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Stabilization of autonomous oscillations and the Hopf bifurcation in the ball and beam

This paper presents a design method for the stabilization of oscillations in a class of nonlinear systems. The method consists of two steps. In the first one, a second-order generalized Hamiltonian subsystem, which presents stable oscillations, is matched. In the second step, the controller is extended to the full system using backstepping. It is shown that the oscillations emerge through a supercritical Hopf bifurcation. The method proposed was first applied to a second order system. In this paper the method is extended to higher order systems by backstepping. The method is introduced by its application to an underactuated mechanical system: the ball and beam.