PID control algorithms are widely used method for smooth control of inertial systems. This can be motor speed control, heating, flight control. Balancing robots is one of fun projects where PID control algorithms are easy to learn. You can change P.I.D parameters and see how they change the behavior of robot. Sean Hodgins has been working on balancing robots and thought why not to build one with PID system. This can be fun to learn and explore.
So he came up with small robot carrying Arduino Nano, 3 potentiometers, 6 DOF IMU. All electronics are mounted on custom PCB once mounted, burn software and you are up to experimenting. You can change each PID parameter by turning any of three potentiometers. Easy and intuitive even for kids.