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Matching and stabilization by the method of controlled Lagrangians

Bloch, Anthony M. and Leonard, Naomi Ehrich and Marsden, Jerrold E. (1998) Matching and stabilization by the method of controlled Lagrangians. In: Proceedings of the 37th IEEE Conference on Decision and Control, 1998. Vol.2. Institute of Electrical and Electronics Engineers , Piscataway, NJ, pp. 1446-1451. ISBN 0-7803-4394-8.

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We describe a class of mechanical systems for which the “method of controlled Lagrangians” provides a family of control laws that stabilize an unstable (relative) equilibrium. The controlled Lagrangian approach involves making modifications to the Lagrangian for the uncontrolled system such that the Euler-Lagrange equations derived from the modified or “controlled” Lagrangian describe the closed-loop system. For the closed-loop equations to be consistent with available control inputs, the modifications to the Lagrangian must satisfy “matching” conditions. Our matching and stabilizability conditions are constructive; they provide the form of the controlled Lagrangian, the control law and, in some cases, conditions on the control gain(s) to ensure stability. The method is applied to stabilization of an inverted spherical pendulum on a cart and to stabilization of steady rotation of a rigid spacecraft about its unstable intermediate axis using an internal rotor.

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Additional Information:“© 1998 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.” Research partially supported by the National Science Foundation grants DMS-9496221 and DMS-9803181, AFOSR grant F49620-96-1-0100 and a Guggenheim Fellowship. 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 AFOSR Grant F49620-95-1-0419.
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