control//feedback control//PID controller//PID tuning//SIMC rules
The SIMC rules are a set of PID tuning formulas, published by Sigurd Skogestad in 2003, that compute a PI controller directly from a first-order-plus-dead-time model of the plant with a single knob, the closed-loop time constant \(\tau_c\) the engineer asks for. They are what a process engineer reaches for after a step test, and they turn ten minutes of testing into two lines of arithmetic.
The SIMC rules are a set of PID tuning formulas, published by Sigurd Skogestad in 2003, that compute a PI controller directly from a first-order-plus-dead-time model of the plant with a single knob, the closed-loop time constant τc\tau_cτc the engineer asks for. They are what a process engineer reaches for after a step test, and they turn ten minutes of testing into two lines of arithmetic.
The model comes from the plant itself. With the loop in manual, move the valve by 5 % and fit the response with a gain KKK, a time constant T1T_1T1 and a dead time θ\thetaθ (FOPDT model, plant). The PI settings are then
Kc=1K T1τc+θ,Ti=min{T1, 4(τc+θ)},K_c=\frac{1}{K}\,\frac{T_1}{\tau_c+\theta},\qquad T_i=\min\{T_1,\;4(\tau_c+\theta)\},Kc=K1τc+θT1,Ti=min{T1,4(τc+θ)},
with KcK_cKc the proportional gain and TiT_iTi the integral (reset) time. The controller gain is the inverse of the plant gain, scaled by how slow the plant is against how fast the loop is asked to be. The dead time sits in the denominator, so it limits the gain directly: a plant that answers late cannot be pushed hard, whatever τc\tau_cτc says. Setting τc=θ\tau_c=\thetaτc=θ is the recommended default, a compromise between speed and robustness. A heated tank with K=2K=2K=2 °C/%, T1=300T_1=300T1=300 s and θ=30\theta=30θ=30 s gives Kc=2.5K_c=2.5Kc=2.5 %/°C and Ti=240T_i=240Ti=240 s.
The integral time has two regimes. For a plant whose lag is short against the loop, Ti=T1T_i=T_1Ti=T1 cancels the plant's lag, as internal model control would. For a plant with a long lag, or one that integrates (a tank level), waiting T1T_1T1 would leave load disturbances uncorrected for far too long, so the integral time is capped at 4(τc+θ)4(\tau_c+\theta)4(τc+θ); that cap was Skogestad's main change to the older IMC rules.
τc\tau_cτc is the one knob, and it means something. Asking for a slower closed loop (a larger τc\tau_cτc) lowers the gain and widens the margins, which is the honest answer to a valve that sticks a little, a plant whose gain moves with load, or a model fitted on a noisy test.
Plants with two lags are first reduced to one by the half rule: half of the faster lag's time constant is added to the dead time and the other half to the dominant lag. It is the same effect a step test shows from the other side, where a fast lag comes back as extra dead time.
Against Ziegler-Nichols, SIMC needs a step test instead of a loop driven to sustained oscillation, gives smooth and robust settings instead of aggressive ones, and tells the engineer which single number to change when the loop is too slow or too nervous.