¹⁹Truth be told, the same principle holds for purely integrating processes as well. A purely integrating process always exhibits a phase shift of −90^o at any frequency, because that is the nature of integration in calculus. A purely first-order lag process will exhibit a phase shift anywhere from 0^o to −90^o depending on frequency, but never more lagging than −90^o, which is not enough to turn negative feedback into positive feedback. In either case, so long as we don’t have process noise to deal with, we can increase the controller’s gain all the way to eleven. If that last sentence (a joke) does not make sense to you, be sure to watch the 1984 movie This is Spinal Tap as soon as possible. Seriously, I have used controller gains as high as 50 on low-noise, first-order processes such as furnace temperature control. With such high gain in the controller, response to setpoint and load changes is quite swift, and integral action is almost unnecessary because the offset is naturally so small.