Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Authors

DOI:

https://doi.org/10.15587/1729-4061.2024.301714

Keywords:

system oscillations, control problem, method of straight lines, functional convergence

Abstract

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

Author Biographies

Kamil Mamtiyev, Azerbaijan State University of Economics (UNEC)

PhD, Associate Professor

Department of Digital Technologies and Applied Informatics

Ulviyya Rzayeva, Azerbaijan State University of Economics (UNEC)

PhD on Mathematics, Associate Professor, Head of Center

Department of Digital Technologies and Applied Informatics

Center of Digital Economy

References

  1. Gabasov, R. (1973). Special optimal controls. Moscow: Nauka, 256.
  2. Akan, M., Geçici, E. (2023). An application of optimal control in medical systems: optimal investment strategy in doctors. Network Modeling Analysis in Health Informatics and Bioinformatics, 12 (1). https://doi.org/10.1007/s13721-022-00408-9
  3. Kononenko, V. O. (1964). Oscillatory systems with limited excitation. Moscow: Nauka.
  4. Butkovsky, G. (1969). Distributed Control Systems. American Elsevier Publishing Company, 446.
  5. Egorov, A. I., Znamenskaya, L. N. (2012). Boundary observability of elastic vibrations in a system of sequentially connected strings. Computational Mathematics and Mathematical Physics, 52 (9), 1233–1238. https://doi.org/10.1134/s0965542512070056
  6. Znamenskaya, L. N. (2007). Two-end observability of elastic vibrations in distributed and lumped parameter systems. Computational Mathematics and Mathematical Physics, 47 (6), 900–914. https://doi.org/10.1134/s0965542507060024
  7. Egorov, A. I., Znamenskaya, L. N. (2010). Observability of elastic oscillations of the network with distributed and concentrated parameters on free boundaries. Trudy Inst. Mat. i Mekh. UrO RAN, 16 (5), 76–81.
  8. Egorov, A. I., Znamenskaya, L. N. (2009). Controllability of vibrations of a net of coupled objects with distributed and lumped parameters. Computational Mathematics and Mathematical Physics, 49 (5), 786–796. https://doi.org/10.1134/s0965542509050054
  9. Egorov, A. I. (1965). Optimal processes in systems containing distributed parameter plants. Avtomatika i Telemekhanika, 26 (6), 977–994.
  10. Sakawa, Y. (1964). Solution of an optimal control problem in a distributed-parameter system. IEEE Transactions on Automatic Control, 9 (4), 420–426. https://doi.org/10.1109/tac.1964.1105753
  11. Mamtiyev, K., Aliyeva, T., Rzayeva, U. (2021). Analysis of one class of optimal control problems for distributed-parameter systems. Eastern-European Journal of Enterprise Technologies, 5 (4 (113)), 26–33. https://doi.org/10.15587/1729-4061.2021.241232
  12. Henrion, D., Kružík, M., Weisser, T. (2019). Optimal control problems with oscillations, concentrations and discontinuities. Automatica, 103, 159–165. https://doi.org/10.1016/j.automatica.2019.01.030
  13. Liu, X., Wang, S., Zeng, L., Yuan, S., Hao, Z., Lu, X. (2023). An Adaptive Additional Control Strategy for Suppressing Low-Frequency Grid Oscillations in Doubly-Fed Wind Farms, 57 (9), 1156–1164. https://doi.org/10.16183/j.cnki.jsjtu.2022.135
  14. Papadopoulos, A., Gavalas, I., Chasalevris, A. (2023). Controlling bifurcations in high-speed rotors utilizing active gas foil bearings. Bulletin of the Polish Academy of Sciences Technical Sciences. https://doi.org/10.24425/bpasts.2023.146796
  15. Arroyo, J., Spiessens, F., Helsen, L. (2022). Comparison of Optimal Control Techniques for Building Energy Management. Frontiers in Built Environment, 8. https://doi.org/10.3389/fbuil.2022.849754
  16. Manuel V., C., Francisco, M. (2021). A Powerful Tool for Optimal Control of Energy Systems in Sustainable Buildings: Distortion Power Bivector. Energies, 14 (8), 2177. https://doi.org/10.3390/en14082177
  17. Semaan, R., Oswald, P., Cornejo Maceda, G. Y., Noack, B. R. (2023). Aerodynamic optimization of a generic light truck under unsteady conditions using gradient-enriched machine learning control. Experiments in Fluids, 64 (3). https://doi.org/10.1007/s00348-023-03587-x
  18. Sowa, S. W., Baldea, M., Contreras, L. M. (2014). Optimizing Metabolite Production Using Periodic Oscillations. PLoS Computational Biology, 10 (6), e1003658. https://doi.org/10.1371/journal.pcbi.1003658
Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

Downloads

Published

2024-04-30

How to Cite

Mamtiyev, K., & Rzayeva, U. (2024). Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system. Eastern-European Journal of Enterprise Technologies, 2(2 (128), 64–74. https://doi.org/10.15587/1729-4061.2024.301714

Issue

Vol. 2 No. 2 (128) (2024): Information technology. Industry control systems

Section

Industry control systems

License

Copyright (c) 2024 Kamil Mamtiyev, Ulviyya Rzayeva

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.

A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.