Ziegler–Nichols method

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, ${\displaystyle K_{p}}$ is then increased (from zero) until it reaches the ultimate gain ${\displaystyle K_{u}}$, which is the largest gain at which the output of the control loop has stable and consistent oscillations; higher gains than the ultimate gain ${\displaystyle K_{u}}$ have diverging oscillation. ${\displaystyle K_{u}}$ and the oscillation period ${\displaystyle T_{u}}$ are then used to set the P, I, and D gains depending on the type of controller used and behaviour desired:

Ziegler–Nichols method[1]

Control Type	${\displaystyle K_{p}}$	${\displaystyle T_{i}}$	${\displaystyle T_{d}}$	${\displaystyle K_{i}}$	${\displaystyle K_{d}}$
P	${\displaystyle 0.5K_{u}}$	–	–	–	–
PI	${\displaystyle 0.45K_{u}}$	${\displaystyle T_{u}/1.2}$	–	${\displaystyle 0.54K_{u}/T_{u}}$	–
PD	${\displaystyle 0.8K_{u}}$	–	${\displaystyle T_{u}/8}$	–	${\displaystyle K_{u}T_{u}/10}$
classic PID[2]	${\displaystyle 0.6K_{u}}$	${\displaystyle T_{u}/2}$	${\displaystyle T_{u}/8}$	${\displaystyle 1.2K_{u}/T_{u}}$	${\displaystyle 3K_{u}T_{u}/40}$
Pessen Integral Rule[2]	${\displaystyle 7K_{u}/10}$	${\displaystyle 2T_{u}/5}$	${\displaystyle 3T_{u}/20}$	${\displaystyle 1.75K_{u}/T_{u}}$	${\displaystyle 21K_{u}T_{u}/200}$
some overshoot[2]	${\displaystyle K_{u}/3}$	${\displaystyle T_{u}/2}$	${\displaystyle T_{u}/3}$	${\displaystyle 0.666K_{u}/T_{u}}$	${\displaystyle K_{u}T_{u}/9}$
no overshoot[2]	${\displaystyle K_{u}/5}$	${\displaystyle T_{u}/2}$	${\displaystyle T_{u}/3}$	${\displaystyle (2/5)K_{u}/T_{u}}$	${\displaystyle K_{u}T_{u}/15}$

The ultimate gain ${\displaystyle (K_{u})}$ is defined as 1/M, where M = the amplitude ratio, ${\displaystyle K_{i}=K_{p}/T_{i}}$ and ${\displaystyle K_{d}=K_{p}T_{d}}$.

These 3 parameters are used to establish the correction ${\displaystyle u(t)}$ from the error ${\displaystyle e(t)}$ via the equation:

${\displaystyle u(t)=K_{p}\left(e(t)+{\frac {1}{T_{i}}}\int _{0}^{t}e(\tau )\,d\tau +T_{d}{\frac {de(t)}{dt}}\right)}$

which has the following transfer function relationship between error and controller output:

${\displaystyle u(s)=K_{p}\left(1+{\frac {1}{T_{i}s}}+T_{d}s\right)e(s)=K_{p}\left({\frac {T_{d}T_{i}s^{2}+T_{i}s+1}{T_{i}s}}\right)e(s)}$