Stabilizing Control for a ball and beam system considering the restricted beam angle and ball speed

Abstract

For a ball and beam system where the beam angle and the ball speed are small, this paper describes a design method of control system transferring the ball to the target position considering especially the constrained beam angle. First, the transfer function of the ball and beam system is represented as the product of a softening filter, an inner function, and a double integrator. Using modal decomposition methods, we decompose the plant into the stable and unstable subsystems, and derive a nested saturating control law globally asymptotically stabilizing the unstable one. Moreover, the state is estimated for feedback by a reduced-order observer with the poles corresponding to the inner function unchanged. Also, it is shown that the regulator and the observer designed in this way give the controller an IMC (Internal Model Control) structure. The control system using this control law has the property that when the ball is transferred, its speed is mostly limited less than the specific value. To reduce the steady-state error in the ball position as well as to alleviate the reset windup, an IMC controller is constructed with the internal model of the control system itself in the outer feedback loop. The effectiveness of the control law is demonstrated numerically and experimentally.