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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.

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

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

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

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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:

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

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