You are looking at the problem all wrong.
Tom Jenkins said:
Nobody derives the equation, and no one really ever did in the sense of deriving it from first principals or laws of physics.
Are you sure? I am not. In fact I doubt it. There have been some pretty smart math geniuses in the past centuries. These differential equations we often of interest because they closely modeled the way things worked in nature. Today, control guys think in terms of poles and zeros and how to place them. This is all derived from the concept of eigen values or roots.
PID is an empirical equation, based on the "three mode controllers"
I think this is due mostly due to a lot of practical limitations because it wasn't really a mathematical limitation.
developed decades ago as pneumatic controllers with orifices, needle valves, and mechanical linkage. The PID equation essentially models the proportional, reset, and rate adjustments of these controllers.
Reset, WTF gets reset? The integrator time constant is just on of the terms that can be use to move a pole in the desired direction.
Controllers were built with vacuum tubes. I bet I can find some very old examples if I look.
The problem is that most of you look at three separate gains and tweak them with understand what it is doing mathematically to the poles ( response ) of the system. As you move the poles closer the the negative real axis the response becomes closer to critically damped. I am sure this knowledge was available long before pneumatic controllers.
I bet Cauchy or Laplace could have derived the formulas for a PID without problem. I can, with a little work, but I stand on their shoulders and have Mathcad. Also, note. The result isn't always going to be a PID. Sometime it will be a PI or a PID with a second derivative. It depends on the system you start with.
It all makes sense when you stop looking at a PID as three gains to tweak and see the gains as coefficients that can be added to a differential equation and used to place the eigen values.
) the history of this so I'd be interested to find out some more. Prior to Z&N I thought most people knew you needed a lump of P, a bit of I and a sprinking of D, but were never quite sure how much of these to use. Z&N put it on some formalised basis.
