The Ball and Beam Apparatus shows the control problems of unstable systems, for example a rocket or missile during launch, which needs active control to prevent the missile going unstable and toppling over.
The apparatus has a steel ball which is free to roll on two parallel tensioned wires. The wires are on a beam that pivots at its centre. A servo motor controls the beam angle and sensors measure the beam angle and ball position.
The basic control problem is to vary the beam angle to control the ball position. The system is a double integrator, so it is naturally unstable. It needs active feedback control using phase-advance methods.