Cohen–Coon PID Tuning Calculator — Kp, Ki, Kd for FOPDT Processes
Calculate P, PI or PID controller settings from signed process gain, time constant and dead time, including Kp, Ki, Kd, Ti, Td, L/T ratio and recommended controller action.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Controller configuration
FOPDT process model
Results
Calculation tools Save, export and manage your results
Save your calculation, export available results, or store a finalized engineering artifact in your account.
Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
Cohen–Coon P, PI and PID tuning formulas
The calculator determines initial controller settings from a first-order-plus-dead-time (FOPDT) process model defined by signed process gain K, process time constant T and process dead time L.
The normalized dead-time ratio is:
R = L / T
The implemented Cohen–Coon correlations first determine proportional gain magnitude Kp and, where applicable, integral time Ti and derivative time Td. The calculator then converts the time-form coefficients to parallel-form integral and derivative gains:
Ki = Kp / Ti
Kd = Kp · Td
The proportional gain is calculated as a magnitude using |K|. The sign of the measured process gain is retained separately to determine the recommended controller action.
P controller
Kp = [1 / (|K| · R)] · (1 + R / 3)
For P-only control:
- Ki = 0
- Kd = 0
- Ti and Td are not used.
PI controller
Kp = [1 / (|K| · R)] · (0.9 + R / 12)
Ti = L · (30 + 3R) / (9 + 20R)
Ki = Kp / Ti
Kd = 0
PID controller
Kp = [1 / (|K| · R)] · (4/3 + R / 4)
Ti = L · (32 + 6R) / (13 + 8R)
Td = 4L / (11 + 2R)
Ki = Kp / Ti
Kd = Kp · Td
Symbols and outputs
- K — signed process gain, ΔPV/ΔCO.
- T — FOPDT process time constant.
- L — FOPDT process dead time.
- R — dead-time-to-time-constant ratio, L/T.
- Kp — proportional gain magnitude, expressed as CO/PV.
- Ki — parallel integral gain magnitude, expressed as CO/(PV·s).
- Kd — parallel derivative gain magnitude, expressed as CO·s/PV.
- Ti — integral time in seconds; not used for P-only control.
- Td — derivative time in seconds; used only for PID control.
Time inputs are normalized to the calculator's canonical time unit before the formulas are evaluated, so T and L remain physically consistent when different supported time units are selected.
When to use this calculator
When to use this Cohen–Coon PID tuning calculator
Use this calculator when a stable, self-regulating process can be represented reasonably well by a FOPDT model and you already know its signed process gain, time constant and dead time. It provides initial tuning values for P, PI or PID controllers.
Typical use starts with an open-loop process test or an identified process model from which the three FOPDT parameters are obtained:
- process gain K — how strongly the process variable changes for a change in controller output,
- time constant T — the characteristic response time of the first-order process,
- dead time L — the delay between an input change and the observed process response.
The calculator is useful when you need a reproducible starting point for PID controller tuning rather than selecting Kp, Ki and Kd by trial and error. It also reports Ti, Td, the L/T ratio, a process-dynamics classification and the recommended direct or reverse controller action from the sign of K.
Use another method or a more detailed model when
- the process cannot be represented adequately by a first-order-plus-dead-time model,
- the process is integrating, unstable or strongly oscillatory rather than self-regulating,
- the operating point changes enough for K, T or L to vary substantially,
- strong multivariable interaction dominates the response,
- actuator saturation, valve stiction or other nonlinearities dominate the loop behavior,
- closed-loop robustness or performance must be demonstrated rather than estimated from a tuning rule.
The calculated values are controller starting settings. The calculator does not simulate the closed-loop response and does not verify overshoot, settling time, stability margins, actuator limits or disturbance rejection.
How to interpret the result
How to interpret the Cohen–Coon tuning results
Kp is the proportional gain magnitude. Higher Kp means a stronger proportional controller response for a given control error. Because this calculator deliberately reports gain magnitude, the required feedback direction is communicated separately as controller action.
Ki is the integral gain in parallel form. It is calculated as Kp/Ti. A non-zero Ki is present for PI and PID modes and accumulates control error over time. P-only mode sets Ki to zero and can therefore leave a steady-state offset.
Kd is the derivative gain in parallel form. It is calculated as Kp·Td and is present only in PID mode. The calculator does not model derivative filtering or measurement noise, so a calculated Kd value is not by itself evidence that the derivative term will be suitable in the real loop.
