Can someone explain to me proportional, integral and derivative control and how it works?

Proportional simply adjusts the process variable according to amount of deviation: If the water is really cold, open the hot water valve wide.





Derivative adjusts according to how fast the variable is changing. If the water *suddenly* gets colder, open the hot water valve.





Integral adjusts according to how long and how much the process has been out-of-whack. The water has been too cold for some time, so open the hot water valve more.





(Illustrations are intended to be meaningful to someone taking a hot shower)Can someone explain to me proportional, integral and derivative control and how it works?
Jerry's explanation seems quite good so I'll just add a few notes.


If purely proportional control was used the desired control point, eg temperature, would never be stable. It would oscillate like a sine wave above and below the desired value.


The derivative and integral controls act as amplifiers and dampers to the proportional control helping to stabilise the output. The combination of all 3 controls results in a stable output but which may be slightly different to the desired value, this is called the offset.


I did Control at college and it seemed confusing at first then in the last couple of weeks everything clicked together and I had my Eureka moment so don't despair.