PID Loop Simulator: PID Equations

Description| How it works| Derivative on PV| Process Model| Robustness| PID Tuning| PID Equations

Every so often I get a query from the visitors on the PID equation used in this app. The equation used in the tool is the parllel form of the PID equation, while the most commonly used PID equation is the ideal form. PID equations used in industrial controllers have many different forms and variations in the way the terms are calculated.

The following equations show the Ideal, Parallel and Series (Interacting) forms of the PID equations, with all terms calculated on error. Here, Kc = Controller Gain, Ti = Integral Time and Td = Derivative Time. e(t) is the control error as a function of time t.

Ideal PID algorithm:

Ideal PID Equation

Parallel PID algorithm:

Parallel PID Equation

Interacting PID algorithm

Series PID Equation

The Series or Interacting form of the PID equation was used when pneumatic controllers were in widespread use. Since the advent of digital controllers, the ideal equation is now very popular. Most controllers offer both the ideal and series forms of the equations. The parallel form that is used in this app is used in academics.

The Advanced PID Loop Simulator can simulate all these equations. A free demo of the advanced version simulates the ideal PID algorithm.