Stabilizability of Controlled Lagrangian Systems of Two Degrees of Freedom and One Degree of Under-Actuation by the Energy-Shaping Method

We provide some criteria for stabilizability by the energy-shaping method for the class of all controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation: a necessary and sufficient condition for Lyapunov stabilizability, two sufficient conditions for asymptotic stabilizability, and a necessary and sufficient condition for exponential stabilizability. As a corollary, we show that some of the asymptotically stabilizing controllers that were designed in old literatures with the energy-shaping method are actually exponentially stabilizing controllers. Examples of such systems are the inverted pendulum on a cart, the Furuta pendulum, the ball and beam system, and the Pendubot.