In 3-state control there are two bounds and it even depends from which side is that bound passed. But it still ramins simple in principle. The best will be show it on example. So we have a pump and we want this pump to hold the pressure in reservoir at 4 atm. First we will set our bounds, “switching” one and the “breaking” one. The switching one we will set to be 4 atm and breaking 5 atm (This two values depends on certain conditions, so this values are not decisive). This means that if the pressure drops down below 4 atm pump will be switched on and it will be turned off just after exceeding 5 atm. If you think about it, that means that between 4 atm and 5 atm there is a band where the pump can occur in two different states (ON/OFF) which depends on from which side is it get to this band. So just the information about actual pressure is not enough, it is important to know if the reservoir is being pressurizes (ON) or it was pressurized in past and the pressure is droping down to 4 atm (OFF), because someone open the valve. In this way one need the memory element (eg. flip-flop, or piece of memory in microprocessor).
If one want to continuously regulate the power one should use the controller with continuous output. The Controller which output is proportional to the error in the measured variable
A proportional-integral-derivative controller (PID controller) is improved kind of proportional controller. With big simplification: The proportional part is the heart of this controller it holds actual difference between measured variable and setpoint. Integral part holds the duration of error and the derivative part holds the tempo with which the error increase.
In math words:
,u is the output of controller. The letters K with indexes are the constants of regulator. The set the “weight” with which each part of controller affect the output of controller. The value of this constats differs and depends on aplication. As you can see the overall output is sum of three controller parts output. One can actually set the K of any part to zero value. This way one can create P, PI or PD version of controller.
More about PID controller can be found on wiki.