PID Controller Tuning Calculator — Ziegler–Nichols FOPDT Method for Kp, Ki and Kd
Calculate Kp, Ki and Kd using process gain, time constant and dead time based on the Ziegler–Nichols FOPDT tuning method.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
FOPDT model parameters
Results
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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
Kp, Ki and Kd formula:
Kp = 1.2 · T / (K · L)
Ti = 2 · L
Td = 0.5 · L
Ki = Kp / Ti
Kd = Kp · Td
R = L / T
where:
- Kp — proportional gain [—]
- Ki — integral gain [1/s]
- Kd — derivative gain [s]
- K — process gain [—]
- T — time constant [s]
- L — dead time / delay [s]
- Ti — integral time constant [s]
- Td — derivative time constant [s]
- R — dead-time ratio [—]
When to use this calculator
When to use this calculator:
- To calculate PID controller gains from a first-order-plus-dead-time process model.
- To tune proportional, integral and derivative action from process gain, time constant and delay.
- To check whether the dead-time ratio is within the Ziegler–Nichols operating region.
- To estimate the expected controller response from the ratio between delay and time constant.
- To compare how changes in process gain, time constant or dead time affect PID gain values.
How to interpret the result
Kp, Ki and Kd are defined as PID controller gains calculated from a FOPDT process model using process gain, time constant and dead time.
The result depends on the relation between delay and time constant. Increasing time constant increases proportional gain, while increasing process gain or dead time decreases proportional gain.
- Safe — dead-time ratio is greater than or equal to 0.05 and less than or equal to 0.5. This is the good operating region for Ziegler–Nichols tuning.
- Warning — dead-time ratio is less than 0.05, where tuning may be overly aggressive, or greater than 0.5 and less than or equal to 1, where oscillatory response is expected.
- Unsafe — dead-time ratio is greater than 1. Dead time dominates and Ziegler–Nichols tuning is likely unstable.
- Invalid — process gain, time constant or dead time is less than or equal to zero, not finite, or the calculated gains are non-physical.
The result is used to evaluate controller gain values and expected response aggressiveness before applying PID tuning.
Calculation example
Example:
A user has identified a FOPDT process model and wants to calculate PID tuning parameters using the Ziegler–Nichols method.
- K — Process gain = 2
- T — Time constant = 10 s
- L — Dead time = 2 s
Kp = 3.0, Ki = 0.75 1/s, Kd = 3.0 s
Assumptions and limitations
- The process is represented by process gain, time constant and dead time.
- Process gain, time constant and dead time must all be greater than zero.
- The dead-time ratio is calculated as dead time divided by time constant.
- Integral time constant is calculated as two times dead time.
- Derivative time constant is calculated as half of dead time.