Stabilization of unstable objects: coupled oscillators

Theoretical mechanics


Аuthors

Semenov M. E.1*, Solovyov A. M.1**, Popov M. A.2***

1. Voronezh State University, 1, Universitetskaya sq., Voronezh, 394006, Russia
2. Voronezh State University of Architecture and Civil Engineering, XX-letiya Oktyabrya St. 84, 394006 Voronezh, Russia

*e-mail: mkl150@mail.ru
**e-mail: darkzite@yandex.ru
***e-mail: soeltic@gmail.com

Abstract

As is known, the problem of inverted pendulum plays a central role in the control theory. In particular, the problem of inverted pendulum (as a test model) provides many challenging problems to control design. Due to their nonlinear nature, pendulums have maintained their usefulness and now they are used to illustrate many ideas emerging in the field of nonlinear control. Typical examples are feedback stabilization, variable structure control, passivity-based control, back-stepping and forwarding, nonlinear observers, friction compensation, and nonlinear model reduction. The challenges of control made the inverted pendulum systems a classic tool in control laboratories. It should also be noted that the problem of such a system stabilization is a classical problem of the dynamics and control theory. Moreover, the model of inverted pendulum is widely used as a standard for testing of control algorithms, such as PID controller, neural networks, fuzzy control, etc.

The article investigates the dynamics of a mechanical system consisting of two inverted pendulums hinged on the moving platform and coupled by a spring. The force applied to the platform causing its horizontal motion is treated as a control. The purpose of this work consists in solving the problem of pendulums stabilization in vertical position using the horizontal motion of the platform at presence of the information on the angles of deviation. To solve this problem, we developed the algorithm of the pendulums stabilization near vertical position, found the stability zones and their dependence on the spring stiffness.

Keywords:

inverted pendulum, coupled oscillators, stabilization, control

References

  1. Stephenson A. On an induced stability, Philosophical Magazine, 1908, no.15, pp. 233-236.

  2. Kapitsa P.L. Uspekhi fizicheskikh nauk, 1951, no. 44, pp. 7–20.

  3. Semenov M.E., Grachikov, D.V., Mishin, M.Y., Shevlyakova, D.V. Stabilization and control models of systems with hysteresis nonlinearities, European Researcher, 2012, no. 20, pp. 523 —528.

  4. Semenov M.E., Shevlyakova D.V., Meleshenko P.A. Inverted pendulum under hysteretic control: stability zones and periodic solutions, Nonlinear Dynamics, 2014, no. 75, pp. 247–256.

  5. Semenov M.E., Solovyov A.M., Meleshenko P.A.. Elastic inverted pendulum with backlash in suspension: stabilization problem, Nonlinear Dynamics, 2015, no. 82, pp. 677-688.

  6. Elmer P.D., Patrick S.F. and Williams D.J. Genetic Algorithm On Line Controller for the Flexi-ble Inverted Pendulum Problem, Journal of Advanced Computational Intelligence and Intelligent Informatics, 2006, vol. 10, no. 2, pp. 65-71.

  7. Tang Jiali, Ren Gexue. Modeling and Simulation of a Flexible Inverted Pendulum System. Tsinghua Science and Technology, 2009, vol. 14, no. 2, pp. 22-26.

  8. Semenov M.E., Solovyov A.M., Semenov A.M., Gorlov V.A. and Meleshenko P.A. Elastic inverted pendulum under hysteretic nonlinearity in suspension: stabilization and optimal control. 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015). Greece. 25–27 May 2015. Athens: National Technical University of Athens, 2015, pp. 683-689.

  9. Mikheev Y.V., Sobolev V.A., Fridman E.M. Asymptotic analysis of digital control systems, Automation and Remote Control, 1988, no. 49, pp. 1175–1180.

  10. Sazhin S., Shakked T., Katoshevski D., Sobolev V. Particle grouping in oscillating flows, European Journal of Mechanics — B/Fluids, 2008, vol. 27, no. 2, pp. 131–149.

  11. Bezglasnyi S.P., Batina E.S., Piyakina E.E. Trudy MAI, 2014, no. 72, available at: http://www.mai.ru/science/trudy/published.php?ID=47314

  12. Bezglasnyi S.P., Krasnov M.V., Mukhametzyanova A.A. Trudy MAI, 2015, no. 79, available at: http://www.mai.ru/science/trudy/published.php?ID=55758


Download

mai.ru — informational site MAI

Copyright © 2000-2024 by MAI

 Вход