PID Tuning Calculator Formula

Understand the math behind the pid tuning calculator. Each variable explained with a worked example.

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Formulas Used

Proportional Gain (Kp)

kp = 0.6 * ku

Integral Gain (Ki)

ki = 1.2 * ku / tu

Derivative Gain (Kd)

kd = 0.075 * ku * tu

Integral Time (Ti)

ti = tu / 2

Derivative Time (Td)

td = tu / 8

Variables

VariableDescriptionDefault
kuUltimate Gain (Ku)10
tuUltimate Period (Tu)(s)2

How It Works

Ziegler-Nichols PID Tuning

The Ziegler-Nichols closed-loop method determines PID gains from two measurements: the ultimate gain Ku (gain at which the system oscillates) and the ultimate period Tu (period of those oscillations).

Formulas

Kp = 0.6 × Ku

Ti = Tu / 2, so Ki = Kp / Ti = 1.2 × Ku / Tu

Td = Tu / 8, so Kd = Kp × Td = 0.075 × Ku × Tu

These gains provide a starting point; fine-tuning is usually needed to balance response speed, overshoot, and stability.

Worked Example

A system oscillates at Ku = 10 with period Tu = 2 seconds.

ku = 10tu = 2
  1. 01Kp = 0.6 × 10 = 6.0
  2. 02Ti = 2 / 2 = 1.0 s, Ki = 1.2 × 10 / 2 = 6.0
  3. 03Td = 2 / 8 = 0.25 s, Kd = 0.075 × 10 × 2 = 1.5

Frequently Asked Questions

What is the Ziegler-Nichols method?

It is a classical tuning technique where you increase the proportional gain until the system oscillates continuously. The gain and period at this point (Ku, Tu) are used to compute PID gains from empirical formulas.

Is Ziegler-Nichols tuning optimal?

No. ZN tuning typically produces aggressive response with 25% overshoot. It provides a good starting point, but most systems benefit from adjustment. Modern methods like IMC, lambda tuning, or auto-tuning often give better results.

What if I only want PI control?

For PI control (no derivative): Kp = 0.45 × Ku, Ti = Tu / 1.2. For P-only control: Kp = 0.5 × Ku. These are also from the Ziegler-Nichols table.

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