Phase shift corrector:
When the system that has to be corrected has high static gain or includes an integrator, one may use this kind of system in order just to get better stability margin and the step response a little faster....
This animation does not constitude a method for how to adjust a PID corrector. It just intend to show a little how the three actions act on frequency and step responses.
Same animation with Nyquist diagram instead of Black diagram. My opinion is that it is somehow less intuitive...