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Linear Interpolation of Program Control with Respect to a Multidimensional Parameter in the Convergence Problem

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Mathematical Optimization Theory and Operations Research: Recent Trends (MOTOR 2023)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1881))

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Abstract

We consider a control system containing a constant three-dimensional vector parameter, the approximate value of which is reported to the control person only at the moment of the movement start. The set of possible values of unknown parameter is known in advance. An convergence problem is posed for this control system. At the same time, it is assumed that in order to construct resolving control, it is impossible to carry out cumbersome calculations based on the pixel representation of reachable sets in real time. Therefore, to solve the convergence problem, we propose to calculate in advance several resolving controls, corresponds to possible parameter values in terms of some grid of nodes. If at the moment of the movement start it turns out that the value of the parameter does not coincide with any of the grid nodes, it is possible to calculate the program control using the linear interpolation formulas. However, this procedure can be effective only if a linear combination of controls corresponding to the same “guide” in the terminology of N.N. Krasovskii’s Extreme Aiming Method is used. In order to be able to effectively apply linear interpolation, for each grid cell, we propose to calculate 8 “nodal” resolving controls and use the method of dividing control into basic control and correcting control in addition. Due to the application of the latter method, the calculated solvability set turns out to be somewhat smaller than the actual one. But the increasing of accuracy of the system state transferring to the target set takes place.

This research was supported by the Russian Science Foundation (grant no. 19-11-00105, https://rscf.ru/en/project/19-11-00105/).

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Acknowledgements

This research was supported by the Russian Science Foundation (grant no. 19-11-00105, https://rscf.ru/en/project/19-11-00105/).

Author information

Authors and Affiliations

  1. N.N. Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Vladimir Nikolaevich Ushakov & Aleksandr Anatol’evich Ershov

  2. Ural Federal University, Yekaterinburg, Russia

    Aleksandr Anatol’evich Ershov & Anna Aleksandrovna Ershova

  3. EMDB “Novator”, Yekaterinburg, Russia

    Aleksandr Vladimirovich Alekseev

Authors
  1. Vladimir Nikolaevich Ushakov
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  2. Aleksandr Anatol’evich Ershov
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  3. Anna Aleksandrovna Ershova
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  4. Aleksandr Vladimirovich Alekseev
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Corresponding author

Correspondence to Aleksandr Anatol’evich Ershov .

Editor information

Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. Sobolev Institute of Mathematics, Omsk, Russia

    Anton Eremeev

  4. Melentiev Energy Systems Institute, Irkutsk, Russia

    Oleg Khamisov

  5. Institute of Applied Mathematical Research, Petrozavodsk, Russia

    Vladimir Mazalov

  6. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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Ushakov, V.N., Ershov, A.A., Ershova, A.A., Alekseev, A.V. (2023). Linear Interpolation of Program Control with Respect to a Multidimensional Parameter in the Convergence Problem. In: Khachay, M., Kochetov, Y., Eremeev, A., Khamisov, O., Mazalov, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research: Recent Trends. MOTOR 2023. Communications in Computer and Information Science, vol 1881. Springer, Cham. https://doi.org/10.1007/978-3-031-43257-6_24

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  • DOI: https://doi.org/10.1007/978-3-031-43257-6_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43256-9

  • Online ISBN: 978-3-031-43257-6

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