Instability: The magnetic levitation system is a classic instability problem. Instability is a special type of equilibrium in a system that is nearly impossible to reach without outside interaction. A classic example of a different type of equilibrium is the pendulum. A pendulum has two equilibrium points, based on the solution to its governing equation of motion, one at the top or apex and one at the bottom. The equilibrium at the top is unstable and the equilibrium at the bottom is stable. This can be observed by playing with a pendulum. The pendulum seems to always be attracted towards the bottom after some time, but will never stay at the top. The reason for this is that at the bottom, there are restoring forces that act in opposite directions to the direction of motion. If the pendulum, at the bottom, moves a little to the right, there is a force that moves it back to the left. If the pendulum moves to the left, there is a restoring force to the right. With the application of some small friction, the system will eventually reach the equilibrium position at the bottom. However, this is not the case for the top point.
At the top, the forces push the pendulum away from the equilibrium position in either direction, such that the pendulum can never reach its equilibrium position. Instability can be shown graphically as a concave-down potential energy graph, or an unstable equilibrium in a phase plot. The goal of this project is to take the magnetic system and “balance” it. In this system (assume that the electromagnet is turned off), the magnetic object is either attracted upwards to the metallic base of the electromagnet, or is pulled down away from equilibrium by gravity.
The PID (Proportional, Integral, Derivative) controller is the device which allows for the counter-balancing of forces. The PID controller will try and push the system towards equilibrium. The PID controller needs a way of sensing the state of the system, like how far above or below the equilibrium position the magnetic object has moved.
This information is called feedback, and it is provided by two sensors that detect magnetic field strength. This will be discussed in more detail later. The PID controller needs some way to apply these “counter” forces to try and maintain equilibrium- this mechanism is commonly called the output or actuator. In this project, the actuator, or output, is an
electromagnet that can vary its output field and, consequentially, the force it exerts on the permanent magnet in the levitated object.