Coupling-energy-based swing-up control for Pendulum–cart system with limited-track length

In this paper, we tackle the swing-up control problem for a pendulum–cart system that drives the pendulum to its upright equilibrium point (with the pendulum vertical and the cart at the target position) while explicitly respecting a track-length constraint. Traditional energy based techniques achieve an increase in the swing on unbounded tracks by regulating the total mechanical energy of the system. Previous coupling-energy controllers exploit the horizontal translational kinetic energy of the system’s center of mass to reinforce cart–pendulum interaction for anti-swing control around the downward equilibrium point (pendulum hanging downward). Here, we extend the coupling-energy method to design a novel swing-up controller suitable for the system with the track-length constraint. We introduce a barrier function to enforce the track-length constraint and refine the Lyapunov function candidate. Based on this, we derive a swing-up controller that ensures the cart remains within the track’s limit. A comprehensive global analysis then proves that the closed-loop state will repeatedly enter arbitrarily small neighborhoods of the upright equilibrium point, where one can switch into a locally stabilizing controller. Finally, experiments on a physical pendulum–cart platform confirm that the proposed controller performs effectively and robustly even when mechanical parameters vary.