Controlled Lagrangian approach to the stabilization of the inverted pendulum system
C. Aguilar–Ibañez*, O. Octavio Gutierrez F., and H. Sossa A.
Centro de Investigación en Computación, Instituto Politécnico Nacional, Av. Juan de Dios Bátiz s/n Esq. Miguel Othón de M. Unidad Profesional Adolfo López Mateos, Col. Nueva Industrial Vallejo, A.P. 75476 México, D.F. 07738, México FAX: +(52) 55 5586–2936
* Corresponding author:
caguilar@cic.ipn.mx.
Recibido el 7 de diciembre de 2006
Aceptado el 4 de junio de 2008
Abstract
A controlled Lagrangian approach is presented for the stabilization of an inverted pendulum mounted on a cart. The stabilization strategy consists in forcing the closed–loop system to behave as an Euler–Lagrange system, with a fixed inertia matrix. For carrying it out, it is necessary to adequately shape the potential and kinetic energies of the closed–loop system. The idea behind this procedure is to make an energy–balance between the overall energy of the pendulum system and the dissipation energy produced by the action of the control force. The resulting closed–loop system is locally asymptotically stable about its unstable equilibrium point with a very large attraction domain.
Keywords: Euler–Lagrange system, energy balance, Lyapunov method.
Resumen
En este trabajo se presenta el problema de la estabilización de un péndulo invertido sobre un carro mediante el lagrangiano controlado. La estrategia de estabilización consiste en forzar al sistema en lazo cerrado a que se comporte como un sistema Euler–Lagrange, donde la matriz característica de inercia es constante. Para lograr esto, es necesario modelar adecuadamente las energías potencial y cinética del sistema en lazo cerrado. La idea fundamental de este procedimiento es hacer un balanceo de energía entre la energía de todo el sistema y la energía disipada mediante los movimientos horizontales del carro. El sistema obtenido en lazo cerrado es localmente estable asintóticamente alrededor de su punto de equilibrio inestable, con un dominio de atracción muy grande.
]]> Descriptores: Sistema Euler–Lagrange, balanceo de energía, método de Lyapunov.
PACS: 45.20.Jj
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Acknowledgments
This research was supported by CIC–IPN and by the Secretaria de Investigación y Posgrado of the Instituto Politécnico Nacional (SIP–IPN) under research grant 20082694. Octavio Gutierrez F. is a scholarship holder of the CONACYT.
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