Step test and the FOPDT model

In short. Put the controller in manual, wait for a steady PV, step the output and record PV until it settles again. The trend gives three numbers: K (how much PV changed per 1 % of output), τ (how fast) and θ (after how long it started to move). The settings are calculated from them (IMC and SIMC). The FOPDT tool finds the model.

Model

τ · dy/dt = −(y − y0) + K · (u(t − θ) − u0)

K = Δy / Δu          [PV unit / output unit, e.g. °C/%]
τ — time constant, s
θ — dead time, s

After a step Δu the model output does not move for θ seconds, then follows an exponential towards y0 + K·Δu and covers 63.2 % of the change in τ.

How to run the test

1. Steady state. PV and output have been constant for at least 2–3 τ (judge from earlier trends).
2. Manual mode; write down u0 and y0.
3. Output step of 5–10 % of span — so that the PV change is 5–10 times the noise and stays within process limits.
4. Do not touch the output or other loops until PV settles (usually θ + 4τ…5τ).
5. Record time, output and PV at a period of at most τ/10.
6. If possible repeat with a step in the opposite direction: a difference in K and τ reveals non-linearity.

Two-point method

Find the times (counted from the step) at which PV covered 28.3 % and 63.2 % of the full change:

τ = 1.5 · (t63 − t28)
θ = t63 − τ
K = (y∞ − y0) / (u1 − u0)

Example

Output 40 → 50 %, PV 55 → 73 °C. Levels: 55 + 0.283·18 = 60.1 °C and 55 + 0.632·18 = 66.4 °C; the trend reaches them at t28 = 27.0 s and t63 = 57.0 s.

K = (73 − 55) / (50 − 40) = 1.8 °C/%
τ = 1.5 · (57.0 − 27.0) = 45 s
θ = 57.0 − 45 = 12 s

This is the model used in all examples of the section. The ratio θ/τ = 0.27: noticeable dead time, but the loop is well controlled by a PI.

Open the example in the FOPDT tool

Fit over the whole trend

Two points are sensitive to noise at the crossing. In CSV mode the tool starts from the two-point estimate and then fits K, τ, θ by least squares over the whole trend, simulating the response to the output actually recorded. If R² < 0.9 or PV has not settled by the end of the record, the tool warns — the model is unreliable.

Typical mistakes

• Test not started from steady state. PV drift before the step ends up in K and τ.
• Step too small — the change drowns in noise; too large — the process leaves its linear region.
• Test in automatic — the response belongs to the loop, not the process.
• A disturbance during the test (load change, a neighbouring loop acting).
• Sensor and filter lag ignored. A PV filter and a slow sensor become part of θ and τ — correct, as long as they are the same in operation.
• Sign mixed up. K < 0 is a normal result for a process whose PV falls when the output rises (cooling); the direction of action is chosen from it (direct and reverse action).

Limits

FOPDT is an approximation. It describes self-regulating processes with a smooth S-shaped response well. For integrating processes, responses with overshoot or inverse response it does not fit — see FOPDT limits. The model is valid near the operating point of the test.