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Automation and Remote Control

, Volume 80, Issue 1, pp 30–42 | Cite as

On Optimal Retention of the Trajectory of Discrete Stochastic System in Tube

  • V. M. AzanovEmail author
  • Yu. S. Kan
Stochastic Systems
  • 3 Downloads

Abstract

Consideration was given to the design of the optimal control of the general discrete stochastic system with a criterion as the probability of the state vector sojourn in the given sets at each time instant. Derived were relations of the dynamic programming enabling one to establish an optimal solution in the class of Markov strategies without extension of the state vector with subsequent reduction to an equivalent problem with the probabilistic terminal criterion. Consideration was given to the problem of one-parameter correction of the flying vehicle trajectory. An analytical solution was established.

Keywords

discrete systems stochastic optimal control probabilistic criterion method of dynamic programming one-parameter pulse correction control of flying vehicle motion 

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References

  1. 1.
    Malyshev, V.V. and Kibzun, A.I., Analiz i sintez vysokotochnogo upravleniya letatel’nymi apparatami (Analysis and Design of Highly-precise Control of Flying Vehicles), Moscow: Mashinostroenie, 1987.Google Scholar
  2. 2.
    Azanov, V.M. and Kan, Yu.S., Design of Optimal Strategies in the Problems of Discrete System Control by the Probabilistic Criterion, Autom. Remote Control, 2017, vol. 78, no. 6, pp. 1006–1027.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azanov, V.M. and Kan, Yu.S., One-parameter Problem of Optimal Correction of the Flying Vehicle Trajectory by the Probability Criterion, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2016, no. 2, pp. 1–13.zbMATHGoogle Scholar
  4. 4.
    Azanov, V.M., and Kan, Yu.S., Optimization of Correction of the Near-Earth Orbit of the Artificial Earth Satellite by the Probabilistic Criterion, Tr. ISA RAN, 2015, no. 2, pp. 18–26.Google Scholar
  5. 5.
    Krasil’shchikov, M.N., Malyshev, V.V., and Fedorov, A.V., Autonomous Realization of Dynamic Operations on the Geostationary Orbit. I. Formalization of the Control Problem, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2015, no. 6, pp. 82–95.zbMATHGoogle Scholar
  6. 6.
    Malyshev, V.V., Starkov, A.V., and Fedorov, A.V., Design of Optimal Control for Retention of the Spacecraft in Orbital Group, Kosmonav. Raketostroen., 2012, no. 4, pp. 150–158.Google Scholar
  7. 7.
    Malyshev, V.V., Starkov, A.V., and Fedorov, A.V., Registering the Problems of Retention and Deviation in the Neighborhood of the Reference Geostationary Orbit, Vestn. Mosk. Gor. Ped. Univ., Ser. Ekonom., 2013, no. 1, pp. 68–74.Google Scholar
  8. 8.
    Kibzun, A.I. and Ignatov, A.N., On the Existence of Optimal Strategies in the Control Problem for a Stochastic Discrete Time System with Respect to the Probability Criterion, Autom. Remote Control, 2017, vol. 78, no. 10, pp. 1845–1856.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kibzun, A.I. and Ignatov, A.N., Reduction of the Two-Step Problem of Stochastic Optimal Control with Bilinear Model to the Problem of Mixed Integer Linear Programming, Autom. Remote Control, 2016, vol. 77, no. 12, pp. 2175–2192.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kibzun, A.I. and Ignatov, A.N., The Two-Step Problem of Investment Portfolio Selection from Two Risk Assets via the Probability Criterion, Autom. Remote Control, 2015, vol. 76, no. 7, pp. 1201–1220.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jasour, A.M., Aybat, N.S., and Lagoa, C.M., Semidefinite Programming for Chance Constrained Optimization over Semialgebraic Sets, SIAM J. Optim., 2015, no. 25 (3), pp. 1411–1440.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jasour, A.M. and Lagoa, C.M., Convex Chance Constrained Model Predictive Control 2016, arXiv preprint arXiv:1603.07413.CrossRefGoogle Scholar
  13. 13.
    Jasour, A.M. and Lagoa, C.M., Convex Relaxations of a Probabilistically Robust Control Design Problem, in 52nd IEEE Conf. on Decision and Control, 2013, pp. 1892–1897.Google Scholar
  14. 14.
    Bosov, A.V., Generalized Problem of Resource Distribution of Program System, Informat. Primen., 2014, vol. 8. no. 2, pp. 39–47.Google Scholar
  15. 15.
    Bosov, A.V., Problems of Analysis and Optimization for the Model of User Activity. Part 3. Optimization of the External Resources, Informat. Primen., 2012, vol. 6, no. 2, pp. 14–21.Google Scholar
  16. 16.
    Bosov, A.V., Problems of Analysis and Optimization for the Model of User Activity. Part 2. Optimization of the Internal Resources, Informat. Primen., 2012, vol. 6, no. 1, pp. 19–26.Google Scholar
  17. 17.
    Yaroshevskii, V.A. and Petukhov, S.V., Optimal One-parameter Correction of Spacecraft Trajectories, Kosm. Issled., 1970, vol. 8, no. 4, pp. 515–525.Google Scholar
  18. 18.
    Yaroshevskii, V.A. and Parysheva, G.V., Optimal Distribution of Corrective Impulses in the One-Parameter Correction, Kosm. Issled., 1965, vol. 3, no. 6; 1966, vol. 4, no. 1.Google Scholar
  19. 19.
    Azanov, V.M. and Kan, Yu.S., Optimal Control for Linear Discrete Systems with Respect to Probabilistic Criteria, Autom. Remote Control, 2014, vol. 75, no. 10, pp. 1743–1753.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Azanov, V.M. and 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, vol. 79, no. 2, pp. 203–215.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Moscow State Aviation InstituteMoscowRussia

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