Here we consider a proportional-integral controller. Set the value of the control gain ; the integral time is fixed ().

The closed-loop transfer function for step point changes is given by (if there is no disturbance).

When a step in the set point is applied (i.e. ), the response is plotted and values of the three integral errors are given (see Snapshot 1).

These integral errors (all truncated at in the calculation shown here) are defined by:

(the integral of the absolute value of the error indicated by the gray area shown in the response versus time plot),

(the integral of the squared error),

and (the integral of the time-weighted absolute error).

In addition, the Demonstration performs controller tuning by computing the values of that minimize the integral errors (see Snapshots 2–4). These are shown by blue dots.

A frequency analysis shows that sustained oscillations (shown in Snapshot 5) take place when and .