kirkpatrick i R. Seidel. ―On the shape of a set of points in the plane‖. V: IEEE Transactions on Information Theory 29.4 (1983), s. 551—559. doi: 10.1109/TIT.1983.1056714Graziano Chesi. ―Domain of attraction: estimates for non-polynomial systems via LMIs‖. В: 2005Graziano Chesi. ―Domain of attraction: estimates for non-polynomial systems via LMIs‖. V: 2005

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