Integral Action analysed

Integral action in a PID controller is the key element in eliminating steady-state error. The reason it does this is that the action is proportional to the integral of the error - if an error persists, then the integral and the resultant control action keeps getting larger.

The algorithm for a pure integral controller (a PID with the proportional and derivative bits switched off) is:

or, expressed in deviation variables:

Since the controller gain and the integral time constant can be freely selected by the engineer (on a real controller just by pressing some buttons), it is convenient for the analysis of pure integral control to replace the two constants with a single, integral gain:

Again let's just use the generic first-order process model:

We need to replace the manipulated variable, u, with the equation for the controller (since the controller output will now determine the value of the manipulated variable):

The integral on the right-hand side of the equation makes things a bit difficult, but we can get rid of it by differentiating both sides w.r.t. time:

Some really interesting things have happened:

1. The process has changed from first-order to second-order when integral control is applied. This is a general result - when integral control is applied to a process the closed-loop response always is of one order higher than the open loop (e.g a four-order process with integral control would have fifth-order closed loop dynamics)
2. The steady-state gain on setpoint changes is 1 - this means that, when steady-state is reached, the setpoint tracking will be exact.
3. The steady-state effect of a disturbance will be zero. Provided the disturbance stops changing at some point (with a step this is immediately), the disturbance derivative will become zero and, when the output response settles to steady-state, the change in the output will be zero. Integral control can produce perfect disturbance rejection.
4. The coefficients of the characteristic equation include the integral gain - the form of the response can be shaped by changing the integral gain.

A movie showing an example of integral control of a first-order process can be found here and the model used in the movie is here.