An interesting mechanical system with a challenging control problem is the ball and beam. It consists of a rigid beam that is free to rotate in the plane of the paper around a center pivot, with a solid ball rolling along a groove in the top of the beam. The control problem is to position the ball at a desired point on the beam using a torque applied to the beam as a control input at the pivot.
A linear model of the system with a measured value of the angle ∅ and its angular velocity d∅/dt = w is available. Select a feedback scheme so that the response of the closed-loop system has an overshoot of 4% and a settling time (with a 2% criterion) of 1 second for a step input.