Published April 30, 2021 | Version v1
Journal article Open

Determining the effect of fuzziness in the parameters of a linear dynamic system on its stability

Creators

  • 1. Ivano-Frankivsk National Technical University of Oil and Gas
  • 2. Technology Ivano-Frankivsk National Technical University of Oil and Gas

Description

This paper considers a relevant issue related to the influence exerted by the fuzziness in linear dynamic system parameters on its stability. It is known that the properties of automated control systems can change under the influence of parametric disturbances. To describe the change in such properties of the system, the concept of roughness is used.

It should be noted that taking into consideration the fuzziness in the parameters of mathematical models could make it possible at the design stage to assess all the risks that may arise as a result of an uncontrolled change in the parameters of dynamic systems during their operation. To prevent negative consequences due to variance in the parameters of mathematical models, automated control systems are designed on the basis of the requirement for ensuring a certain margin of stability of the system in terms of its amplitude and phase. At the same time, it remains an open question whether such a system would satisfy the conditions of roughness.

Parameters of the mathematical model of a system are considered as fuzzy quantities that have a triangular membership function. This function is inconvenient for practical use, so it is approximated by the Gaussian function. That has made it possible to obtain formulas for calculating the characteristic polynomial and the transfer function of the open system, taking into consideration the fuzziness of their parameters.

When investigating the system according to Mikhailov’s criterion, it was established that the dynamic system retains stability in the case when the parameters of the characteristic equation are considered as fuzzy quantities. It has been determined that the quality of the system significantly deteriorated in terms of its stability that could make it enter a non-steady state. When using the Nyquist criterion, it was established that taking into consideration the fuzziness in the parameters of the transfer function did not affect the stability of the closed system but there was a noticeable decrease in the system stability reserve both in terms of phase and amplitude. The relative decrease in the margin of stability for amplitude was 16 %, and for phase ‒ 17.4 %.

Files

15-21_Determining the effect of fuzziness in the parameters of a linear dynamic system on its stability.pdf

Files (757.8 kB)

Additional details

References

  • Kovryho, Yu. M., Stepanets, O. V., Bahan, T. H., Bunke, O. S. (2018). Suchasna teoriya upravlinnia. Chastyna 2. Prykladni aspekty suchasnoi teoriyi upravlinnia. Kyiv: KPI im. Ihoria Sikorskoho, 155. Available at: https://ela.kpi.ua/bitstream/123456789/23117/1/Suchasna-teoriia-upravlinnia_Kovrygo_et.al.pdf
  • Khrabatyn, R. I., Samaniv, L. V., Krykhivskyi, M. V. (2011). Systematyzuvannia matematychnykh modelei system upravlinnia u vyhliadi peredavalnykh funktsiy. Naftohazova enerhetyka, 1 (14), 99–101. Available at: http://elar.nung.edu.ua/handle/123456789/2867
  • Maksymov, V. H., Nitsevych, O. D., Droma, I. A. (2013). Osnovy metodiv diahnostuvannia elektronnykh system keruvannia avtomobilem. Pratsi Odeskoho politekhnichnoho universytetu, 3 (42), 60–65. Available at: http://dspace.opu.ua/jspui/handle/123456789/5179
  • Strutinsky, S., Gurgiy, A. (2011). Application of theory of fuzzy sets is for description of clearance of connections in occasions and transmissions. Materialy mizhnarodnoi naukovo-tekhnichnoi konferentsiyi "Problemy suchasnykh tekhnolohiy vyhotovlennia ta nadiynosti peredach z hnuchkym zviazkom". Ternopil, 88–89. Available at: http://elartu.tntu.edu.ua/handle/123456789/12252
  • Wong, L. K., Leung, F. H. F., Tam, P. K. S. (2000). Improved stability analysis method for fuzzy logic control systems. Electronics Letters, 36 (12), 1085. doi: https://doi.org/10.1049/el:20000787
  • Albers, D. J., Sprott, J. C. (2005). Structural Stability and Hyperbolicity Violation in High-Dimensional Dynamical Systems, 38. Available at: http://tuvalu.santafe.edu/~albers/research/papers/hyp.pdf
  • Garashenko, F. G., Soproniyk, O. L. (2016). Analysis of the practical stability and sensitivity of linear dynamical systems with change of phase space measurability. System research and information technologies, 3, 86–100. doi: https://doi.org/10.20535/srit.2308-8893.2016.3.08
  • Yukhymchuk, M. S., Osipenko, A. A. (2015). Method of stability analysis of automatic systems with logic control devices when exposed to parametric perturbations on the basis of the sensitivity functions. Visnyk Vinnytskoho politekhnichnoho instytutu, 2, 92–100. Available at: http://nbuv.gov.ua/UJRN/vvpi_2015_2_15
  • Astrom, K. I., Hagglund, T. (2006). Advanced PID Control. The Instrumentation, Systems And Automation Society, 460.
  • Raskin, L. G., Seraya, O. V. (2008). Nechetkaya matematika. Osnovy teorii. Prilozheniya. Kharkiv: Parus, 352. Available at: http://library.kpi.kharkov.ua/files/new_postupleniya/raskin.pdf
  • Verzhbitskiy, V. M. (2002). Osnovy chislennyh metodov. Moscow: Vysshaya shkola, 840.
  • Horbiychuk, M. I., Pistun, Ye. P. (2010). Chyslovi metody i modeliuvannia na EOM. Ivano-Frankivsk: Fakel, 403.
  • Gorbiychuk, M. I., Skripka, O. A., Pashkovskyi, B. V. (2016). Optimum distribution of gas compressor units in uncertainty for a given volume on pumping natural gas. East European Scientific Journal, 2 (3 (7)), 53–58.
  • Gorbiychuk, M. I., Lazoriv, O. T., Zaiachuk, Y. I. (2020). Depth-Optimal Distribution of Drilling Meterage Under Uncertainty. Cybernetics and Systems Analysis, 56 (2), 269–277. doi: https://doi.org/10.1007/s10559-020-00242-9