Abstract
We consider a system of a planar inverted pendulum in a gravitational field. First, we assume that the pivot point of the pendulum is moving along a horizontal line with a given law of motion. We prove that, if the law of motion is periodic, then there always exists a periodic solution along which the pendulum never becomes horizontal (never falls). We also consider the case when the pendulum with a moving pivot point is a control system, in which the mass point is constrained to be strictly above the pivot point (the rod cannot fall ‘below the horizon’). We show that global stabilization of the vertical upward position of the pendulum cannot be obtained for any smooth control law, provided some natural assumptions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Srzednicki, R.: On periodic solutions in the Whitney’s inverted pendulum problem. arXiv:1709.08254v1
Polekhin, I.: On topological obstructions to global stabilization of an inverted pendulum. arXiv:1704.03698v2
Ważewski, T.: Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires. Ann. Soc. Polon. Math. 20, 279–313 (1947)
Reissig, R., Sansone, G., Conti, R.: Qualitative Theorie nichtlinearer Differentialgleichungen. Edizioni Cremonese, Rome (1963)
Srzednicki, R.: Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations. Nonlinear Anal. Theor. Methods Appl. 22, 707–737 (1994)
Srzednicki, R., Wójcik, K., Zgliczyński, P.: Fixed point results based on the Ważewski method. In: Brown, R.F., Furi, M., Górniewicz, L., Jiang, B. (eds.) Handbook of Topological Fixed Point Theory, pp. 905–943. Springer, Dordrecht (2005). doi:10.1007/1-4020-3222-6_23
Demidovich, B.P.: Lectures on the Mathematical Theory of Stability. Nauka, Moscow (1967)
Polekhin, I.: Forced oscillations of a massive point on a compact surface with a boundary. Nonlinear Anal. Theor. Methods Appl. 128, 100–105 (2015)
Bolotin, S.V., Kozlov, V.V.: Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney’s inverted pendulum problem. Izv. Math. 79(5), 894–901 (2015)
Polekhin, I.: On forced oscillations in groups of interacting nonlinear systems. Nonlinear Anal. Theor. Methods Appl. 135, 120–128 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Polekhin, I. (2017). A Topological View on Forced Oscillations and Control of an Inverted Pendulum. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_38
Download citation
DOI: https://doi.org/10.1007/978-3-319-68445-1_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68444-4
Online ISBN: 978-3-319-68445-1
eBook Packages: Computer ScienceComputer Science (R0)