Інтегрування нестаціонарних систем оптимального керування із післядією та нестабільним спектром

Петро  Самусенко; Тетяна Новік; Микола Рашевський

Authors

Petro Samusenko✉ National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” Author https://orcid.org/0000-0002-4241-6173
Tetiana Novik✉ National Aviation University Author https://orcid.org/0009-0003-3146-6292
Mykola Rashevskyi✉ National Aviation University Author https://orcid.org/0000-0003-1136-2691

Keywords:

system of automatic control, systems of differential equations with degenerations, asymptotic solution, turning point, systems with delays

Abstract

Purpose. We considered a non-stationary optimal control system with delay. Non-stationary optimal control systems are described by systems of differential or differential-algebraic equations. Variable coefficients do not allow, in general, to construct a solution to such systems in quadratures. Numerical or asymptotic methods are used to solve such systems. Design / Method / Approach. In this paper, asymptotic methods are used, in particular, the Feshchenko-Shkil' method for integrating singularly perturbed systems and the Wasow’s method for systems with an unstable spectrum. Findings. In this paper, we construct a transformation that reduces the optimal delay control system to a system that does not contain terms without rejecting the argument. This transformation makes it possible to integrate the system by the method of steps without solving systems of differential equations at each step. Theoretical Implications. The system of equations obtained as a result of the transformation of the original system is somewhat easier to study in terms of building a solution. However, the problem of optimizing the control of both systems requires both a separate mathematical study and clarification of the practical reality of spectrum instability in such systems. Practical Implications. If the instability of the spectrum is caused by the degeneracy of the main matrix, this leads to the unboundedness of the system solution as the small parameter approaches zero. The aforementioned growth of the solution can create emergency situations in real systems. Originality / Value. The delayed control systems in the described formulation are studied for the first time. Research Limitations / Future Research. Future research concerns solving the problem of optimal control of systems with an unstable spectrum and studying the question of the reality and physical meaning of turning points in specific systems. Paper Type. Conceptual Paper.

PURL: https://purl.org/cims/2403.015

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References

Boykov, I. V., & Krivulin, N. P. (2021). Methods for Control of Dynamical Systems with Delayed Feedback. Journal of Mathematical Sciences, 255(5), 561–573. https://doi.org/10.1007/s10958-021-05393-4

Michiels, W. (2021). Control of linear systems with delays. In Encyclopedia of Systems and Control (pp. 338-345). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-44184-5_16

Stanzhytskyi, O. M., Kichmarenko, O. D., Mogylova, V. V., & Koval’chuk, T. V. (2023). Optimal Control Over Systems of Functional-Differential Equations With Infinite Delay. Ukrainian Mathematical Journal, 75(1), 157–173. https://doi.org/10.1007/s11253-023-02191-w

Самусенко, П., Даниліна, Г., & Рашевський, М. (2024). Про асимптотичне інтегрування лінійних систем диференціальних рівнянь з відхиленням аргументу. Збірник Наукових Праць Фізико-Математичного Факультету ДДПУ, 14, 015–023. https://doi.org/10.31865/2413-26672415-3079142024311278