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Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

Bloch, Anthony M. and Leonard, Naomi Ehrich and Marsden, Jerrold E. (2000) Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem. IEEE Transactions on Automatic Control, 45 (12). pp. 2253-2270. ISSN 0018-9286. http://resolver.caltech.edu/CaltechAUTHORS:BLOieeetac00

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Abstract

We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. We use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane.


Item Type:Article
Additional Information:“© 2000 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.” Manuscript received September 10, 1998; revised December 20, 1999 and January 13, 2000. Recommended by Associate Editor, O. Egeland. This work was supported in part by the NSF under Grants DMS 9496221, DMS-9803181, BES 9502477, the AFOSR under Grants F49620-96-1-0100 and F49620-95-1-0419, a Guggenheim Fellowship, Institute for Advanced Study, and by the Office of Naval Research under Grants N00014-96-1-0052 and N00014-98-1-0649. The authors would like to thank J. Baillieul, F. Bullo, J. Burdick, N. Getz, D. Koditschek, P. S. Krishnaprasad, R. Murray, T. Ratiu, A. Ruina, G. Sánchez de Alvarez, and C. Woolsey for helpful comments. They would also like to thank the referees for their suggestions.
Subject Keywords:Lagrangian mechanics, nonlinear systems, stabilization
Record Number:CaltechAUTHORS:BLOieeetac00
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:BLOieeetac00
Alternative URL:http://dx.doi.org/10.1109/9.895562
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:534
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:14 Jul 2005
Last Modified:26 Dec 2012 08:40

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