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Mathematical Problems in Engineering
Volume 2011, Article ID 856015, 14 pages
Research Article

On the Stabilization of the Inverted-Cart Pendulum Using the Saturation Function Approach

1CIC-IPN, Avenida Juan de Dios Bátiz S/N Casi Esq. Miguel Othón de Mendizábal, Unidad Profesional Adolfo López Mateos Col., Nueva Industrial Vallejo Delegación Gustavo A. Madero, 07738 México, DF, Mexico
2Departamento de Control Automático, CINVESTAV-IPN, A.P. 14-740, 07000 México, DF, Mexico
3SEPI-ESIME-IPN Azcapotzalco, Avenida de las Ganjas No. 682, Santa Catarina, 02250 México, DF, Mexico

Received 23 May 2011; Revised 4 August 2011; Accepted 5 August 2011

Academic Editor: Angelo Luongo

Copyright © 2011 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 simple stabilizing controller for the cart-pendulum system is designed in this paper. Our control strategy describes the underactuated system as a chain of integrators with a high-order smooth nonlinear perturbation and assumes initialization of the system in the upper half plane. The design procedure involves two sequentially associated control actions: one linear and one bounded quasilinear. The first control action brings the nonactuated coordinate near to the upright position and keeps it inside of a well-characterized small vicinity, whereas the second control action asymptotically brings the whole state of the system to the origin. The corresponding closed-loop stability analysis uses standard linear stability arguments as well as the traditional Lyapunov method and the LaSalle's theorem. Our proposed control law ensures global stability of the system in the upper half plane. We illustrate the effectiveness of the proposed control strategy via numerical simulations.