Ziegler–Nichols PID Tuning

The 1942 Ziegler–Nichols rules are where practical PID tuning began, and both of their experiments are here. The CLOSED-LOOP (ultimate-gain) method: with integral and derivative off, raise the proportional gain until the loop oscillates steadily; that gain is Ku and the oscillation period is Pu, and the table scales them into controller settings. The OPEN-LOOP (reaction-curve) method: with the loop open, step the actuator and fit the S-shaped response with a first-order-plus-dead-time model — process gain K, time constant T, dead time L — and a second table converts the fit.

Both tables target roughly quarter-amplitude decay, which is deliberately aggressive: each overshoot peak is about a quarter of the previous one, meaning visible ringing and modest margins. That aggressiveness is the point of the method (disturbance rejection) and its notorious limitation (setpoint overshoot) — the interpretation panel says which you are getting, and the IMC/lambda tool is the modern alternative when you want to dial robustness explicitly.

Gains are reported in BOTH controller forms — standard/ISA (Kp, Ti, Td), which is how the original tables are written, and parallel (Kp, Ki, Kd) via Ki = Kp/Ti, Kd = Kp·Td — because applying an ISA table to a parallel-form controller without converting is the single most common tuning error in the field.

For the reaction-curve method the closed-loop step-response preview and the open-loop margins are computed on the fitted FOPDT plant, with the dead time represented by a first-order Padé approximation (stated on the chart). For the ultimate-gain method the plant itself is unknown — take the gains to the PID Controller Analyzer with your plant model for the full four-panel picture.