On the Controlled Lagrangian of an inverted pendulum on a force-driven cart

The Controlled Lagrangian method is a strategy for the design of control systems for a class of Lagrangian systems. This method has resolved the stabilization problem of some well-known underactuated mechanical systems, including an inverted pendulum on a force-driven cart benchmark. This paper goes deeply into the analysis of this control system by studying in detail asymptotic stability invoking the Barbashin-Krasovskii's theorem and giving an explicit novel estimate of the domain of attraction. As a consequence, an estimate of the domain of attraction that resolves the practical motivated issue of maintaining the travel of the cart within prescribed physical limits of displacement is obtained. Numerical simulations are presented to illustrate these results.