Balancing an inverted pendulum is the typical example when demonstrating a control system. A weight on an arm above the rotation pivot is an unstable system. It has one stable point when the weight is perfectly balanced. If the weight is slightly on either side of this point it will begin to move away from this stable point. On the other hand, a weight on an arm below the rotation pivot is a stable system. The weight is stable at the point exaclty below the pivot as well, but if it is moved slightly away from this point it will move back towards it.
A screenshot of the simulation. Click to do your own simulations.