PID controllers
PID controllers

The common property of ordinary P, PI and PID controllers is linearity. In case of proportional controller (P), the control variable u is directly proportional to the error e.

The control variable of proportional-integral controller (PI) is a sum of two components – the proportional one (which is, as well as in case of pure P controller, directly proportional to the error), and an integral one, which is proportional to accumulated value of the error, i.e. to its integral. Integral action is able to reach zero error in some cases, where it is impossible with pure proportional controller, e.g. if Couloumb friction is present. It is of inertial character, sometimes it can cause unstable or oscillatory response of the control system.

The output (control variable) of proportional-integral-derivative controller (PID) contains an additional derivative action. It has "anticipating" behavior and brings faster response to sudden changes. Its disadvantage is that it amplifies high frequency noise, present in measurement, what may cause random, erratic operation of the system.