Advertisement

Automation and Remote Control

, Volume 80, Issue 7, pp 1252–1264 | Cite as

Necessary and Sufficient Conditions for Optimal Stabilization of Quasi-Linear Stochastic Systems

  • M. M. KhrustalevEmail author
  • E. E. OneginEmail author
Stochastic Systems

Abstract

A wide class of admissible control strategies that guarantee the mean-square stabilization of a stochastic system is considered. Necessary and sufficient conditions for the opti-mality of a linear time-invariant controller are established. The difference between the stated problem and the optimal control problem on an infinite time interval is demonstrated. The obtained optimality conditions are illustrated by the example of stabilization of an artificial Earth satellite in the neighborhood of a circular orbit.

Keywords

quasi-linear stochastic systems optimal stabilization linear controller 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was carried out within the state task no. 9.7555.2017/BCh.

References

  1. 1.
    Panossian, H.V., Review of Linear Stochastic Optimal Control Systems and Applications, J. Vib. Acoust. Stress Reliab. Des., 1989, vol. 111, no. 4, pp. 399–403.CrossRefGoogle Scholar
  2. 2.
    Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley-Interscience, 1972. Translated under the titlezbMATHGoogle Scholar
  3. 2a.
    Kwakernaak, H. and Sivan, R., Lineinye optimal’nye sistemy upravleniya, Moscow: Mir, 1972.Google Scholar
  4. 3.
    Chen, S., Li, X., and Zhou, X.-Y., Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs, SIAM J. Control Optim., 1998, vol. 36, no. 5, pp. 1685–1702.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 4.
    Chen, S. and Zhou, X.-Y., Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs, SIAM J. Control Optim., 2000, vol. 39, no. 4, pp. 1065–1081.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 5.
    Ait Rami, M., Moore, J.B., and Zhou, X.-Y., Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation, SIAM J. Control Optim., 2002, vol. 40, no. 4, pp. 1296–1311.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 6.
    Arnold, L., Stochastic Differential Equations: Theory and Applications, New York: Wiley, 1974.zbMATHGoogle Scholar
  8. 7.
    Wonham, W.M., Optimal Stationary Control of a Linear System with State-Dependent Noise, SIAM J. Control, 1967, vol. 5, no. 3, pp. 486–500.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 8.
    Haussmann, U.G., Optimal Stationary Control with State Control Dependent Noise, SIAM J. Control, 1971, vol. 9, no. 2, pp. 184–198.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 9.
    McLane, P.J., Optimal Stochastic Control of Linear Systems with State- and Control-Dependent Disturbances, IEEE Trans. Automat. Control, 1971, vol. 16, no. 6, pp. 793–798.CrossRefGoogle Scholar
  11. 10.
    El Ghaoui, L., State-Feedback Control of Systems with Multiplicative Noise via Linear Matrix Inequalities, Syst. Control Lett., 1995, vol. 24, no. 3, pp. 223–228.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 11.
    Verriest, E.I. and Florchinger, P., Stability of Stochastic Systems with Uncertain Time Delays, Syst. Control Lett., 1995, vol. 24, no. 1, pp. 41–47.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 12.
    Paraev, Yu.I., Vvedenie v statisticheskuyu dinamiku protsessov upravleniya i fil’tratsii. Biblioteka tekhni-cheskoi kibernetiki (Introduction to Statistical Dynamics of Control Processes and Filtering. Library of Engineering Cybernetics), Moscow: Sovetskoe Radio, 1976.Google Scholar
  14. 13.
    Rumyantsev, D.S., Khrustalev, M.M., and Tsarkov, K.A., An Algorithm for Synthesis of the Suboptimal Control Law for Quasi-linear Stochastic Dynamical Systems, J. Comput. Syst. Sci. Int., 2014, vol. 53, no. 1, pp. 71–83.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 14.
    Willems, J.L. and Willems, J.C., Feedback Stabilizability for Stochastic Systems with State and Control Dependent Noise, Automatica, 1976, vol. 12, no. 3, pp. 277–283.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 15.
    Kleinman, D.L., Optimal Stationary Control of Linear Systems with Control-Dependent Noise, IEEE Trans. Automat. Control, 1969, vol. 14, no. 6, pp. 673–677.MathSciNetCrossRefGoogle Scholar
  17. 16.
    Damm, T., Rational Matrix Equations in Stochastic Control, Berlin-Heidelberg: Springer, 2004.zbMATHGoogle Scholar
  18. 17.
    Khrustalev, M.M. and Onegin, E.E., Analytical Design of Optimal Controllers for Quasi-linear Stochastic Systems on the Infinite Time Interval, Program. Sist.: Teor. Prilozheniya, 2015, vol. 6, no. 2, pp. 29–44.Google Scholar
  19. 18.
    Øksendal, B., Stochastic Differential Equations, Berlin-Heidelberg: Springer, 2003.CrossRefzbMATHGoogle Scholar
  20. 19.
    Khrustalev, M.M., Nash Equilibrium Conditions in Stochastic Differential Games where Players Information about a State Is Incomplete. I, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 1995, vol. 34, no. 6, pp. 194–208.Google Scholar
  21. 20.
    Khrustalev, M.M., Nash Equilibrium Conditions in Stochastic Differential Games where Players Information about a State Is Incomplete. II. Lagrange Method, J. Comput. Syst. Sci. Int., 1996, vol. 35, no. 1, pp. 67–73.zbMATHGoogle Scholar
  22. 21.
    Khalina, A.S. and Khrustalev, M.M., System Shape Optimization and Stabilization of Controlled Quasi-linear Stochastic Systems that Operate on an Infinite Time Interval, J. Comput. Syst. Sci. Int., 2017, vol. 56, no. 1, pp. 64–86.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 22.
    Lebedev, A.A., Krasil’shchikov, M.N., and Malyshev, V.V., Optimal’noe upravlenie dvizheniem kosmich-eskikh letatel’nykh apparatov (Optimal Control of Motion of Spacecrafts), Moscow: Mashinostroenie, 1974.Google Scholar
  24. 23.
    Lebedev, A.A., Bobronnikov, V.T., Krasil’shchikov, M.N., and Malyshev, V.V., Statisticheskaya di-namika i optimizatsiya upravleniya letatel’nykh apparatov (Statistical Dynamics and Optimization of Control of Aircrafts), Moscow: Mashinostroenie, 1985.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia
  2. 2.Moscow Aviation InstituteMoscowRussia

Personalised recommendations