keep it simple...
The controller doesn't know anything about the process, that's the job of the engineer applying it.
Controllers generally operate on an error signal, which is the difference between measured and desired values of whatever's being controlled.
a PID calculates three functions of that error:
its value multiplied by some gain (
Proportional)
its time integral (
Integral)
and its approximate rate of change (
Derivative)
The PID controller sums the three and applies its output to an actuating device that physically affects the process being controlled.
Analog controllers are just analog computers that use resistors and capacitors to realize their transfer function, which can be expressed as a Laplace transform. Digital controllers use computer programs to do the same calculations and produce same transfer function..
Figure 3 here is a slick analog PID controller that uses just one op-amp.
http://www.postreh.com/vmichal/papers/PID-Radio.pdf
I think it's based on a 1940's design by Philbrick Nexus... a google search turns up dozens of other circuits.
Filters are similarly implemented in either analog hardware or computer software.
hope this helps