Ti and Td are the integral and derivative times from the classical Cohen–Coon correlations. They are also provided because many industrial PID implementations are configured using Kp, Ti and Td rather than independent Ki and Kd.
L/T describes the relative importance of delay:
- L/T < 1: lag-dominant process, T > L.
- L/T = 1: process time constant and dead time are equal.
- L/T > 1: dead-time-dominant process, L > T.
When L/T ≥ 1, the calculator reports a warning and recommends explicit verification of robustness and actuator limits before commissioning.
The sign of the process gain determines the reported controller-action recommendation:
- K > 0: reverse-acting controller recommendation.
- K < 0: direct-acting controller recommendation.
Controller manufacturers do not all use identical gain-sign and error conventions. Verify the action definition of the actual PID block before applying the settings.
The calculator does not issue a general PASS or FAIL decision for the control loop. A numerically valid Cohen–Coon result means that the implemented tuning equations could be evaluated from the supplied FOPDT parameters; it does not demonstrate closed-loop stability, robustness or acceptable process performance.
Calculation example
Cohen–Coon PID tuning example
Consider a self-regulating process represented by the following FOPDT parameters:
- Controller type: PID
- K = +2.0 — signed process gain
- T = 100 s — process time constant
- L = 20 s — process dead time
1. Calculate the dead-time ratio
R = L / T = 20 / 100 = 0.20
Because R is below 1, the calculator classifies the process as lag-dominant (T > L).
2. Calculate proportional gain
1 / (|K| · R) = 1 / (2 · 0.20) = 2.5
Kp = 2.5 · (4/3 + 0.20/4) = 2.5 · 1.383333... = 3.4583
3. Calculate integral time
Ti = 20 · (32 + 6 · 0.20) / (13 + 8 · 0.20)
Ti = 20 · 33.2 / 14.6 = 45.4795 s
4. Calculate derivative time
Td = 4 · 20 / (11 + 2 · 0.20) = 80 / 11.4 = 7.0175 s
5. Convert to parallel-form Ki and Kd
Ki = Kp / Ti = 3.4583 / 45.4795 = 0.07604
Kd = Kp · Td = 3.4583 · 7.0175 = 24.2690
Calculated PID settings
| Parameter | Result |
|---|---|
| Kp | 3.4583 CO/PV |
| Ki | 0.07604 CO/(PV·s) |
| Kd | 24.2690 CO·s/PV |
| Ti | 45.4795 s |
| Td | 7.0175 s |
| L/T | 0.20 |
Since the process gain is positive, the calculator recommends reverse-acting controller action. The numerical gains are magnitudes; the required sign or action setting must be implemented according to the conventions of the actual PLC, DCS or PID controller.
These values are initial Cohen–Coon tuning settings. Closed-loop behavior must still be checked before the tuning is accepted for operation.
Assumptions and limitations
Assumptions and limitations
- The process is assumed to be adequately represented by a stable, self-regulating first-order-plus-dead-time model over the operating region of interest.
- Signed process gain K must be finite and non-zero. Negative K is valid and is used to determine the controller-action recommendation.
- Process time constant T must be greater than zero.
- Process dead time L must be greater than zero.
- T and L must describe the same physical process response and are converted to a common internal time unit before calculation.
- The calculator assumes the supplied K, T and L values are representative of the actual process. It does not identify these parameters from raw step-response data.
- Cohen–Coon tuning is used as an initial open-loop tuning rule. The calculated values should be checked and, where necessary, adjusted using the real closed-loop response.
- The calculator converts Ti and Td to parallel-form Ki and Kd. A target controller that uses another PID structure or parameter convention requires an appropriate conversion.
- The calculation does not model actuator saturation, valve stiction, deadband, anti-windup, derivative filtering, signal noise or output-rate limitations.
- The calculation does not include controller sample time, discretization effects, setpoint weighting or implementation-specific PID features.
- Processes with substantial nonlinear behavior, operating-point dependence, interactions or higher-order dynamics may require system identification and a tuning method more appropriate to the actual process model.
- A dead-time-dominant result with L/T ≥ 1 is specifically flagged for additional engineering review; the calculator does not convert that warning into an automatic rejection of the tuning.
- P-only control can retain steady-state offset because no integral action is present.