Editing Proportional–integral–derivative controller (section)

===Ziegler–Nichols method===
{{Further|Ziegler–Nichols method}}

Another heuristic tuning method is known as the [[Ziegler–Nichols method]], introduced by [[John G. Ziegler]] and [[Nathaniel B. Nichols]] in the 1940s. As in the method above, the <math>K_i</math> and <math>K_d</math> gains are first set to zero. The proportional gain is increased until it reaches the ultimate gain <math>K_u</math> at which the output of the loop starts to oscillate constantly. <math>K_u</math> and the oscillation period <math>T_u</math> are used to set the gains as follows:
{| class="wikitable" style="text-align:center;"
|+ Ziegler–Nichols method
|-
! Control type
! <math>K_p</math>
! <math>K_i</math>
! <math>K_d</math>
|-
! ''P''
| <math>0.50{K_u}</math>
| —
| —
|-
! ''PI''
| <math>0.45{K_u}</math>
| <math>0.54{K_u}/T_u</math>
| —
|- style="text-align:right;"
! ''PID''
| <math>0.60{K_u}</math>
| <math>1.2{K_u}/T_u</math>
| <math>3{K_u}{T_u}/40</math>
|}

The oscillation frequency is often measured instead, and the reciprocals of each multiplication yields the same result.

These gains apply to the ideal, parallel form of the PID controller. When applied to the standard PID form, only the integral and derivative gains <math>K_i</math> and <math>K_d</math> are dependent on the oscillation period <math>T_u</math>.