Controlling a Bouncing Ball

A ball bouncing on a vibrating plate is surely one of the simplest physical systems that can produce chaotic motion, yet it is a prototype for a large class of mechanical systems. We wish to design a control algorithm for the plate motion so that the ball, starting at rest on the plate, can be bounced up to a prescribed periodic orbit, and maintained bouncing there. Keeping the amplitude of the vibrating plate fixed, we use the frequency of the plate as a control input. We examine two approaches for designing the controller. The first is classical LQR design and the second is a variant of a "greedy" method which only looks a short distance ahead in time. Using both of these methods we attempt control by linearizing an analytic model, the so-called high-bounce approximation of the ball map, and also by using a data-derived approximation to the true system. These approximate controllers are applied to a more accurate continuous system model of the bouncing ball. Encouragingly, both controllers perform well in spite of the approximations in their construction. The greedy control method appears to be more robust in certain situations.