PID Tuner

PID controller settings from a process step test: Ziegler–Nichols, Cohen–Coon, lambda and SIMC tuning for a first-order process with dead time, with a simulated step response.

How it works

Most industrial processes, from a heated tank to a motor speed, respond to a step in the controller output with a delay followed by a smooth rise to a new value. This is described by three numbers (the first-order-plus-dead-time, FOPDT, model): the process gain K (final change of the measurement ÷ step in the output), the time constant T (the time to reach 63 % after the delay) and the dead time L. You find them by an open-loop step test: put the controller in manual, change the output by a few percent, and record the measurement.

u(t) = Kc [ e + (1 ÷ Ti) ∫ e dt + Td de/dt ], Ziegler–Nichols PID: Kc = 1.2 T ÷ (K L), Ti = 2 L, Td = 0.5 L

The Ziegler–Nichols rule gives a fast response with about 25 percent overshoot and little robustness; Cohen–Coon suits processes with a large delay; lambda (IMC) and SIMC let you choose the closed-loop speed λ and give a smooth response without overshoot. The simulation shows the response of the closed loop (ideal controller with derivative on the measurement filtered with N = 10, no noise, no output limits) to a setpoint step. The control hardware and the PLC side are in PLC and Logic Control and the motor and axis loops in Motion Control and Actuators.

Worked example

A step test on a heated process gives K = 2 (°C per % output), T = 50 s and L = 10 s.

  • Ziegler–Nichols PID: Kc = 1.2 × 50 ÷ (2 × 10) = 3.00, Ti = 2 × 10 = 20 s, Td = 0.5 × 10 = 5 s. Parallel gains: Ki = 0.150 per second and Kd = 15.0.
  • Z–N PI: Kc = 2.25, Ti = 33.3 s. Cohen–Coon PID: Kc = 3.458, Ti = 22.74 s, Td = 3.51 s.
  • Lambda PI with λ = 20 s: Kc = 50 ÷ (2 × (20 + 10)) = 0.833, Ti = T = 50 s. The simulated step response overshoots by 0.0 % and settles to 2 % in about 77 s.
  • SIMC PI (τc = 20 s): Kc = 0.833, Ti = min(50, 4 × 30) = 50 s.

Things to know

The tuning rules are starting points. Test the loop with a small setpoint change and adjust: raise Kc for a faster response, lengthen Ti if it oscillates, and add derivative only if the measurement is clean and the process lags.

The results assume that the controller and the process are in the same units: Kc is the output (percent or volts) per unit of measurement. Many controllers use the proportional band 100 ÷ Kc and the integral rate (repeats per minute = 60 ÷ Ti in seconds).

Derivative action amplifies noise. Use a filter (N between 5 and 20), derivative on the measurement rather than on the error, and consider PI for noisy or fast loops.

Integrating processes (tanks, positions) and processes with a high order or a long delay need other models and rules. Open-loop step tests also need the process to be stable and settled before the test.

Anti-windup, output limits and bumpless transfer matter as much as the tuning in a real controller. Check how your controller implements them.

Safety: tune only with the process safe to disturb, and with protective limits in place.

Common questions

How do I get K, T and L from a step test?

Change the controller output by a step in manual mode. K = (final change in the measurement) ÷ (step size). L is the time from the step until the measurement starts to move. T is the time from there to reach 63 % of the final change.

Which tuning rule should I use?

Lambda or SIMC for most loops, because you choose the speed and get no overshoot. Ziegler–Nichols when a fast response with some overshoot is acceptable; Cohen–Coon when the dead time is large compared with the time constant.

What does each term do?

Proportional reacts to the present error, integral removes the steady offset by accumulating the error, and derivative reacts to the rate of change and adds damping.

Why is my loop oscillating?

The gain is too high or the integral time too short for the dead time. Reduce Kc by half and lengthen Ti.

Sources

  • Ziegler, J. G. and Nichols, N. B. (1942), “Optimum settings for automatic controllers”, Transactions of the ASME 64; Cohen, G. H. and Coon, G. A. (1953), Transactions of the ASME 75.
  • Skogestad, S. (2003), “Simple analytic rules for model reduction and PID controller tuning”, Journal of Process Control 13; Seborg, D. E. et al., Process Dynamics and Control, Wiley (IMC tuning).

Last reviewed: November 2, 2026