Mathematical models for balancing tasks on a see-saw with reaction time delay

Abstract

Mechanical models of balancing a ball rolling on a see-saw (“ball and beam” system) and balancing an inverted pendulum attached to a cart rolling on a see-saw (“pendulum-cart and beam” system) are analyzed. A delayed proportional-derivative controller is modeled with four different actuation schemes. The angular position, the angular velocity, the angular acceleration of the see-saw and the torque acting on the see-saw are considered to be the variables manipulated by the control system. The corresponding mathematical models take the form of retarded, neutral and advanced functional differential equations. Stabilizability analysis shows that the ball and beam system can only be stabilized in the presence of feedback delay if the manipulated variable is the angular position of the see-saw or the torque acting on the see-saw. The pendulum-cart and beam system can only be stabilized in the presence of feedback delay if the manipulated variable is the torque acting on the see-saw.