Swinging up multiple parallel pendulums on a cart via energy control

This paper concerns the problem of swinging up multiple parallel pendulums on a cart by controlling only the energy of pendulums. Although the analysis of the energy control for swinging up single pendulum and two pendulums on the cart has been achieved in existing literature, a few analysis results have been reported for the case of multiple (greater than 2) pendulums. In this paper, we aim to provide some new results on swinging up n pendulums (n ¿ 3) by using a notion of normalized energy of a pendulum. We find that such notion enables us to facilitate the analysis of closed-loop behavior of all pendulums and establish some relationships between the motion of pendulums and their natural frequencies. Indeed, for the case of three pendulums with the different natural frequencies, irrespective of initial state of the three pendulums, we show that the energy of each pendulum converges to either of the desired value or the minimal value, and the motion of the three pendulums can be described by eight invariant sets, among which the invariant set corresponding to the convergent value of the energy of each pendulum being zero is asymptotically stable, and the rest of the invariant sets are unstable. This paper gives further insight on the energy control for cart-pendulum systems.