Controlling a ball to bounce at a fixed height

A ball bouncing on a vibrating plate is perhaps one of the simplest physical systems which can produce chaotic motion. The objective here is to design a control algorithm for the plate so that the ball, started at rest on the plate, can be bounced up to and maintained bouncing at a prescribed maximum height. Complications include the fact that not all heights represent stable motion, more that one cycle of the plate may be necessary to achieve a given height, multiple height solutions are possible, and chaotic motion can result at certain height-frequency combinations. Keeping the amplitude of the vibrating plate fixed, we use the frequency of the plate as a control input. Two state variables define the motion. One state variable is the phase angle of the plate at the time of bounce and the other is related to the maximum height of the ball. Motion is initiated by controlling the plate, open loop, in such a fashion so that chaotic motion is produced. Due to the random like nature of chaotic motion, many phase-height combinations are obtained. By switching to closed loop control in the neighborhood of the desired phase-height combination it is possible to stabilize the ball to the desired bounce height specification. A basic requirement with this approach is knowledge of when to turn on the closed loop control. The following method is used. The system is first linearized about the desired periodic solution. If necessary, a feedback controller is designed so that this reference solution has suitable stability properties. A Lyapunov function is then obtained based on this stable linear system.