Stabilization of Mechanical Systems with Controlled Lagrangians

A.M. Bloch, N.E. Leonard and J.E. Marsden

In Proceedings of the 1997 IEEE Conference on Decision and Control, December 1997, p. 2356-2361.


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 {\em 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.

Postscript Version (6 pages postcript)

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