The three recipes
ZN closed loop: with integral and derivative off, increase gain until the loop oscillates with constant amplitude. That gain is Ku; the oscillation period is Pu. Then:
ZN open loop: make a manual step change and fit the response with process gain K, dead time L and time constant T:
Cohen–Coon uses the same three parameters but keeps accuracy as dead time grows (r = L/T):
Worked example
Step test on a temperature loop: K = 2, dead time L = 10 s, time constant T = 50 s. ZN open-loop PID:
- Kc = 1.2 × 50 ÷ (2 × 10) = 3.0
- Ti = 2 × 10 = 20 s, Td = 0.5 × 10 = 5 s
- Cohen–Coon for comparison: Kc 3.46, Ti 22.7 s, Td 3.5 s — same ballpark, slightly different shape
Field notes — read before typing these into a live controller
- These are starting points, not answers. Both methods target quarter-amplitude damping — fast but jittery and fragile. A common refinement: halve Kc and re-observe.
- Check your controller's units: gain vs proportional band (PB = 100/Kc), Ti in seconds vs minutes vs repeats/min, and interactive vs parallel PID forms all change what number to enter.
- The ultimate-cycling test disturbs the process — many plants forbid it on critical loops. Relay auto-tune (built into most modern controllers) achieves the same identification with a bounded oscillation.
- Derivative on noisy measurements amplifies noise — flow loops almost always run PI only.
Frequently asked questions
What is Ziegler-Nichols tuning?
Two classic recipes from 1942: the closed-loop method raises controller gain until the loop oscillates steadily (ultimate gain Ku, period Pu), then sets PID constants as fractions of those; the open-loop method fits a step response with gain, dead time and time constant and computes constants from them.
When should I use Cohen-Coon instead of Ziegler-Nichols?
Cohen-Coon was designed for processes with significant dead time (dead-time-to-time-constant ratios up to ~2), where ZN open-loop rules degrade. For small dead time both give similar, aggressive results.
Why do these formulas give aggressive tuning?
Both target quarter-amplitude damping — each oscillation peak a quarter of the previous — which is fast but oscillatory and not robust. Standard practice is to use these values as a starting point and then reduce gain (or increase integral time) for smoother response.
What are Kc, Ti and Td?
Controller gain Kc (proportional action), integral (reset) time Ti in time units, and derivative (rate) time Td. Some systems use proportional band PB = 100/Kc and reset in repeats/minute = 1/Ti — check your controller's convention before entering values.
Provided for reference and education. Tune live loops under proper management-of-change and supervision. See our disclaimer.