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Stabilization of mechanical systems using controlled Lagrangians

Bloch, Anthony M. and Leonard, Naomi Ehrich and Marsden, Jerrold E. (1997) Stabilization of mechanical systems using controlled Lagrangians. In: IEEE Conference on Decision and Control, 36th, San Diego, CA, 10-12 December 1997. Vol.3. IEEE , Piscataway, NJ, pp. 2356-2361. ISBN 0-7803-4187-2.

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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.

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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.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)F49620-96-1-0100
John Simon Guggenheim FoundationUNSPECIFIED
Institute for Adavanced StudyUNSPECIFIED
Office of Naval Research (ONR)N00014-96-1-0052
Department of Energy (DOE)DE-FG0395-ER25251
Subject Keywords:closed loop systems; mechanical stability; Euler-Lagrange equations; admissible modifications; cart-pole system; closed-loop stability; controlled Lagrangians; energy methods; internal rotor; inverted pendulum; mechanical system stabilization; rigid spacecraft rotation; underwater vehicle dynamics; unstable intermediate axis
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ID Code:11515
Deposited On:28 Aug 2008 02:17
Last Modified:08 Nov 2021 22:00

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