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Optimal in the Mean Control of Deterministic Switchable Systems Given Discrete Inexact Measurements

  • A. S. BortakovskiiEmail author
  • G. I. Nemychenkov
CONTROL IN STOCHASTIC SYSTEMS AND UNDER UNCERTAINTY CONDITIONS
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Abstract

We consider the problem of the optimal-in-the-mean control of a switchable system whose continuous change of state is described by differential equations, whereas instantaneous discrete changes of the state (switches) are described by recurrent equations. Discrete changes in the control process simulate the operation of an automaton (with a memory) that switches modes of the continuous motion of a control plant. Switching moments and their number are not set in advance. The quality of the control is characterized by a functional that takes into account the cost of each switch. The state of the control plant is not known exactly; however, this state is refined as a result of discrete inexact measurements. Therefore, in addition to the problem of the optimal control, the problem of the optimal-in-the-mean control of bundles of trajectories is also studied. Sufficient conditions for the optimality of control are obtained; based on them, algorithms for constructing the suboptimal control of bundles of trajectories of switchable systems given discrete inexact measurements are proposed. The use of the algorithms is demonstrated by academic examples.

Notes

ACKNOWLEDGMENTS

This work was performed by assignment no. 1.7983.2017/VU of the Ministry of Education and Science of the Russian Federation.

REFERENCES

  1. 1.
    S. N. Vasil’ev and A. I. Malikov, “Some results on the stability of switchable and hybrid systems,” in Actual Problems of Mechanics of Continuous Media, To 20 Years of Inst. Mech. Eng. Kazan Sci. Center of RAS (Foliant, Kazan, 2011), vol. 1, pp. 23–81 [in Russian].Google Scholar
  2. 2.
    A. S. Bortakovskii, “Sufficient optimality conditions for controlling switched systems,” J. Comput. Syst. Sci. Int. 56, 636 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. S. Bortakovskii, Optimization of Switching Systems (Mosk. Aviats. Inst., Moscow, 2016) [in Russian].zbMATHGoogle Scholar
  4. 4.
    A. S. Bortakovskii, “Synthesis of optimal control systems with a change of the models of motion,” J. Comput. Syst. Sci. Int. 57, 543 (2018).CrossRefzbMATHGoogle Scholar
  5. 5.
    R. Bellman, Dynamic Programming, Dover Books on Computer Science (Dover, New York, 2003).zbMATHGoogle Scholar
  6. 6.
    D. A. Ovsyannikov, Mathematical Methods of Beam Control (Leningr. Gos. Univ., Leningrad, 1980) [in Russian].Google Scholar
  7. 7.
    T. F. Anan’ina, “Incomplete data control task,” Differ. Uravn. 12, 612–620 (1976).MathSciNetGoogle Scholar
  8. 8.
    W. M. Wonham, “On the separation theorem of stochastic control,” SIAM J. Control 6, 312–326 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    F. L. Chernous’ko and A. A. Melikyan, Game-Theoretical Problems of Control and Search (Nauka, Moscow, 1978) [in Russian].zbMATHGoogle Scholar
  10. 10.
    F. L. Chernous’ko, Estimation of the Phase State of Dynamic Systems: Ellipsoid Method (Nauka, Moscow, 1988) [in Russian].Google Scholar
  11. 11.
    A. S. Bortakovskii, “Optimal and suboptimal control for sets of trajectories of deterministic continuous-discrete systems,” J. Comput. Syst. Sci. Int. 48, 14 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. S. Bortakovskii and G. I. Nemychenkov, “Suboptimal control of bunches of trajectories of discrete deterministic automaton time-invariant systems,” J. Comput. Syst. Sci. Int. 56, 914 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. S. Bortakovskii, “Optimal and suboptimal control over bunches of trajectories of automaton-type deterministic systems,” J. Comput. Syst. Sci. Int. 55, 1 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control (Nauka, Moscow, 1973) [in Russian].Google Scholar
  15. 15.
    A. S. Bortakovskii, “Synthesis of optimal processes with the change of motion equations,” in Proceedings of the International Conference on Topological Methods in Dynamics and Related Topics, Nizh. Novgorod (2019, in press).Google Scholar
  16. 16.
    V. V. Aleksandrov, V. G. Boltyanskii, S. S. Lemak, et al., Optimal Control of Motion (Fizmatlit, Moscow, 2005) [in Russian].Google Scholar
  17. 17.
    M. M. Khrustalev, “Necessary and sufficient conditions for optimality in the form of the bellman equation,” Dokl. Akad. Nauk SSSR 242, 1023–1026 (1978).MathSciNetGoogle Scholar
  18. 18.
    Yu. G. Evtushenko, Methods of Solving Extreme Problems and their Application in Optimization Systems (Nauka, Moscow, 1982) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Moscow Institute of Aviation (National Research University)MoscowRussia

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