Control SystemsPreliminary engineering
Cohen–Coon PID Tuning
Cohen–Coon open-loop tuning from an FOPDT fit (K, T, L) — designed for dead-time-dominant processes — with gains in both forms and a closed-loop preview.
About this calculator
Cohen and Coon published their 1953 corrections to Ziegler–Nichols for one specific weakness: processes where the dead time is a large fraction of the time constant. Where the ZN reaction-curve table quietly degrades as L/T grows, the Cohen–Coon table carries the ratio R = L/T inside every entry, keeping the intended quarter-amplitude-decay behavior out to dead-time-dominant fits (R approaching 1 and beyond, with the usual caution that ALL FOPDT rules weaken as R grows large).
The method starts from the same experiment as ZN open-loop tuning: a manual step on the actuator, an S-shaped reaction curve, and a three-number fit — process gain K, time constant T, dead time L. The table then produces the controller settings in standard/ISA form (Kp, Ti, Td); the parallel-form Ki = Kp/Ti and Kd = Kp·Td conversions are computed and shown, as everywhere in this toolkit.
Like Ziegler–Nichols, Cohen–Coon aims at roughly quarter-amplitude decay — aggressive, disturbance-rejection-first behavior with visible ringing. The closed-loop preview (FOPDT plant, dead time as a first-order Padé approximation, stated on the chart) shows exactly how much, and the margins quantify what robustness is left. When the answer is 'not enough', the IMC/lambda tool trades speed for robustness explicitly.
Every table entry is listed in FORMULAS-FOR-VERIFICATION.md with its algebra so it can be checked against a published table line by line.
Design notes & common mistakes
- Fit K, T, L from an OPEN-loop step (the reaction curve). The tangent-line fit at the inflection point is classical; two-point (28.3%/63.2%) fits are usually more repeatable on noisy data.
- Cohen–Coon (like ZN) targets quarter-amplitude decay — treat the output as a deliberately aggressive starting point, not a finished tune.
- The table's R-corrections help most for 0.1 < R < 1; far beyond that, no FOPDT rule is trustworthy and dead-time compensation (Smith predictor) is the real answer.
- As always: the table is ISA-form. Convert to Kp/Ki/Kd before typing into a parallel-form controller.
Assumptions
- The process is adequately described by the FOPDT fit K·e^(−Ls)/(Ts+1) from an open-loop reaction curve.
- Table entries are the classical Cohen–Coon values targeting ≈ quarter-amplitude decay.
- The preview approximates e^(−Ls) by first-order Padé; accuracy degrades for L ≳ T.
- Ideal derivative; standard/ISA table with parallel conversions shown.
When to use this calculator
Appropriate for
- First-cut PID tuning of a dead-time-affected process from an FOPDT reaction-curve fit
- Dead-time-dominant loops (R = L/T roughly 0.1 to 1) where the Cohen–Coon corrections help
- Comparing against Ziegler–Nichols on the same fit
Not suitable for
- Processes not well described by first-order-plus-dead-time (integrating, strongly underdamped, inverse-response)
- Applications needing a robust, low-overshoot response — Cohen–Coon also targets quarter-amplitude decay
- Final settings without detuning and validation on the real process
What this calculator does not cover
- FOPDT-only: integrating, strongly underdamped, or inverse-response processes are outside the method's model and its table.
- Quarter-amplitude decay is aggressive by design — most servo and mechanical applications will detune from these values.
- The published Cohen–Coon coefficients vary slightly between textbook reprintings; the exact algebra used here is listed in FORMULAS-FOR-VERIFICATION.md for line-by-line checking.
- The Padé(1) preview under-represents long dead times; for L > T treat the preview as optimistic.
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
When should I choose Cohen–Coon over Ziegler–Nichols?
When the dead-time ratio R = L/T is significant — roughly 0.1 to 1. Cohen–Coon's table carries R inside every coefficient, so it holds its intended behavior on dead-time-dominant processes where the plain ZN reaction-curve table over-tunes. For tiny R they give nearly identical settings.
How do I get K, T, and L from a real process?
Open the loop, make a modest actuator step, and record the S-curve. K is the total output change divided by the step size; classically, a tangent at the inflection point gives L (where the tangent crosses the initial value) and T (the tangent's rise time). Two-point fits at 28.3% and 63.2% of the change are more robust on noisy data.
Why is my Cohen–Coon tune still oscillatory?
Because it is meant to be — the design target is quarter-amplitude decay, where each overshoot is a quarter of the last. That prioritizes disturbance recovery over smoothness. If the ringing is unacceptable, reduce Kp 20–40%, or switch to IMC/lambda tuning where the response speed is an explicit choice.
References
- Cohen, G. H. & Coon, G. A., 'Theoretical Consideration of Retarded Control', Trans. ASME 75 (1953)
- Seborg, D., Edgar, T., Mellichamp, D. & Doyle, F., Process Dynamics and Control, 4th ed. (Cohen–Coon table and FOPDT fitting)
- Åström, K. J. & Hägglund, T., PID Controllers: Theory, Design, and Tuning, 2nd ed.
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