Parametric constrained control of motions of a two-mass pendulum

Аuthors

Bezglasnyi S. P.^*, Batina E. S.^**, Piyakina E. E.^***

Samara National Research University named after Academician S.P. Korolev, 34, Moskovskoye shosse, Samara, 443086, Russia

*e-mail: bezglasnsp@rambler.ru
**e-mail: katja4-2@mail.ru
***e-mail: snait2009@yandex.ru

Abstract

This paper presents the problem of the control of plane motions of a two-mass pendulum (swing). The swing is modeled as a weightless rod with two points of mass. The one end of the rod is hinged to a fixed point and can perform rotational or oscillatory motions in a vertical plane around the pivot. The first mass is fixed on the rod, and the second one is bound to slide along the rod. The swing moves in a uniform gravitational field. The two-mass pendulum has two equilibrium positions. The lower one is stable and upper one is instable. Friction forces are neglected.
The swing parametric control by the means of swing excitation and swing damping in the vicinity of the lower equilibrium position are discussed. The control is carried out by the continuously varying of the length from the pendulum pivot to the moving point mass. The control is a function that depends on the representative point of the moving point mass in the phase plane. The bounded and continuous derivative is required for the control.
The aim of this paper is to construct the new moving point mass control laws with a set of specified properties that implement excitation and damping swing processes near the lower equilibrium.
The problem is solved using the Lyapunov functions method of the classical stability theory.
In this paper two control laws of excitation and damping swing processes with the assumption of the restrictions on the motion of the moving point mass are constructed. The Lyapunov functions are found out that prove the asymptotic stability and instability of the pendulum lower position in cases of the pendulum damping and excitation. It is shown that the asymptotic damping oscillation amplitude swings happen for any initial conditions in case of the controlled moving point mass motion along the rod according to the first law. As for the controlling according to the second law the amplitude growth and the transition from oscillatory to rotational motions occur. The theoretical results are illustrated by the graphic demonstration of the numerical results.
The results of this paper can be used for modeling and controling of plane pendulum motions in the various mechanical systems.

Keywords:

inverted pendulum, limited control, Lyapunov's function, asymptotical stability

References

1. Krasil’nikov P.S. Prikladnaja matematika i mehanika, 2012, vol. 76, no. 1, pp. 36–51.
2. Markeev A.P. Nelinejnaja dinamika, 2010, vol. 6, no. 3, pp. 605–622.
3. Andreev A.S. Zhurnal Srednevolzhskogo matematicheskogo obshhestva, 2010, vol. 12, no. 4, pp. 64–73.
4. Andreev A.S. Prikladnaja matematika i mehanika, 1996, vol. 60, no. 3, pp. 388–396.
5. Strizhak T.G. Metody issledovanija dinamicheskih sistem tipa «majatnik» (Methods for Studying «Pendulum»-Type Dynamical Systems), Alma-Ata, Nauka, 1981, 253 p.
6. Magnus K. Kolebaniya (Pendulum), Мoscow, Мir, 1982, 304 p.
7. Chechurin S.L. Parametricheskie kolebanija i ustojchivost’ periodicheskogo dvizhenija (Parametric Vibrations and Stability of Periodic Motion), Leningrad, LGU, 1983, 219 p.
8. Sejranjan A.P. Prikladnaja matematika i mehanika, 2004, vol. 68, no. 5, pp. 847–856.
9. Zevin A.A., Filonenko L.A. Prikladnaja matematika i mehanika, 2007, vol. 71, no. 6, pp. 989–1003.
10. Akulenko L.D., Nesterov S.V. Prikladnaja matematika i mehanika, 2009, vol. 73, no. 6, pp. 893–901.
11. Akulenko L.D. Prikladnaja matematika i mehanika, 1993, vol. 57, no. 2, pp. 82–91.
12. Lavrovskij Je.K., Formal’skij A.M. Prikladnaja matematika i mehanika, 1993, vol. 57, no. 2, pp. 92–101.
13. Aslanov V.S., Bezglasnyi S.P. Prikladnaja matematika i mehanika, 2012, vol. 76, no. 4, pp. 565–575.
14. Volosov V.M., Morgunov B.I. Metod osrednenija v teorii nelinejnyh kolebatel’nyh system (Averaging Method in the Theory of Nonlinear Oscillatory Systems), Moscow, MGU, 1971, 508 p.
15. Pontrjagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaja teorija optimal’nyh processov (Mathematical Theory of Optimal Processes), Moscow, Nauka, 1969, 384 p.
16. Aslanov V.S., Bezglasnyi S.P. Mehanika tverdogo tela, 2012, no. 3, pp. 32–46.
17. Bezglasnyi S.P., Pijakina E.E., Talipova A.A. Avtomatizacija processov upravlenija, 2013, vol. 34, no. 4.
18. Malkin I.G. Teorija ustojchivosti dvizhenija (Theory of Motion Stability), Moscow, Nauka, 1966, 530 p.

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