With integral action, the controller output is proportional to the amount of time the error is present. Integral action eliminates offset that remains when proportional control is used.

controller output = (1/Ti)*int(error)

where the parameter Ti is called the integral time. Integral action is also know as reset and the parameter Ti as reset time.

Integral action gives the controller a large gain at low frequencies that results in eliminating offset. Integrals give information concerning the past. That is why integrals are always late. Integrals provide stability but have a tendency to get stuck in the past. In most controllers the proportional and integral action are combined. The output of the combined proportional and integral action (in s-domain) is then:

with E equal to SP - PV.