ZieglerNichols

Ziegler–Nichols is a set of tuning rules for PID controllers developed by John G. Ziegler and Nijko Nichols in 1942. The method provides practical formulas to convert simple process measurements into controller parameters, and it has become one of the most widely taught tuning approaches in control engineering.

There are two standard Ziegler–Nichols tuning procedures: the reaction curve (step response) method and the ultimate gain (closed-loop oscillation) method. The reaction curve method uses the step response of the process to estimate a process gain K and a time delay or lag L. From these parameters, classic Ziegler–Nichols formulas yield initial PID, PI, or P controller settings. For a PID controller, typical values are Kp = 1.2 (L/K), Ti = 2L, and Td = 0.5L, with Ki = Kp/Ti and Kd = Kp Td. For a PI controller, Kp = 0.9 (L/K) and Ti = 3L. For a P controller, Kp = 0.5 (L/K). The exact constants vary by presentation, but the general idea is to map the reaction curve quadrature into proportional, integral, and derivative terms.

The ultimate gain method involves increasing the proportional gain of a closed-loop system until sustained oscillations occur, yielding the ultimate gain Ku and the oscillation period Pu. The PID settings are then Kp = 0.6 Ku, Ti = Pu/2, and Td = Pu/8, with PI and P variants using corresponding scaled formulas (e.g., Kp = 0.45 Ku for PI).

Ziegler–Nichols tuning is valued for its simplicity and quick results but can produce aggressive responses and overshoot. It is often used as a starting point, with further refinement through empirical tuning or more robust methods in modern practice.