Limits of the FOPDT model
When a first-order-plus-dead-time model fits and when it does not: integrating and oscillatory processes, inverse response, non-linearity, lag-dominant processes. Skogestad's half rule and the integrating process model.
In short. FOPDT describes a self-regulating process with a smooth S-shaped response. If PV keeps moving after a step (level), oscillates, or first goes the wrong way, the FOPDT IMC/SIMC formulas give wrong settings. Check the shape of the response before calculating.
What shape the process response has
| Response to an output step | Process type | What to do |
|---|---|---|
| Moves smoothly to a new value | self-regulating | FOPDT, IMC/SIMC |
| Ramps at a constant rate and never stops | integrating (level, pressure in a closed vessel) | integrator-plus-delay model (below) |
| Almost an integrator: τ much larger than θ (τ > 8θ) | lag dominant | FOPDT acceptable, but Ti = 4(τc + θ) instead of τ |
| Oscillates on its own | oscillatory (second order) | FOPDT does not fit; second-order model or careful manual tuning |
| First goes the opposite way | inverse response (boiler drum level) | add the inverse response to θ; tune slower |
| Different K or τ for steps up and down | non-linear | test at several points, tune for the worst case or use gain scheduling |
Half rule
For a process with several time constants (τ1 ≥ τ2 ≥ …) and dead time θ0 Skogestad proposes this FOPDT approximation: add half of the second largest time constant to τ1 and the other half to the dead time, and count the smaller time constants entirely as dead time.
τ = τ1 + τ2/2
θ = θ0 + τ2/2 + τ3 + …
Example: τ1 = 40 s, τ2 = 10 s, θ0 = 5 s → τ = 45 s, θ = 10 s.
Integrating process
Model: PV changes at a rate proportional to the output change, after a dead time θ.
dy/dt = k′ · (u(t − θ) − u0) k′ — [PV unit / (output unit · s)]
SIMC PI: Kc = 1 / (k′ · (τc + θ)), Ti = 4 · (τc + θ)
Example: the level rises 0.6 %/min when the valve opens by 10 % → k′ = 0.6 / 60 / 10 = 0.001 1/s; θ = 10 s. For averaging level control choose τc = 100 s:
Kc = 1 / (0.001 · 110) = 9.09 %/% Ti = 4 · 110 = 440 s
The tuning tool in the app handles only self-regulating FOPDT; calculate an integrating process with the formula above.
Lag-dominant process
If τ ≫ θ, IMC with Ti = τ rejects disturbances at the process input slowly: PV takes a long time to return to set point. SIMC limits Ti = min(τ, 4·(τc + θ)).
Example: K = 1.8 °C/%, τ = 400 s, θ = 12 s, τc = θ → Kc = 400 / (1.8·24) = 9.26 %/°C; Ti = min(400, 96) = 96 s, not 400 s.
Typical mistakes
- Fitting FOPDT to a level: the test is cut short, the "steady" value is accidental, K and τ are meaningless.
- Assuming θ = 0 because the response started at once: as θ → 0 the formulas give a very large Kc. Count the sample time, filter and sensor.
- Not checking non-linearity: settings from a test at low load make the loop oscillate at high load.
Limits
The half rule and the integrator model are approximations for tuning, not exact models. Oscillatory processes and processes with strong inverse response need models and methods beyond this section.
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Updated 2026-10-12