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Revista mexicana de física
Print version ISSN 0035-001X
Rev. mex. fis. vol.54 n.4 México Aug. 2008
Instrumentación
Controlled Lagrangian approach to the stabilization of the inverted pendulum system
C. AguilarIbañez*, 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 55862936
* 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 closedloop system to behave as an EulerLagrange system, with a fixed inertia matrix. For carrying it out, it is necessary to adequately shape the potential and kinetic energies of the closedloop system. The idea behind this procedure is to make an energybalance between the overall energy of the pendulum system and the dissipation energy produced by the action of the control force. The resulting closedloop system is locally asymptotically stable about its unstable equilibrium point with a very large attraction domain.
Keywords: EulerLagrange 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 EulerLagrange, 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 EulerLagrange, balanceo de energía, método de Lyapunov.
PACS: 45.20.Jj
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Acknowledgments
This research was supported by CICIPN and by the Secretaria de Investigación y Posgrado of the Instituto Politécnico Nacional (SIPIPN) under research grant 20082694. Octavio Gutierrez F. is a scholarship holder of the CONACYT.
References
1. A.M. Bloch, D.E. Chang, N.E. Leonard, and J.E. Marsden, IEEE Trans. on Auto. Control 46 (2001) 1556. [ Links ]
2. G. Blankenstein, R. Ortega, and A. van der Schaft, International Journal Control 75 (2002) 645. [ Links ]
3. D. Auckly and L. Kapitanski, SIAM J Control Optim 41 (2002) 1372. [ Links ]
4. A.M. Bloch, N.E Leonard, and J.E. Marsden, IEEE Trans Automatic Control 45 (2000) 2253. [ Links ]
5. C.C. Chung, J. Hauser, Automática 36 (2000) 287. [ Links ]
6. I. Fantoni and R. Lozano, Nonlinear control for the underactuacted mechanical system (SpringerVerlag, London 2002). [ Links ]
7. K.J. Astrom and K. Furuta, Automática 36 (2000) 287. [ Links ]
8. A.S. Shiriaev A.S, A. Pogromsky, H. Ludvigsen, and O. Egeland, International Journal of Robust and Nonlinear Contol 10 (2000)283. [ Links ]
9. H.S. Ramirez, S. K. Agrawal, Differential flat system (Marcel Dekker, New York, 2004). [ Links ]
10. D.E. Chang, A.M. Bloch, N.E. Leonard, J.E. Marsden, and C.A. Woolsey, ESAIM: Control, Optimisation, and Calculus of Variations 8 (2002) 393. [ Links ]
11. C. AguilarIbañez, O. GutierrezF. , M. S. Suarez, and C. Lyapunov, Nonlinear Dyanamics 40 (2005) 367. [ Links ]
12. Ö.T. Altinöz, International Journal of Computer and Information Science and Engineering 1 (4). [ Links ]
13. O. Fujita and K. Uchimura, Proceedings of the 3rd International Conference on Fuzzy Logic, Neural Nets and Soft Computing (Iizuka, Japan, August 17, 1994) 271. [ Links ]
14. Y. Ledeneva and C.A. REYES G. 5th Mexican International Conference on Artificial Intelligence (MICAI06), (LNAI 4293, México, SpringerVerlag, 2006. ISSN 03029743) 146. [ Links ]
15. Y. Ledeneva and C.A. REYES G. Proceedings of 4th International Conference on Informatics in Control, Automation and Robotics (ICINCO) (France, 2007) 398. [ Links ]
16. H.K. Khalil, Nonlinear Systems, (Prentice Hall, 2nd Edition, New Jersey, 1996). [ Links ]