# What is the original source of the common Ziegler-Nichols PID tuning coefficients?

In many places, including but not limited to the Wikipedia page about PID controllers, I see the following PID coefficients: $$K_P = 0.6K_U$$, $$K_I = 1.2 K_U / T_U$$, and $$K_D = 0.075 K_UT_U$$. When were these constants first introduced? The sources point to the papers Optimum Settings for Automatic Controllers and Rule-Based Autotuning Based on Frequency Domain Identification, but I cannot find these values there.

Table I in the second mentioned paper introduces $$K_C = 0.6K_U$$, $$T_i = 0.5T_U$$, and $$T_D = 0.125T_U$$, but I do not understand what those coefficients mean and how I go to $$K_P$$, $$K_I$$, and $$K_D$$ from there.

For PID, Ziegler, Nichols (Optimum Settings for Automatic Controllers) cites:

$$K_p = 0.6K_u$$, $$T_i = 0.5 T_u$$, and $$T_d = 0.125 T_u$$.

Which McCormack (Rule-based autotuning based on frequency domain identification) and Ziegler-Nichols Tuning Rules for PID agree with.

OP cites Wikipedia PID coefficients: $$K_p = 0.6K_u$$, $$K_i = 1.2 K_u / T_u$$, and $$K_d = 0.075 K_u T_u$$

The appropriate math:

$$K_i = \frac{K_p}{T_i} = \frac{0.6 K_u}{0.5 T_u} = 1.2 \frac{K_u}{T_u}$$ $$K_d = K_p T_d = (0.6 K_u) (0.125 T_u) = 0.075 K_u T_u$$

Wikipedia: The Ziegler–Nichols tuning method is a heuristic method of tuning a PID controller. It was developed by John G. Ziegler and Nathaniel B. Nichols. It is performed by setting the I (integral) and D (derivative) gains to zero. The "P" (proportional) gain, $$K_{p}$$ is then increased (from zero) until it reaches the ultimate gain $$K_{u}$$, at which the output of the control loop has stable and consistent oscillations. $$K_{u}$$ and the oscillation period $$T_{u}$$ are then used to set the P, I, and D gains depending on the type of controller used and behaviour desired:

• Thank you so much for the explanation, it’s clear now and I really appreciate it very much! Dec 5, 2022 at 21:42