Skip to main content
SpringerLink
Log in

Application of Conditional-Optimal Filter for Synthesis of Suboptimal Control in the Problem of Optimizing the Output of a Nonlinear Differential Stochastic System

  • topical-issue
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We propose a suboptimal solution for the control problem for the Ito diffusion process and linear controlled output with a quadratic quality criterion for the case of indirect observations of the state. We use the previously obtained solution of the problem with complete information, the concept of separation between control and filtering problems, and the method of conditionally optimal filtering developed by V.S. Pugachev. We propose an alternative to the traditional practical approach for the synthesis of suboptimal control in a problem with incomplete information, which consists in a formal substitution in the solution of a state with its estimate. Instead of the problem of optimizing the output generated by the original model of the differential equation, we use as the state an estimate of a conditionally optimal filter. We develop one version of the numerical implementation of the proposed algorithm based on the Monte Carlo method and computer simulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kushner, H. J. & Dupuis, P. G. Numerical Methods for Stochastic Control Problems in Continuous Time. (Springer-Verlag, New York, 2001).

    Book  Google Scholar 

  2. Bertsekas, D. P. Dynamic Programming and Optimal Control. (Athena Scientific, Cambridge, 2013).

    Google Scholar 

  3. Fleming, W. H. & Rishel, R. W. Deterministic and Stochastic Optimal Control. (Springer-Verlag, New York, 1975). Translated under the title Optimal’noe upravlenie determinirovannymi i stokhasticheskimi sistemami, Moscow: Mir, 1978.

    Book  Google Scholar 

  4. Athans, M. Editorial on the LQG Problem. IEEE Trans. Automat. Control 16(no. 6), 528–552 (1971).

    Article  MathSciNet  Google Scholar 

  5. Bortakovskii, A. S. Sufficient Optimality Conditions for Controlled Switched Systems. J. Comput. Syst. Sci. 56(no. 4), 636–651 (2017).

    Article  MathSciNet  Google Scholar 

  6. Khrustalev, M. M. & Onegin, E. E. Necessary and Sufficient Conditions for Optimal Stabilization of Quasi-linear stochastic Systems. Autom. Remote Control 80(no. 7), 1252–1264 (2019).

    Article  MathSciNet  Google Scholar 

  7. Panteleev, A. V. & Rybakov, K. A. Synthesis of Optimal Nonlinear Stochastic Control Systems with the Spectral Method. Informat. Primen. 5(no. 2), 69–81 (2011).

    Google Scholar 

  8. Fleming, W. H. Stochastic Control of Partially Observable Diffusions. SIAM J. Control 6(no. 2), 194–214 (1968).

    Article  MathSciNet  Google Scholar 

  9. Khrustalev, M. M. Optimal and Stable Controllable Stochastic Systems Synthesis with Incomplete State Information on an Unbounded Time Interval. Autom. Remote Control 72(no. 11), 2379–2394 (2011).

    Article  MathSciNet  Google Scholar 

  10. Panteleev, A. V. & Rybakov, K. A. Optimal Continuous Stochastic Control Systems with Incomplete Feedback: Approximate Synthesis. Autom. Remote Control 79(no. 1), 103–116 (2018).

    Article  MathSciNet  Google Scholar 

  11. Wonham, W. M. On the Separation Theorem of Stochastic Control. SIAM J. Control 6(no. 2), 312–326 (1968).

    Article  MathSciNet  Google Scholar 

  12. Georgiou, T. T. & Lindquist, A. The Separation Principle in Stochastic Control, Redux. IEEE Trans. Automat. Control 58(no. 10), 2481–2494 (2013).

    Article  MathSciNet  Google Scholar 

  13. Bortakovskii, A. S. & Nemychenkov, G. I. Optimal in the Mean Control of Deterministic Switchable Systems Given Discrete Inexact Measurements. J. Comput. Syst. Sci. 58(no. 1), 50–74 (2019).

    Article  Google Scholar 

  14. Davtyan, L. G. & Panteleev, A. V. Method of Parametric Optimization of Nonlinear Continuous Systems of Joint Estimation and Control. J. Comput. Syst. Sci. 58(no. 3), 360–373 (2019).

    Article  Google Scholar 

  15. Mortensen, R. E. Stochastic Optimal Control with Noisy Observations. Int. J. Control 4(no. 5), 455–464 (1966).

    Article  MathSciNet  Google Scholar 

  16. Bensoussan, A. Stochastic Control of Partially Observable Systems. (Cambridge Univ. Press, Cambridge, 1992).

    Book  Google Scholar 

  17. Davis, M. H. A. & Varaiya, P. P. Dynamic Programming Conditions for Partially Observable Stochastic Systems. SIAM J. Control 11(no. 2), 226–262 (1973).

    Article  MathSciNet  Google Scholar 

  18. Benes, V. E. & Karatzas, I. On the Relation of Zakaias and Mortensenas Eq.s. SIAM J. Control Optim. 21(no. 3), 472–489 (1983).

