Robust stabilization using time-scaling and Lyapunov redesign: The ball-beam system

This paper proposes a nonlinear stabilizing controller for a ball on an end-actuated beam system, which is robust to an uncertainty in the mass of the ball. To this end, the dynamics is time-scaled into two subsystems termed as the `Outer-Loop´ and the `Inner-Loop´ dynamics. An Outer-Loop controller generates a reference trajectory for the beam´s pitch angle, which if faithfully followed, would result in the stabilization of all states of the system. A robust Inner-Loop controller is synthesized using the Lyapunov redesign technique, which forces the actual pitch angle to stabilize to the trajectory of the Outer-Loop controller. It is shown that the effects of the uncertainty in the mass of the ball are eliminated by the Inner-Loop controller. The uncertainty tolerance limit of this robust controller is also characterized through a necessary condition. Experiments validate the effectiveness of this strategy in stabilizing the system.