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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 810597, 13 pages
Research Article

Stabilization of the Ball on the Beam System by Means of the Inverse Lyapunov Approach

1CIC-IPN, Unidad Profesional Adolfo López Mateos, Avendia Juan de Dios Bátiz S/N, Casi Esquire Miguel Othón de Mendizábal, Colonia Nueva Industrial Vallejo, Delegación Gustavo A. Madero, 07738 Mexico City, DF, Mexico
2ESCOM-IPN, 07738 Mexico City, DF, Mexico
3SEPI-ESIME Azcapotzalco, 02250 Mexico City, DF, Mexico

Received 15 November 2011; Accepted 4 January 2012

Academic Editor: Alexander P. Seyranian

Copyright © 2012 Carlos Aguilar-Ibañez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.