Simulation of the Optimal Control of the Established Motion of the Oscillatory System in Random Excitation
DOI:
https://doi.org/10.31649/mccs2022.24Keywords:
vibration system, pendulum, oscillations, modeling, optimal control, random excitation, white noise, torqueAbstract
An approach to modeling the motion dynamics of an oscillatory system with external random excitation is proposed. This made it possible to determine the optimal control modes for the established movement of the system. In the oscillatory system under consideration, as an example of one of the types of vibration machine, random periodic force excitation is presented in the form of "white noise". Also, the motion of the system is described by stochastic differential equations. The periodic component of the excitation is represented as an expansion in terms of cosines. It is accepted that random and deterministic excitations have the same effect on the motion of the system. It is determined that in oscillatory systems excited by white noise, the value of the drift coefficients in the functional is formed only by averaging the deterministic components. This made it possible to write down the averaged dynamic programming equation and build a control synthesis. The used principle of dynamic programming determined the synthesis of control and the stochastic principle of maximum. This made it possible to build program management. The function of optimal control of the external application of the moment of forces to the suspension (executive body) is determined. This is necessary to stabilize the oscillatory system in case of random force excitation of the system as a whole. Based on the optimal control equation, special cases are considered, namely: parametric excitation includes only the first harmonic, and there is no parametric resonance; external force excitation does not contain the first harmonic, there is no external resonance; there are no external force random excitations. It is shown that for any admissible control with the help of the torque applied to the suspension, the process is reduced to diffusion. An optimal search is also performed on the trajectories (modes) of the limiting diffusion system.
References
Iskovych–Lototskyi R. D. Analiz vykorystannia hidroimpulsnykh vibrorozvantazhuvalnykh prystroiv na avtomobilnomu transporti / R. D. Iskovych–Lototskyi, Ya. V. Ivanchuk // Visnyk Vinnytskoho politekhnichnoho instytutu. – 2011, – № 6. –
S. 228 – 231.
Iskovych–Lototskyi R. D. Osnovy rezonansno–strukturnoi teorii vibroudarnoho rozvantazhennia transportnykh zasobiv / R. D. Iskovych–Lototskyi, Ya. V. Ivanchuk, Ya. P. Veselovskyi // Nauka ta prohres transportu. Visnyk Dnipropetrovskoho natsionalnoho universytetu zaliznychnoho transportu im. akademika V. Lazariana. – D., 2014. – № 5(53) – S. 109 – 118.
doi: 10.15802/stp2014/30458.
Kovaleva A. S. Upravlenye kolebatelnimy y vybroudarnimy systemamy. – M.: Nauka. Hl. red. Fyz. mat. lyt., 1990. –
s.
Ivanchuk Ya. V. Matematychnyi metod vyznachennia stiikosti kolyvalnykh system pid diieiu zovnishnoho vibratsiinoho navantazhennia / Ya. V. Ivanchuk / Tekhnichni nauky ta tekhnolohii : naukovyi zhurnal / Chernihiv. nats. tekhn. un-t. – Chernihiv : ChNTU, 2018. – № 2 (12). – S. 25 – 33. doi: 10.25140/2411-5363-2018-2(12)-25-33.
Svetlytskyi V. A. Sluchainie kolebanyia mekhanycheskykh system. – M.: Mashynostroenye, 1976. – 462 s.
Hussman U. G. On the approximation of optimal stochastic control. J. Optimiz. Th. Appl. 1983. Vol. 40, No. 3.
P. 433–450.
Rostislav D. Iskovych-Lototsky, Yaroslav V. Ivanchuk, Natalia R. Veselovska, Wojciech Surtel, Samat Sundetov. "Automatic system for modeling vibro-impact unloading bulk cargo on vehicles", Proc. SPIE 10808, Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2018, 1080860 (1 October 2018).
doi: 10.1117/12.2501526.
Ito K. On stochastic differential equations. Memoirs Amer. Math. Soc. 1992. Vol. 4. No. 1. P. 234–251.
Rostislav Iskovich-Lototsky, Ivan Kots, Yaroslav Ivanchuk, Yevheniy Ivashko, Konrad Gromaszek, Assel Mussabekova, Mashat Kalimoldayev. "Terms of the stability for the control valve of the hydraulic impulse drive of vibrating and vibro-impact machines // Przeglad Elektrotechniczny. – 2019. Vol. 4, no. 19. – P. 19-23. doi: 10.15199/48.2019.04.04.
Rozenvasser E. N. Peryodycheskye nestatsyonarnie systemi upravlenyia. – M.: Nauka, 1973. – 512 s.
Ivanchuk Ya. V. Matematychne modeliuvannia robochykh protsesiv v keruiuchii aparaturi hidroimpulsnoho pryvoda /
Ya. V. Ivanchuk, R. D. Iskovych-Lototskyi, I. V. Sevostianov, N. R. Veselovska ta in. // Mechanics and Advanced Technologies/ Tom 5, №2 (2021) – S. 47-56. doi: https://doi.org/10.20535/2521-1943.2021.5.2.243661.
