MODAL CONTROLLABILITY OF ONE TWO-DIMENSIONAL DELAYED SYSTEM WITH THREE DELAYS

UDC 517.977

Yakimenka Andrei Aliaksandravich – PhD (Physics and Mathematics), Associate Professor, Assistant Professor, the Department of Higher Mathematics. Belarusian State Technological University (13a Sverdlova str., 220006, Minsk, Republic of Belarus). E-mail: yakimenko@belstu.by

DOI: https://doi.org/ 10.52065/2520-6141-2025-290-2.

Key words: delayed systems, modal control, regulators, feedback control, delay, commensurate delays.

For citation: Yakimenka A. A. Modal controllability of one two-dimensional delayed system with three delays. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2025, no. 1 (290), pp. 11–15 (In Russian). DOI: 10.52065/2520-6141-2025-290-2.

Abstract

The publication considers the solution of the modal controllability problem for a two-dimensional stationary dynamic system with a retarded argument with one input and three commensurate delays in one special case. The definition of the modal control problem for the system under study is given. The modal control problem is one of the main problems of control theory. It consists in reducing the coefficients of the characteristic quasi-polynomial of a closed system to a given form. Such a problem has been well studied for systems without delay. For systems with a retarded argument and neutral type systems, the solution of the modal control problem is much more complicated. In the article, a solution to the problem is obtained under certain conditions on the values of the parameters of the system with delay. Also, feedback- type controllers are obtained that solve the modal control problem for the system under study. These controllers are found in the frequency domain as elementary functions of the coefficients of the original system. The rules according to which the obtained regulators are transferred from the frequency domain to the feedback type regulators for the system under study are also given. An illustrative example of solving the modal control problem for the system under consideration is considered. A list of literature isgiven in which the modal control problem is solved for other systems with delay and neutral type systems.

Download

References

  1. Marchenko V. M. On problem of modal control in linear systems with delay. Doklady Akademii nauk BSSR [Reports of the BSSR Academy of Science], 1978, no. 5, pp. 401–404 (In Russian).
  2. Yakimenka A. A. Modal control for one delayed system. Trudy BGTU [Proceedings of BSTU], 2013, no. 6: Physics and Mathematics. Informatics, pp. 3–7 (In Russian).
  3. Yakimenka A. A. Modal control for one neutral type system. Trudy BGTU [Proceedings of BSTU], 2016, no. 6: Physics and Mathematics. Informatics, pp. 18–21 (In Russian).
  4. Yakimenka A. A. Modal control for one neutral type system in general cyclic case. Trudy BGTU [Proceedings of BSTU], issue 3, Physics and Mathematics. Informatics, 2017, no. 2, pp. 25–27 (In Russian).
  5. Yakimenka A. A. Modal control for one neutral type system in general cyclic case with double roots. Trudy BGTU [Proceedings of BSTU], issue 3, Physics and Mathematics. Informatics, 2018, no. 1, pp. 5–8 (In Russian).
  6. Yakimenka A. A. Modal controllability of one two-dimensional delayed system. Trudy BGTU [Proceedings of BSTU], issue 3, Physics and Mathematics. Informatics, 2023, no. 1 (266), pp. 15–19. DOI: 10.52065/ 2520-6141-2023-266-1-3 (In Russian).
  7. Yakimenka A. A. Modal controllability of one two-dimensional delayed system in the case of multiple roots. Trudy BGTU [Proceedings of BSTU], issue 3, Physics and Mathematics. Informatics, 2023, no. 2 (272), pp. 18–22. DOI: 10.52065/2520-6141-2023-272-2-3 (In Russian).
  8. Yakimenka A. A. Modal controllability of one two-dimensional delayed system in a special case. Trudy BGTU [Proceedings of BSTU], issue 3, Physics and Mathematics. Informatics, 2024, no. 2 (284), pp. 5–9. DOI: 10.52065/2520-6141-2024-284-1 (In Russian).

17.11.2024