PID tuning calculator
Classical tuning rules turn three numbers from a step test — gain, dead time and time constant — into PID settings.
Change in output divided by change in input, both as a fraction of span.
s
s
Proportional gain, Ziegler-Nichols
4.8000
- Integral time
- 10.000 s
- Derivative time
- 2.500 s
- Integral gain
- 0.480 00
- Derivative gain
- 12.0000
- Proportional gain, Cohen-Coon
- 5.5389
- Proportional gain, lambda tuning
- 0.4722
- Integral time, lambda tuning
- 42.500 s
- Dead time to time constant ratio
- 0.1250 Easy to control
Show the working
- Process gain = 2
- Dead time = 5
- Time constant = 40
- Controller type = 3
- ratio = 5 / 40 = 0.125
- Proportional gain, Ziegler-Nichols = 3 == 1 ? 40/(2*5) : 3 == 2 ? 0.9*40/(2*5) : 1.2*40/(2*5) = 4.8
- Integral time = 3 == 2 ? 3.33*5 : 3 == 3 ? 2*5 : 0 = 10
- Derivative time = 3 == 3 ? 0.5*5 : 0 = 2.5
- Proportional gain, Cohen-Coon = 3 == 3 ? (1.35/2)*(0.125^(-1))*(1 + 0.18*0.125/(1-0.125)) : (0.9/2)*(0.125^(-1))*(1 + 0.125/11) = 5.538 86
- Proportional gain, lambda tuning = (2*40 + 5) / (2 * 2 * (40 + 5)) = 0.472 222
- Integral time, lambda tuning = 40 + 5/2 = 42.5
- Dead time to time constant ratio = 0.125 = 0.125
- Integral gain = 10 > 0 ? 4.8/10 : 0 = 0.48
- Derivative gain = 4.8 * 2.5 = 12
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- Ziegler-Nichols PID: Kp = 1.2 T / (K L), Ti = 2 L, Td = 0.5 L
- lambda tuning gives a slower, more robust loop with less overshoot
Ziegler-Nichols is deliberately aggressive and gives about 25% overshoot. It is a starting point, not a finished tune.
A process with dead time longer than its time constant is genuinely hard to control by PID alone.