Model-based tuning: IMC and SIMC

In short. With a model K, τ, θ the settings follow from formulas. The only choice is τc — how fast the closed loop should follow a set-point change. A smaller τc is faster but more sensitive to model errors. To start: SIMC with τc = θ. The tuning tool does the calculation.

Formulas

Model: K — process gain, τ — time constant, θ — dead time (see step test). Results in the ideal form, time in seconds, Kc in units of 1/K.

SIMC PI (Skogestad, 2003):
    Kc = τ / (K · (τc + θ))
    Ti = min(τ, 4 · (τc + θ))
    recommended τc = θ

IMC PI:
    Kc = τ / (K · (θ + τc))         Ti = τ

IMC PID:
    Kc = (τ + θ/2) / (K · (τc + θ/2))
    Ti = τ + θ/2                    Td = τ·θ / (2τ + θ)

τc presets for IMC (APMonitor): aggressive max(0.1τ, 0.8θ), moderate max(τ, 8θ), conservative max(10τ, 80θ).

Example

Section model: K = 1.8 °C/%, τ = 45 s, θ = 12 s.

Responses are for a set-point step simulated with Ts = 1 s. SIMC check: Kc = 45 / (1.8 · (12 + 12)) = 45 / 43.2 = 1.042.

Open the model in the tuning tool · Check SIMC in the simulator

Disturbance rejection: for a 1 % step at the process input (without control PV would move 1.8 °C) SIMC with τc = θ keeps the PV deviation below 0.65 °C, with the peak after 40 s.

How to choose τc

• τc = θ — tight control, a good compromise between speed and stability margin according to Skogestad.
• Larger τc (2θ…τ and above) — if the model is uncertain, the process is non-linear, the output is noisy or smoothness matters (levels, interacting loops).
• τc below θ — not recommended: the dead time cannot be removed, and the loop oscillates for a small model error.

Kc is inversely proportional to K: if the real process gain is twice the model one, the overshoot of a loop tuned with τc = θ grows from 5.5 % to 55 % (troubleshooting).

Typical mistakes

• Model in one set of units, block in another. K in °C/% gives Kc in %/°C; for a %/% block convert (normalisation).
• Wrong PID form. The formulas give the ideal form; for parallel Ki = Kc/Ti, Kd = Kc·Td (forms).
• IMC PID without a D filter — noise and output jumps (D filter).
• FOPDT formulas for an integrating process — see FOPDT limits.

Limits

The formulas assume a linear FOPDT process and a continuous controller. The sample time adds about Ts/2 to the dead time (discretisation); output limits and saturation are not considered. The result is a starting point: check it in the simulator and on the plant.