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.

[1]  F. Andreev,et al.  Matching, linear systems, and the ball and beam , 2000, Autom..

[2]  Naif B. Almutairi,et al.  On the sliding mode control of a Ball on a Beam system , 2009 .

[3]  En Li,et al.  Energy-based balance control approach to the ball and beam system , 2009, Int. J. Control.

[4]  J. Aracil,et al.  Stabilization of autonomous oscillations and the Hopf bifurcation in the ball and beam , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[5]  Wen Yu,et al.  Stability analysis of PD regulation for ball and beam system , 2005, Proceedings of 2005 IEEE Conference on Control Applications, 2005. CCA 2005..

[6]  Rogelio Lozano,et al.  Lyapunov function for the ball and beam: robustness property , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).