- PII
- 10.31857/S0005231023120036-1
- DOI
- 10.31857/S0005231023120036
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume / Issue number 12
- Pages
- 18-37
- 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 calculatin 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.
- Keywords
- спектральные разложения грамианов сингулярные числа обратная матрица грамиана устойчивость с учетом взаимодействия мод матрицы Сяо уравнение Ляпунова
- Date of publication
- 15.12.2023
- Year of publication
- 2023
- Number of purchasers
- 0
- Views
- 44
References
- 1. Antoulas A.C. Approximation of Large-Scale Dynamical Systems. SIAM. Philadephia, 2005.
- 2. Поляк Б.Т., Хлебников М.В., Рапопорт Л.Б. Теория автоматического управления. Учеб. пособие. М.: ЛЕНАНД, 2019. 504 с.
- 3. Зубов Н.Е., Зыбин Е.Ю., Микрин Е.А., Мисриханов М.Ш., Рябченко В.Н. Общие аналитические формы решения уравнений Сильвестра и Ляпунова для непрерывных и дискретных динамических систем // Известия РАН. Теория и системы управления. 2017. № 1. С. 3-20.
- 4. Гантмахер Ф.Р. Теория матриц. М.: Наука, 1966.
- 5. Икрамов Х.Д. Численное решение матричных уравнений. М.: Наука, 1984. 192 с.
- 6. Фаддеев Д.К., Фаддеева В.Н. Вычислительные методы линейной алгебры. Учебник-М: Изд-во Лань, 2009. 726 с.
- 7. Квакернаак Х., Сиван Р. Линейные оптимальные системы управления. М.: Мир, 1977.
- 8. Андреев Ю.Н. Управление конечномерными линейными объектами М.: Наука, 1976. 424 c.
- 9. Годунов С.К. Лекции по современным аспектам линейной алгебры. Новосибирск: Научная книга, 2002. 216 с.
- 10. Проскурников А.В., Фрадков А.Л. Задачи и методы сетевого управления // АиТ. 2016. № 10. С. 3-39.
- 11. Жабко А.П., Харитонов В.Л. Методы линейной алгебры в задачах управления: учебное пособие / СПбГУ СПб.: Изд-во СПб. универ-та, 1993. 318 с.
- 12. Sreeram V., Agathoklis P. Solution of Lyapunov equation with system matrix in companion form // IEE Proc. D. Control. Theory Appl. 1991. V. 138. No. 6. P. 529-534. https://doi.org/10.1049/ip-d.1991.0074
- 13. Xiao C., Feng Z., Shan X. On the Solution of the Continuous-Time Lyapunov Matrix Equation in Two Canonical Forms // IEE Proc. 1992. V. 139. No. 3. P. 286-290. https://doi.org/10.1049/ip-d.1992.0038
- 14. Hauksdottir A., Sigurdsson S. The continuous closed form controllability Gramian and its inverse // 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009. P. 5345-5351. https://doi.org/978-1-4244-4524-0/09
- 15. Yadykin I.B. Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms // Mathematics. 2022. V. 10. No. 13. P. 2339. https://doi.org/10.3390/math10132339
- 16. Dilip A.S.A. The controllability Gramian, the Hadamard product and the optimal actuator // Leader Sensor Select. Problem Nature Phys. 2015. V. 11. P. 779-786. https://doi.org/10.1109/LCSYS.2019.2919278
- 17. Bianchin G., Pasqualetti F. Gramian-Based Optimization for the Analysis and Control of Traffic Networks // IEEE Transactions on Intelligent Transportation Systems. 2022. V. 21. No. 7. P. 3013-3024. https://doi.org/10.1109/TITS.2019.2922900
- 18. Himpe C. The Empirical Gramian Framework // Algorithms. 2018. V. 11. No. 91. https://doi.org/10.3390/a11070091
- 19. Benner P., Goyal P., Duff I.P. Gramians, Energy Functionals, and Balanced Truncation for Linear Dynamical Systems With Quadratic Outputs // IEEE Transact. Autom. Control. 2022. V. 67. No. 2. P. 886-893. https://doi.org/10.1109/TAC.2021.3086319
- 20. Ядыкин И.Б. О свойствах грамианов непрерывных систем управления // АиТ. 2010. № 6. С. 39-50. https://doi.org/10.1134/S0005117910060032
- 21. Yadykin I.B., Galyaev A.A. On the methods for calculation of grammians and their use in analysis of linear dynamic systems // Automation and Remote Control. 2013. V. 74. No. 2. P. 207-224.
- 22. Ядыкин И.Б., Искаков А.Б. Энергетический подход к анализу устойчивости линейных стационарных динамических систем // АиТ. 2016. № 12. С. 37-58.
- 23. Gardner M.F., Barns J.L. Transients in linear systems studied by the Laplace transformation / V. 1. Lumped-constant systems. New York, London. Wiley, Chapman and Hall. 1942.
