PID Tuning Estimate Calculator

Work out pid tuning estimate instantly with clear inputs, formula shown and shareable results.

About PID Tuning Estimate Calculator

First-order-plus-dead-time models let a controller be tuned from three numbers: process gain, time constant and dead time. Ziegler-Nichols gives fast but oscillatory response, Cohen-Coon suits dead-time-dominant loops, and IMC/lambda tuning gives a smooth non-oscillatory response that most process plants prefer.

Formula

FOPDT tuning

Ziegler-Nichols: Kp = 1.2 tau / (K theta), Ti = 2 theta, Td = 0.5 theta; IMC uses a lambda closed-loop time constant

Worked example

With the default values below, PID Tuning Estimate Calculator returns Proportional gain of 0.41667. Change any input above and every figure here is recalculated in your browser.

Inputs used

Process gain: 2.4 units/%
Process time constant: 12 s
Dead time: 3 s
Tuning ruleone of Ziegler-Nichols (aggressive), Cohen-Coon, IMC / lambda (conservative): IMC / lambda (conservative)

Results

Proportional gain: 0.41667
Integral time: 13.50 s
Derivative time: 1.33 s
Controllability ratio: 4.000

Step by step

1Values used
2FOPDT tuning
3Proportional gain= 0.41667
4Integral time= 13.50 s
5Derivative time= 1.33 s
6Controllability ratio= 4.000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Which rule should I start with?

IMC or lambda tuning. It is stable, has one intuitive knob (desired closed-loop speed) and will not leave the loop oscillating.

What does the controllability ratio tell me?

Tau/theta above about 4 is easy to control; below 1 the loop is dead-time dominated and needs conservative tuning or a model-based controller.