Analysis of the energy-based control for swinging up two pendulums

This note studies the energy-based control for swinging up two pendulums on a cart and it presents an original analysis of the convergence of the energy and the motion of the two pendulums. For the two pendulums with the different natural frequencies, irrespective of initial state of the two pendulums, it is shown that the energy of each pendulum converges to either of two values, and the motion of the two pendulums can be described by four invariant sets. Moreover, the stability analysis of the motion of the two pendulums is performed via the concept of stability of an invariant set. The results obtained in this note not only show theoretically the effectiveness of the energy-based control for swinging up the two pendulums, but also pave one way for analyzing and designing the energy-based control for more complicated underactuated mechanical systems.