Matching and Stabilization by the Method of Controlled Lagrangians
A.M. Bloch, N.E. Leonard and J.E. Marsden
In Proceedings of the 1998 IEEE Conference on Decision and Control,
December 1998, to appear.
Abstract
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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