    Article  MathSciNet  Google Scholar 

  19. Miller, B. M., Avrachenkov, K. E., Stepanyan, K. V. & Miller, G. B. The Problem of Optimal Stochastic Data Flow Control Based Upon Incomplete Information. Probl. Inform. Transmis. 41(no. 2), 150–170 (2005).

    Article  Google Scholar 

  20. Bosov, A. V. & Stefanovich, A. I. Control over the Output of a Stochastic Differential System with the Quadratic Criterion. I. Optimal Solution with Dynamic Programming. Informat. Primen. 12(no. 3), 99–106 (2018).

    Google Scholar 

  21. Liptser, R. Sh & Shiryaev, A. N. Statistics of Random Processes. II. Applications. (Springer-Verlag, Berlin, 2001).

    Book  Google Scholar 

  22. Pugachev, V. S. & Sinitsyn, I. N. Stochastic Differential Systems. Analysis and Filtering. (Wiley, New York, 1987).

    MATH  Google Scholar 

  23. Gilbert, E.N.Capacity of a Burst-noise Channel, Bell Syst. Tech. J., 1960, vol. 39, pp. 1253–1265.

  24. Elliott, E. O. Estimates of Error Rates for Codes on Burst-noise Channels. Bell Syst. Tech. J. 42, 1977–1997 (1963).

    Article  Google Scholar 

  25. Altman, E., Avrachenkov, K. & Barakat, C. TCP in Presence of Bursty Losses. Perform. Evaluat. 42, 129–147 (2000).

    Article  Google Scholar 

  26. Borisov, A., Bosov, A. & Miller, G. Modeling and Monitoring of RTP Link on the Receiver Side. Lect. Notes Comput. Sci. 9247, 229–241 (2015).

    Article  Google Scholar 

  27. Borisov, A. V. Application of Optimal Filtering Methods for On-line of Queueing Network States. Autom. Remote Control 77(no. 2), 277–296 (2016).

    Article  MathSciNet  Google Scholar 

  28. Whitt, W. Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and their Application to Queues. (Springer, New York, 2002).

    Book  Google Scholar 

  29. Bohacek, S., A Stochastic Model of TCP and Fair Video Transmission, 22nd Annual Joint Conf. IEEE Computer and Communications (INFOCOM), Proc. IEEE, 2003, vol. 2, pp. 1134–1144.

  30. Tanenbaum, A.S. and Woodhull, A.S.Operating Systems: Design and Implementation, Upper Saddle River: Prentice Hall, 2006, 3rd ed. Translated under the title Operatsionnye sistemy. Razrabotka i realizatsiya, St. Petersburg: Piter, 2007.

  31. Date, C.J.An Introduction to Database Systems, Reading: Addison-Wesley, 2004, 8th ed. Translated under the title Vvedenie v sistemy baz dannykh, Moscow: Vil’yams, 2005.

  32. Elsässer, R., Monien, B. & Preis, R. Diffusion Schemes for Load Balancing on Heterogeneous Networks. Theor. Comput. Syst. 35(no. 3), 305–320 (2002).

    Article  MathSciNet  Google Scholar 

  33. Welzl, M. Network Congestion Control. (Wiley, New York, 2005).

    Book  Google Scholar 

  34. Borovkov, A. A. Asymptotic Methods in Queuing Theory. (Wiley, New York, 1984).

    Google Scholar 

  35. Bosov, A. V. Discrete Stochastic System Linear Output Control with Respect to a Quadratic Criterion. J. Comput. Syst. Sci. 55(no. 3), 349–364 (2016).

    Article  MathSciNet  Google Scholar 

  36. Fel’dbaum, A. A. Osnovy teorii optimalanykh avtomaticheskikh sistem (Foundations of the Theory of Optimal Automatic Systems). 2nd ed (Nauka, Moscow, 1966).

    Google Scholar 

  37. Kloden, P. E. & Platen, E. Numerical Solution of Stochastic Differential Equations. (Springer-Verlag, Berlin-Heidelberg, 1992).

    Book  Google Scholar 

  38. Bosov, A. V. & Stefanovich, A. I. Control over the Output of a Stochastic Differential System with the Quadratic Criterion. IV. Alternative Numerical Solution. Informat. Primen. 14(no. 14), 24–30 (2020).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Informatics Problems of Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, Russia

    A. V. Bosov

  2. Moscow Aviation Institute, Moscow, Russia

    A. V. Bosov

Authors
  1. A. V. Bosov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bosov, A. Application of Conditional-Optimal Filter for Synthesis of Suboptimal Control in the Problem of Optimizing the Output of a Nonlinear Differential Stochastic System. Autom Remote Control 81, 1963–1973 (2020). https://doi.org/10.1134/S0005117920110028

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117920110028

Keywords

Search

Navigation