Stabilization of the linear controlled output of an autonomous stochastic differential system on an infinite horizon

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Abstract

The control problem of the linear output of an autonomous nonlinear stochastic differential system is considered. The infinite horizon and the quadratic objective make it possible to interpret the control goal as stabilization of the output near the position determined by the state, which is described by a nonlinear stochastic differential equation. The solution is obtained for two variants of the model: with accurate measurements and under the assumption that the linear output represents indirect observations of the state. In the case of indirect observations, a continuous Markov chain is used as a state model, which makes it possible to separate the control and filtering tasks and apply the Wonham filter. In both variants, sufficient conditions for the existence of an optimal solution consist of typical requirements for linear systems that ensure the existence of a limiting solution to the Riccati equation. Additional requirements due to nonlinear elements are the ergodicity of nonlinear dynamics and the existence of a limit in the Feynman-Katz formula for the coefficients of the nonlinear part of the control. The results of the numerical experiment are presented and analyzed.

About the authors

A. V. Bosov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Author for correspondence.
Email: ABosov@frccsc.ru
Russian Federation, Moscow

References

  1. Athans M. The Role and Use of the Stochastic Linear-Quadratic-Gaussian Problem in Control System Design // IEEE T. Automat. Contr. 1971. V. 16. № 6. P. 529–552.
  2. Astrom K.J. Introduction to Stochastic Control Theory. N.Y.: Acad. Press, 1970.
  3. Lindquist A. On Feedback Control of Linear Stochastic Systems // SIAM J. Control. 1973. V. 11. № 2. P. 323–343.
  4. Elliott R.J., Aggoun L., Moore J.B. Hidden Markov Models: Estimation and Control. N.Y.: Springer-Verlag, 1995.
  5. Bar-Shalom Y., Willett P.K., Tian X. Tracking and Data Fusion: a Handbook of Algo-rithms. Storrs, Conn.: YBS Publishing, 2011.
  6. Wonham W.M. Linear Multivariable Control. A Geometric Approach. Lecture Notes in Economics and Mathematical Systems, V. 101. Berlin: Springer-Verlag, 1974.
  7. Девис М.Х.А. Линейное оценивание и стохастическое управление / Пер. с англ. М.: Наука, 1984.
  8. Kalman R.E., Bucy R.S. New Results in Linear Filtering and Prediction Theory // Trans. ASME J. Basic Eng. 1965. № 83. P. 95–107.
  9. Босов А.В. Задача управления линейным выходом нелинейной неуправляемой сто-хастической дифференциальной системы по квадратичному критерию // Изв. РАН. ТиСУ. 2021. № 5. C. 52–73.
  10. Липцер Р.Ш., Ширяев А.Н. Статистика случайных процессов (нелинейная филь-трация и смежные вопросы). М.: Наука, 1974.
  11. Rishel R. A Strong Separation Principle for Stochastic Control Systems Driven by a Hidden Markov Model // SIAM J. Control and Optimization. 1994. V. 32. P. 1008–1020.
  12. Wonham W.M. Some Applications of Stochastic Differential Equations to Optimal Non-linear Filtering // SIAM J. Control. 1965. V. 2. P. 347–369.
  13. Ширяев А.Н. Вероятность. 2-е изд. М.: Наука, 1989.
  14. Borisov A., Bosov A., Miller G. Optimal Stabilization of Linear Stochastic System with Statistically Uncertain Piecewise Constant Drift // Mathematics. 2022. V. 10. № 2 (184).
  15. Флеминг У., Ришел Р. Оптимальное управление детерминированными и стохасти-ческими системами / Пер. с англ. М.: Мир, 1978.
  16. Bosov A., Borisov A. Comparative Study of Markov Chain Filtering Schemas for Stabi-lization of Stochastic Systems under Incomplete Information // Mathematics. 2022. V. 10. № 18 (338).
  17. Босов А.В., Стефанович А.И. Управление выходом стохастической дифференци-альной системы по квадратичному критерию. II. Численное решение уравнений динамического программирования // Информатика и ее применения. 2019. Вып. 1. Т. 13. С. 9–15.
  18. Босов А.В., Стефанович А.И. Управление выходом стохастической дифференци-альной системы по квадратичному критерию. IV. Альтернативное численное ре-шение // Информатика и ее применения. 2020. Вып. 1. Т. 14. С. 24–30.
  19. Cox J.C., Ingersoll J.E., Ross S.A. A Theory of the Term Structure of Interest Rates // Econometrica. 1985. V. 53. Iss. 2. P. 385–407.
  20. Bohacek S., Rozovskii B. A Diffusion Model of Roundtrip Time // Computational Statis-tics & Data Analysis. 2004. V. 45. Iss. 1. P. 25–50.
  21. Босов А.В. Стабилизация и слежение за траекторией линейной системы со скачко-образно изменяющимся дрейфом // АиТ. 2022. № 4. С. 27–46.
  22. Øksendal B. Stochastic Differential Equations. An Introduction with Applications. N.Y.: Springer-Verlag, 2003.

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