Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Control of a Movable Platform with a Pendulum

Author(s):

Alexander V. Bratishchev

Don State Technical University,
Professor of Applied Mathematics Department ,
Rostov-on-Don, Gagarin sq.,1

avbratishchev@spark-mail.ru

Abstract:

The connection between the concepts of stability of a variety in the sense of A. A. Lyapunov, F. R. Gantmacher and A. A. Kolesnikov is established. The concept of partial equilibrium position according to a given mapping is introduced. Within framework of the synergetic control theory, the formula of the control vector is found when the aggregated variables are the first integrals of a dynamic system, and the control vector enters the controller in a linear manner. These concepts and results are used in the task of control a platform with a fixed pendulum, when it is necessary to stabilize the platform in a predetermined position, and the pendulum in the lower equilibrium position.

Keywords

References:

  1. Bratichshev A. V. Control of oscillation of the pendulum on a movable platform. Differential equations and control processes. [Electronic resource]: electron. Journal. / St. Petersburg State University, 2020 (4): 75-86, (In Russ. ) - Access mode: http://www.math.spbu.ru/ diffjournal
  2. A V Bratishchev First integrals of the system " platform with pendulum" and controlling of it. 2021 J. Phys. : Conf. Ser. Volume 2131 Mathematical modeling and computational methods in problems of hydro-aerodynamics, magnetohydrodynamics, plasma physics and astrophysics. 022020. P. 1-11
  3. A A Kolesnikov Sinergeticheskiye metodi upravleniya slodznimi sistemami. Teoriya sistemnogo analiza [Synergetic methods of managing complex systems. Theory of system Analysis]. Moscow: KomKniga, 2006. - 240 p
  4. A V Bratichshev Qualitative analysis of the equation of motion of the " Mobile platform with a pendulum" system. Differential equations and control processes. [Electronic resource]: electron. Journal. / St. Petersburg State University, 2021 (2): 49-64, (In Russ. ) - Access mode:http://www. math. spbu. ru/diffjournal
  5. Dabney J.. Harman T. Mastering SIMULINK 4. New Jersey 07458 : Prentice Hall, 2001. 412 p
  6. Lyapunov A. M. Obshchaya zadacha ustoychivosti dvizheniya [The general problem of motion stability] . M. -L. : GITTL, 1950. - 473 p
  7. F R Gantmakher Lecii po analiticheskoy mekhanike [Lectures on analytical mechanics]. 3rd ed. M. : FIZMATLIT, 2005. - 300 p
  8. V V Rumyantsev, A S Oziraner Ustoichevost’ i stabilizaciya dvidzeniy po otosheniyu k chasti peremennikh [Stability and stabilization of movements with respect to a part of variables]. Moscow: Nauka, 1987. - 256 p
  9. I V Miroshnik, I O Nikiforov, A L Fradkov Nelineynoe i adaptivnoe upravlenie slodznikh dinamicheskikh sistem [Nonlinear and adaptive control of complex dynamic systems]. St. Petersburg: Nauka, 2000. - 552 p
  10. V I Vorotnikov, V V Rumyantsev Ustoychivost’ i upravleniye po chasti koordinat phazovogo vektora dinamicheskikh system: teoriya, metodi i prilodzeniya [Stability and control in terms of the coordinates of the phase vector of dynamic systems: theory, methods and applications]. M. : Scientific World, 2001. - 320 p
  11. Levi-Civita T., Amal’di U. Kurs teoreticheskoy mekhaniki [Course in theoretical mechanics]. Vol. 2. Part 2. Moscow: Foreign Literature Publ., 1951. 556 p
  12. V I Zubov Ustoychevost’ dvidzeniya. Metodi Lyapunova i ikh prilodzeniya [Stability of movement. Lyapunov's methods and their application]. Moskow: Vysshaya shkola, 1984. - 232 p
  13. A V Bratichshev On the characteristic polynomial of the equilibrium state of an autonomous system having an attracting invariant manifold. Differential equations and control processes. [Electronic resource]: electron. Journal. / St. Petersburg State University, 2017 (2):15-23. (In Russ. ) - Access mode:http://www. math. spbu. ru/diffjournal
  14. [A V Bratichshev. Factorization of the characteristic polynomial of the equilibrium state an autonomous system that has an attractive invariant manifold. Differential equations and control processes . Differential equations and control processes. [Electronic resource]: electron. Journal. / St. Petersburg State University, 2018 (4):1-17. (In Russ. ) - Access mode:http://www. math. spbu. ru/diffjournal
  15. I Yu Tyukin, V A Terekhov Adaptaciya v nelineynikh dinamisheskikh sistemakh. [Adaptation in nonlinear dynamical systems]. Moscow: LKI Publishing House, 2014. - 384 p

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