Abstract
The energy based control approach has been shown effective for designing and analyzing the control systems for underactuated systems with underactuation degree (the difference between the numbers of generalized coordinates and controlinputs) being one. For underactuated systems of underactuation degree greater than one, it is still an open and challenging problem to analyze the behavior of the resulting closed loop system governed by a controller designed by using the energy based control approach. In this study, we investigate an unsolved problem of how to analyze energy based swing-up control a double pendulum on a cart, which is an example of an underactuated system of underactuation two. We present an original analysis of the convergence of the energy of the cart-double pendulum system. We show that for all initial states of the cart-double pendulum system, if the convergent value of the energy is not equal to the energy at the up-up equilibrium point, then the cart-double pendulum remains at its up-down, down-up, and down-down equilibrium points. Moreover, we show that these three equilibrium points are unstable. This shows that for almost all initial states of the cart-double pendulum system, as time approaches infinity, the energy of the cart-double pendulum system can be controlled to its energy at the upright equilibrium point. This paper provides insight into the energy based control approach to underactuated systems of underactuation degree greater than one.
