Ziegler–Nichols, Cohen–Coon and lambda tuning all read the same three numbers off a process — gain, time constant and dead time — and differ only in how close to the stability edge they are willing to sit. On a τ=2 s, θ=0.5 s loop they give Kp of 5.6, 4.8 and 1.0.
The whole tuning question is set by θ/τ, how much of the response is delay. Ziegler–Nichols targets quarter-amplitude decay, which is deliberately oscillatory and close to the edge. Cohen–Coon was built for loops where dead time is a large fraction of the time constant. Lambda tuning replaces the fixed opinion with a dial: pick the closed-loop speed you actually want and it backs the gain off to suit. A factor of five between them is not disagreement about the maths — it is disagreement about how much margin is worth buying.
All three assume the process really is first order plus dead time. On an oscillatory or integrating plant, or one whose gain changes with operating point, the fitted parameters are a fiction and the numbers that come out are precise but wrong. Above θ/τ ≈ 1 no PID tuning rescues the loop — that is a different problem.
Estimate with the rule, then check it against the calculator that models it properly.
Open PID Control →Ziegler–Nichols, Cohen–Coon and lambda tuning all read the same three numbers off a process — gain, time constant and dead time — and differ only in how close to the stability edge they are willing to sit. On a τ=2 s, θ=0.5 s loop they give Kp of 5.6, 4.8 and 1.0. The whole tuning question is set by θ/τ, how much of the response is delay.