Ziegler–Nichols and other classic methods

In short. Ziegler and Nichols (1942) proposed calculating settings from two numbers: the ultimate gain Ku, at which a P controller brings the loop to sustained oscillation, and the oscillation period Tu. The method is simple but gives aggressive settings (decay of about 1/4 per period) and requires driving the plant into oscillation. For process loops SIMC is usually better.

Ultimate-gain method

1. Turn off I and D, controller in automatic.
2. Increase Kc in steps, applying a small disturbance after each change.
3. Find the Kc at which PV oscillations neither decay nor grow: this is Ku; the oscillation period is Tu.

P:    Kc = 0.5·Ku
PI:   Kc = 0.45·Ku     Ti = Tu / 1.2
PID:  Kc = 0.6·Ku      Ti = Tu / 2      Td = Tu / 8

The result is in the ideal form.

Example

For the section model (K = 1.8 °C/%, τ = 45 s, θ = 12 s) the ultimate values can be calculated: the process phase is −180° at the frequency where θ·ω + arctan(τ·ω) = π. This gives Ku = 3.63 %/°C, Tu = 43.7 s.

P steady-state error: 1 / (1 + Kc·K) = 1 / (1 + 1.815·1.8) = 0.234.

Open the example in the tuning tool

Step-response method

The same authors proposed a second variant based on an open-loop step test. A tangent at the inflection point gives the delay L and the slope; in FOPDT terms a = K·θ/τ:

P:    Kc = 1/a
PI:   Kc = 0.9/a     Ti = 3·θ
PID:  Kc = 1.2/a     Ti = 2·θ     Td = θ/2

For the section model a = 1.8·12/45 = 0.48 °C/% → PI Kc = 1.875 %/°C, Ti = 36 s — just as aggressive. This variant is not implemented in the tuning tool.

Relay test

Instead of searching for Ku by hand, the output is switched between u0 ± d depending on the sign of the error. The loop settles into an oscillation with a period ≈ Tu and a PV amplitude a:

Ku ≈ 4·d / (π·a)

The oscillation is limited by the amplitude d, so the test is safer. Many PLC autotuners work this way; which algorithm a particular block uses — check the documentation of its version.

Why the settings are aggressive

The Ziegler–Nichols criterion is a decay of about 4 times per period. For process loops this is a lot: 40–60 % overshoot and a small stability margin. Skogestad shows that SIMC on the same model gives a more robust loop with comparable disturbance rejection.

Typical mistakes

• Driving the plant to the stability limit where oscillation is unacceptable (pressure, reactor temperature).
• Finding Ku with a saturated output: saturation "limits" the oscillation and Ku comes out too high.
• Using Z–N PID without a D filter — Td = Tu/8 with a noisy PV makes the output jump.
• Treating the result as final: it is a starting point; Kc usually has to be reduced.

Limits

Cohen–Coon, CHR, Tyreus–Luyben and other table methods are not given in this section: they have different assumptions and table editions; use them only from the primary source.