How do I find Ku and Pu for Ziegler–Nichols tuning?

1) Disable integral and derivative action (Ki = 0, Kd = 0). 2) Slowly increase the proportional gain Kp from a low value until the output oscillates with a constant amplitude. This gain is the ultimate gain Ku. 3) Measure the time between two consecutive peaks of the oscillation; this is the ultimate period Pu.

When should I avoid using Ziegler–Nichols tuning?

Avoid Ziegler–Nichols if your process is unstable, highly nonlinear, has long dead time, or if sustained oscillations are unsafe or can damage equipment. In those cases, use safer methods such as relay auto-tuning, model-based tuning, or conservative manual tuning.

What is the difference between classic and modified Ziegler–Nichols rules?

Classic Ziegler–Nichols rules target a quarter-amplitude damping response, which is often aggressive and overshoots. Modified rules reduce Kp and/or Kd to improve robustness and reduce overshoot, at the cost of slower response. This calculator shows both sets of parameters so you can choose.

Ziegler–Nichols PID Tuning Calculator

Q: What is the Ziegler–Nichols tuning method?

The Ziegler–Nichols tuning method is a classical empirical procedure to tune PID controllers. You first increase the proportional gain until the closed-loop system oscillates with constant amplitude (ultimate gain Ku and period Pu). Then you compute Kp, Ki, and Kd from simple formulas based on Ku and Pu.

Compute PID gains from ultimate gain K_u and oscillation period P_u using classic and modified Ziegler–Nichols rules.

Enter Ultimate Gain and Period

Ultimate gain K_u

Gain at which the loop oscillates with constant amplitude.

Ultimate period P_u (seconds)

Oscillation period at K_u (time between peaks).

PID Parameters from Ziegler–Nichols

Controller Type	Kp	Ti (s)	Td (s)	Ki	Kd

Note: Ki = Kp / Ti and Kd = Kp · Td. Units of Ki and Kd depend on your implementation.

How to use this Ziegler–Nichols calculator

Disable integral and derivative in your controller: set Ki = 0, Kd = 0.
Increase Kp slowly until the output oscillates with constant amplitude. Record this gain as K_u.
Measure the time between two consecutive peaks of the oscillation. This is the ultimate period P_u.
Enter K_u and P_u above and click Compute PID Gains.
Choose between Classic and Modified rules depending on how much overshoot you can tolerate.

Formulas used (Ziegler–Nichols closed-loop method)

Classic Ziegler–Nichols rules:

P controller: Kp = 0.5·K_u
PI controller: Kp = 0.45·K_u, Ti = P_u/1.2
PID controller: Kp = 0.6·K_u, Ti = P_u/2, Td = P_u/8

Integral and derivative gains are computed as Ki = Kp / Ti and Kd = Kp · Td.

Modified (less aggressive) rules used here:

P controller: Kp = 0.33·K_u
PI controller: Kp = 0.35·K_u, Ti = P_u/1.2
PID controller: Kp = 0.5·K_u, Ti = P_u/2, Td = P_u/8

These values are commonly used variants that trade some speed for better robustness and lower overshoot.

Example calculation

Suppose you experimentally find:

Ultimate gain: K_u = 4.0
Ultimate period: P_u = 2.0 s

For a classic PID controller:

Kp = 0.6 · 4.0 = 2.4
Ti = P_u / 2 = 1.0 s → Ki = Kp / Ti = 2.4 / 1.0 = 2.4
Td = P_u / 8 = 0.25 s → Kd = Kp · Td = 2.4 · 0.25 = 0.6

You would configure your PID as Kp = 2.4, Ki = 2.4, Kd = 0.6 (adjusting for your controller’s units).

Limitations and best practices

Ziegler–Nichols assumes a reasonably linear, stable process without large dead time.
The method intentionally produces a fairly oscillatory response (quarter-amplitude damping).
Always validate the tuned gains at low setpoints or in simulation before applying to critical equipment.
Use the Modified tab as a safer starting point, then fine-tune manually.

Ziegler–Nichols PID Tuning – FAQ

What is the Ziegler–Nichols tuning method?

It is a classical empirical method to tune PID controllers. You first find the ultimate gain K_u and ultimate period P_u by forcing the loop to oscillate with only proportional control. Then you compute Kp, Ki, and Kd from simple formulas. It is quick and requires no process model, but can be aggressive.

How do I safely find K_u and P_u?

Start from a low Kp with Ki = 0 and Kd = 0. Increase Kp gradually while monitoring the output and actuator limits. Stop increasing as soon as you see sustained oscillations. Measure the period between peaks as P_u. If oscillations grow or the system behaves dangerously, stop and use a different tuning method.

What is the difference between P, PI, and PID in this table?

P uses only proportional action (Ki = 0, Kd = 0).
PI adds integral action to remove steady-state error (Kd = 0).
PID adds derivative action to improve damping and response speed. Most industrial loops use PI or PID.

Why do my units for Ki and Kd look different from other sources?

Some controllers use Ki as 1/Ti, others use Ki = Kp/Ti, and some specify integral and derivative times directly (Ti, Td). This calculator outputs both Ti, Td and Ki, Kd. Always check your controller’s documentation and convert accordingly.