Matching and stabilization by the method of controlled Lagrangians

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.