Bit of a yawn, this. Boiler energy management systems have been trying to get it right for a lot longer than I can remember. I've been trying to get it right since modelling the output of a nuclear reactor 30 years ago - as controlled by lowering in the control rods, which has all sorts of other problems.
The idea IS to limit the overtemperature excursion ("wasted energy") and get the control as near as possible to "Critical Damping".
Go to the interactive demonstration on oilman's link page two, and click the graph. Then change the "Simulation Parameters" from ON/OFF to PID and you'll see the objective.
You get the ideal , "Critical" damping by applying heat as shown by the blue line at the bottom.
Trouble is, you can't DO IT with conventional boiler controls because they don't turn up and down, they ONLY go on and off.
So what you have to do is turn the boiler on and off rapidly - like if it's on for a minute, off for a minute that's half power, right? Yeah sure -
unless it's a Netaheat which takes a minute to light, then you're screwed, or -
similar sort of problem with the motorised valve, though in practice they often stay in the position where they were last used (makes another long post) so it isn't such a problem. Or,-
the boiler takes a while to ramp up, get warm or whatever, and also will keep putting energy into the system after you've turned it off.
The answer doesn't lie in conventional controls but in modulating boilers. They OUGHT to know what their own parameters are, can "learn" to refine them anyway, and can be made to react to changing temperature readings coming from sensors in the house, with a varying heat output. They can learn the response of the house with changing outside temp, and all that. Some are going that way (WB for one, there must be others) but nowhere near as fast or as loudly as I would have expected. I daresay before long this sort of proportional feedback will be mandatory under the building regs.
For anyone who doesn't understand what the hell this is all about, try flying the helicopter round the obstacles in "Chopper 2", 8 down the list, here:
http://www.trustedtradesmen.com/forum/arcade/arcade.asp
You don't have proportional control, only on/off. By clicking the mouse constantly you can get a horizontal-ish flight, but when you go round an obstacle you have to go early and control the overshoot...