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Time-Optimal Control Problem of Two Non-Synchronous Oscillators

Author

Listed:
  • Leonid Berlin

    (Institute of Control Sciences of RAS, 117997 Moscow, Russia)

  • Andrey Galyaev

    (Institute of Control Sciences of RAS, 117997 Moscow, Russia)

  • Pavel Lysenko

    (Institute of Control Sciences of RAS, 117997 Moscow, Russia)

Abstract

The time-optimal control problem for a system consisting of two non-synchronous oscillators is considered. Each oscillator is controlled with a shared limited scalar control. The objective of the control is to accelerate the oscillatory system to a given specific position, where the first oscillator must have non-zero phase coordinates, but the second one must remain motionless at the terminal moment. For an arbitrary number of unknown switching moments that determine the optimal relay control, the necessary extremum conditions in the form of nonlinear matrix equalities are proposed. The study of the necessary/sufficient conditions of the extremum made it possible to describe the reachability set in the phase space of the first oscillator, to find an analytical form of the curve corresponding to the two-switching control class, which also separates the reachability set of the three switching-control class. The corresponding theorems are proved and the dependence of the criteria on control constraints is shown. Matrix conditions for different classes of control switchings are found. All of the obtained analytical results are numerically validated and illustrated with mathematical modeling.

Suggested Citation

  • Leonid Berlin & Andrey Galyaev & Pavel Lysenko, 2022. "Time-Optimal Control Problem of Two Non-Synchronous Oscillators," Mathematics, MDPI, vol. 10(19), pages 1-19, September.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3552-:d:928934
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    References listed on IDEAS

    as
    1. Jing Li & Yuying Chen & Shaotao Zhu, 2022. "Periodic Solutions and Stability Analysis for Two-Coupled-Oscillator Structure in Optics of Chiral Molecules," Mathematics, MDPI, vol. 10(11), pages 1-24, June.
    2. Zhao, Y. & Chen, G.H., 2003. "Two oscillators in a dissipative bath," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(1), pages 13-40.
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