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Published December 1997 | Published
Book Section - Chapter Open

Stabilization of mechanical systems using controlled Lagrangians

Abstract

We propose an algorithmic approach to stabilization of Lagrangian systems. The first step involves making admissible modifications to the Lagrangian for the uncontrolled system, thereby constructing what we call the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system where new terms 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. The procedure is demonstrated for the problem of stabilization of an inverted pendulum on a cart and for the problem of stabilization of rotation of a rigid spacecraft about its unstable intermediate axis using a single internal rotor. Similar results hold for the dynamics of an underwater vehicle.

Additional Information

© Copyright 1997 IEEE. Reprinted with permission. Publication Date: 10-12 December 1997. Research partially supported by the National Science Foundation PYI grant DMS-91-57556, AFOSR grant F49620-96-1- 0100, a Guggenheim Fellowship and the Inst. for Advanced Study. Research partially supported by the National Science Foundation under grant BES-9502477 and by the Office of Naval Research under grant N00014-96-1-0052. Research partially supported by the National Science Foundation under Grant DMS-9302992 and the Department of Energy under Contract DE-FG0395-ER25251. We thank John Ballieul, Francesco Bullo, Joel Burdick, Neil Getz, Dan Koditschek, P.S. Krishnaprasad, Richard Murray, Tudor Ratiu and Gloria Sdnchez de Alvarez for helpful comments.

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