Energy-based balance control approach to the ball and beam system

Abstract

This article investigates the set-point balance control problem of the ball and beam system acting on the ball which is an underactuated system. The control law is designed, based on the total mechanical energy and the passivity properties of the system. According to the desired set-point, the zero equilibrium point and non-zero equilibrium points are studied respectively. For the zero case, it is proved that a single PD feedback controller is sufficient to bring the state to zero from any initial condition, on condition that the control parameters satisfy an inequality. For the non-zero case, the control problem is much more complicated. Unlike the previous energy-based control laws, a new form of Lyapunov function candidate is constructed. A complete analysis of the convergence of the energy and the dynamics is given, and the characteristics of the closed-loop system with the proposed feedback control law are illustrated. Moreover, it is proved that with the parameter choice rules proposed in this article, the ball and beam system will eventually converge exactly to the desired non-zero equilibrium point. Furthermore, since the length of the beam is not unlimited, the trajectory of the ball should be restricted within a limited range. The balance control laws are modified to avoid the ball running beyond the joint range limitation for the zero case and the non-zero case respectively. Simulation results show that the control laws proposed in this article are effective for the set-point balance control problem of the ball and beam system.