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

  • 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.

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REFERENCES

  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].

  2. A. S. Bortakovskii, “Sufficient optimality conditions for controlling switched systems,” J. Comput. Syst. Sci. Int. 56, 636 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. S. Bortakovskii, Optimization of Switching Systems (Mosk. Aviats. Inst., Moscow, 2016) [in Russian].

    MATH  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  5. R. Bellman, Dynamic Programming, Dover Books on Computer Science (Dover, New York, 2003).

    MATH  Google Scholar 

  6. D. A. Ovsyannikov, Mathematical Methods of Beam Control (Leningr. Gos. Univ., Leningrad, 1980) [in Russian].

    Google Scholar 

  7. T. F. Anan’ina, “Incomplete data control task,” Differ. Uravn. 12, 612–620 (1976).

    MathSciNet  Google Scholar 

  8. W. M. Wonham, “On the separation theorem of stochastic control,” SIAM J. Control 6, 312–326 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  9. F. L. Chernous’ko and A. A. Melikyan, Game-Theoretical Problems of Control and Search (Nauka, Moscow, 1978) [in Russian].

    MATH  Google Scholar 

  10. F. L. Chernous’ko, Estimation of the Phase State of Dynamic Systems: Ellipsoid Method (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  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).

  16. V. V. Aleksandrov, V. G. Boltyanskii, S. S. Lemak, et al., Optimal Control of Motion (Fizmatlit, Moscow, 2005) [in Russian].

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

  18. Yu. G. Evtushenko, Methods of Solving Extreme Problems and their Application in Optimization Systems (Nauka, Moscow, 1982) [in Russian].

    Google Scholar 

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ACKNOWLEDGMENTS

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

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Authors and Affiliations

  1. Moscow Institute of Aviation (National Research University), 125993, Moscow, Russia

    A. S. Bortakovskii & G. I. Nemychenkov

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  1. A. S. Bortakovskii
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  2. G. I. Nemychenkov
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Correspondence to A. S. Bortakovskii.

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Translated by L. Kartvelishvili

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Bortakovskii, A.S., Nemychenkov, G.I. Optimal in the Mean Control of Deterministic Switchable Systems Given Discrete Inexact Measurements. J. Comput. Syst. Sci. Int. 58, 50–74 (2019). https://doi.org/10.1134/S1064230719010052

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  • DOI: https://doi.org/10.1134/S1064230719010052

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