Matching and Stabilization by the Method of Controlled Lagrangians
Bloch, A. M., N. Leonard and J. E. Marsden
Proc. CDC, 37, 1446-1451
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.