- PII
- 10.31857/S0005231023010038-1
- DOI
- 10.31857/S0005231023010038
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume / Issue number 1
- Pages
- 63-83
- Abstract
- This paper considers an optimal control problem for a time-invariant linear stochastic system with discrete time, scalar unbounded control, additive noise, and a probabilistic criterion for retaining its trajectories in a given neighborhood of zero. We use dynamic programming and two-sided Bellman function estimates to derive analytical expressions for the optimal control at two time steps and a suboptimal control on any control horizon. The effectiveness of these controls is illustrated on a numerical example.
- Keywords
- дискретные системы стохастическое оптимальное управление вероятностный критерий метод динамического программирования функция Беллмана стационарные системы неограниченное управление
- Date of publication
- 15.01.2023
- Year of publication
- 2023
- Number of purchasers
- 0
- Views
- 7
References
- 1. Малышев В.В., Кибзун А.И. Анализ и синтез высокоточного управления летательными аппаратами. М.: Машиностроение, 1987.
- 2. Lesser K., Oishi M., Erwin R. Stochastic reachability for control of spacecraft relative motion // Proc. IEEE Conf. Dec. and Ctrl. 2013. P. 4705-4712.
- 3. Kan Y.S. Control Optimization by the Quantile Criterion // Autom. Remote Control. 2001. V. 62. No. 6. P. 746-757.
- 4. Azanov V.M., Kan Yu.S. Design of Optimal Strategies in the Problems of Discrete System Control by the Probabilistic Criterion // Autom. Remote Control. 2017. V. 78. No. 6. P. 1006-1027.
- 5. Кузьмин В.П., Ярошевский В.А. Оценка предельных отклонений фазовых координат динамической системы при случайных возмущениях. М.: Наука, 1995.
- 6. Soudjani S., Abate A. Probabilistic reach-avoid computation for partially degenerate stochastic processes // IEEE Trans. Autom. Ctrl. IEEE Trans. Autom. Ctrl., 2014. V. 59. No. 2. P. 528-534.
- 7. Summers S., Lygeros J. Verification of discrete time stochastic hybrid systems: A stochastic reach-avoid decision problem // Automatica. 2010. V. 46. No. 12. P. 1951-1961.
- 8. Vinod A., Oishi M. Scalable underapproximation for the stochastic reach-avoid problem for highdimensional LTI systems using Fourier transforms // IEEE Lett.-Contr. Syst. Soc. 2017. V. 1. No. 2. P. 316-321.
- 9. Jasour A.M., Aybat N.S., Lagoa C.M. Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets // SIAM J. Optimization. 2015. V. 25. No. 3. P. 1411-1440.
- 10. Jasour A.M., Lagoa C.M. Convex constrained semialgebraic volume optimization: Application in systems and control. arXiv:1701.08910, 2017.
- 11. Grigor'ev P.V., Kan Y.S Optimal Control of the Investment Portfolio with Respect to the Quantile Criterion // Autom. Remote Control. 2004. V. 65. No. 2. P. 319-336.
- 12. Bunto T.V., Kan Y.S. Quantile criterion-based control of the securities portfolio with a nonzero ruin probability // Autom. Remote Control. 2013. V. 74. No. 5. P. 811-828.
- 13. Maidens J.N., Kaynama S., Mitchell I.M., Oishi M.M., Dumont G.A. Lagrangian methods for approximating the viability kernelin high-dimensional systems // Automatica. 2013. V. 49. No. 7. P. 2017-2029.
- 14. Kariotoglou N., Raimondo D.M., Summers S., Lygeros J. Astochastic reachability framework for autonomous surveillance withpan-tilt-zoom cameras // Proc. European Ctrl. Conf. 2011. P. 1411-1416.
- 15. Doyen L., De Lara M. Stochastic viability and dynamic programming // Systems and Control Letters. 2010. V. 59. No. 10. P. 629-634.
- 16. Кибзун А.И., Иванов С.В., Степанова А.С. Построение доверительного множества поглощения в задачах анализа статических стохастических систем // АиТ. 2020. № 4. С. 21-36.
- 17. Azanov V.M., Kan Yu.S. Bilateral Estimation of the Bellman Function in the Problems of Optimal Stochastic Control of Discrete Systems by the Probabilistic Performance Criterion // Autom. Remote Control. 2018. V. 79. No. 2. P. 203-215.
- 18. Azanov V.M., Kan Yu.S. Refined Estimation of the Bellman Function for Stochastic Optimal Control Problems with Probabilistic Performance Criterion // Autom. Remote Control. 2019. V. 90. No. 4. P. 634-647.
- 19. Vinod A.P., Oishi M.M. Stochastic reachability of a target tube: Theory and computation // Automatica. 2021. V. 125.
- 20. Афанасьев В.Н., Колмановский В.Б., Носов В.Р. Математическая теория конструирования систем управления. М.: Высш. шк., 2003.
- 21. Azanov V.M., Tarasov A.N. Probabilistic criterion-based optimal retention of trajectories of a discrete-time stochastic system in a given tube: bilateral estimation of the Bellman function // Autom. Remote Control. 2020. V. 81. P. 1819-1839.
- 22. Konrad Schmudgen. The moment problem. Vol. 9. Springer, 2017.
- 23. Azanov V.M., Kan Yu.S. On Optimal Retention of the Trajectory of Discrete Stochastic System in Tube // Autom. Remote Control. 2019. V. 80. No. 1. P. 30-42.
- 24. Кан Ю.С., Кибзун А.И. Задачи стохастического программирования с вероятностными критериями. М.: ФИЗМАТЛИТ, 2009.
