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    Calculates locally in your browser · PID tuning from a step test
    Process / control

    PID tuning

    Three rules, three different answers, and the disagreement is the point. Ziegler-Nichols is aggressive and tuned for quarter-amplitude damping; Cohen-Coon corrects it for a large dead-time ratio; IMC is deliberately conservative and lets you choose how fast the loop should recover.

    ModePIDKc (ZN)3.00Kc (IMC)1.25
    Ziegler-Nichols gain
    3.00
    Ti 4.0 s, Td 1.0 s
    IMC gain
    1.25
    Ti 10.0 s, Td 1.0 s
    Dead-time ratio
    0.20
    Easy, lag dominated
    Proportional band
    33%
    at the Ziegler-Nichols gain
    Three rules on the same process
    The disagreement is the useful part: aggressive, corrected, and conservative.
    Kc 3.00
    Three rules on the same process
    RuleKcTi (s)
    Ziegler-Nichols3.0004.00
    Cohen-Coon3.5104.62
    IMC (lambda)1.25010.00
    Ziegler-Nichols Td—1.00
    Cohen-Coon Td—0.71
    IMC Td—1.00
    Process gain K2.000output per input
    Time constant τ10.0 s63% of the final value
    Dead time θ2.0 stransport delay
    θ/τ0.200Easy, lag dominated

    Ziegler-Nichols assumes quarter-amplitude damping, which is visibly oscillatory. Cohen-Coon corrects for a large dead-time ratio. IMC lets you pick the closed-loop speed directly through lambda, so a larger lambda means a smaller gain and a smoother response.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Gain against dead time
    Both rules fall as the delay grows, but Ziegler-Nichols falls faster because it is chasing a tighter response.
    Gain against dead time0.0002.695.388.0610.7513.440.5 s0.5 s1 s1 s2 s2 s4 s4 s6 s6 s8 s8 sZiegler-NicholsIMC
    Gain by rule
    The spread is typically 2–3× on the same process.
    Gain by rule0.0000.8281.662.493.314.143.00Z-NZ-N3.51Cohen-CoonCohen-Coon1.25IMCIMCKc
    Canonical equations
    The exact relationships implemented by this engine.

    Kc = 1.2 τ/(K θ), Ti = 2θ, Td = 0.5θ · PID form.

    Kc = (τ/(Kθ))(1.35 + 0.27θ/τ) · Corrects for θ/τ.

    Kc = τ/(K(λ + θ)), Ti = τ · λ sets the speed.

    Ki = Kc / Ti · Per second.

    engineering methodology

    A clear path from inputs to results.

    Three rules, three different answers, and the disagreement is the point. Ziegler-Nichols is aggressive and tuned for quarter-amplitude damping; Cohen-Coon corrects it for a large dead-time ratio; IMC is deliberately conservative and lets you choose how fast the loop should recover.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the process gain, time constant and dead time from a step test.
    2. Choose the controller form.
    3. Set the IMC lambda if you want a specific closed-loop speed.
    4. Compare the three rules and pick the one that matches the plant.

    The formula

    Kc=1.2 τKθ  ;  Ti=2θ  ;  Td=0.5θK_c = 1.2\,\frac{\tau}{K\theta} \;;\; T_i = 2\theta \;;\; T_d = 0.5\theta

    Reference: Ziegler & Nichols (1942); Cohen & Coon (1953); Rivera, Morari & Skogestad (1986) for IMC.

    Worked example

    A process with K = 2, τ = 10 s and θ = 2 s

    Inputs
    • Gain 2, time constant 10 s, dead time 2 s.
    • PID form.
    • IMC lambda 2 s.
    Calculation path
    1. 1

      The dead-time ratio is 2/10 = 0.2, a moderate loop.

    2. 2

      Ziegler-Nichols gives Kc = 1.2 × 10/(2 × 2) = 3.0, Ti = 4 s, Td = 1 s.

    3. 3

      Cohen-Coon corrects for the ratio: Kc = 3.51, Ti = 4.92 s, Td = 0.71 s.

    4. 4

      IMC with lambda = 2 s gives Kc = 10/(2 × 4) = 1.25 with Ti = 10 s.

    5. 5

      The Ziegler-Nichols gain is 2.4 times the IMC gain on the same process.

    Result

    Z-N Kc 3.0 · Cohen-Coon 3.51 · IMC 1.25.

    Interpreting the result

    A 2.4× spread on the same process is normal, and it is not a disagreement about the maths — it is a disagreement about the objective. Ziegler-Nichols accepts overshoot to get speed; IMC trades speed for robustness. The plant decides which is appropriate.

    Assumptions & limitations

    • A first-order-plus-dead-time process identified from a step test.
    • No valve or sensor dynamics beyond the identified model.
    • A non-integrating, self-regulating process.
    • No feedforward, cascade or gain scheduling.

    Frequently asked questions

    Which rule should I use?

    IMC when overshoot is unacceptable or the model is uncertain; Ziegler-Nichols when speed matters and the plant tolerates oscillation. Cohen-Coon sits between them.

    What if the dead-time ratio is above 1?

    The loop is delay-dominated and no tuning rule will make it fast. The answer is usually to reduce the delay — move the sensor, shorten the piping — not to retune.

    Does this work for an integrating process?

    No. Level loops and other integrating processes need a different model and a different set of rules, because they have no steady state to identify.

    What is lambda?

    The closed-loop time constant you are willing to accept. A small lambda gives a fast, aggressive loop; a large one gives a slow, robust one.

    How do I get K, τ and θ?

    From a step test: K is the steady-state output change per unit input change, τ is the time to reach 63% of it, and θ is the delay before the output moves.

    Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.