control//system identification//FOPDT model

A FOPDT model, first order plus dead time, is a three-parameter process model (a gain \(K\), a time constant \(T_1\) and a dead time \(\theta\)) fitted from a single step test, and it is the daily basis of PID tuning in process plants. Most temperature, level, pressure and flow loops are tuned from it: put the controller in manual, move the valve 5 %, record the response, and read three numbers. Its transfer function is

A FOPDT model, first order plus dead time, is a three-parameter process model (a gain KKK, a time constant T1T_1T1​ and a dead time θ\thetaθ) fitted from a single step test, and it is the daily basis of PID tuning in process plants. Most temperature, level, pressure and flow loops are tuned from it: put the controller in manual, move the valve 5 %, record the response, and read three numbers. Its transfer function is

G(s)=K e−θsT1s+1.G(s)=\frac{K\,e^{-\theta s}}{T_1 s+1}.G(s)=T1​s+1Ke−θs​.

KKK is how far the output moves per unit of input at steady state (°C per % of valve), T1T_1T1​ how fast it gets there once it starts (time constant), and θ\thetaθ how long nothing happens at all, the transport and measurement delay (loop delay). A real process is rarely first order, but higher-order lags look, from outside, much like a first-order response plus some extra delay, so the model fits surprisingly many plants.

A heated tank shows the whole workflow. A 5 % step gives K=2K=2K=2 °C/%, T1=300T_1=300T1​=300 s and θ=30\theta=30θ=30 s. The SIMC rules then give a PI with gain Kc=2.5K_c=2.5Kc​=2.5 %/°C and integral time Ti=240T_i=240Ti​=240 s, taking the requested closed-loop time constant equal to θ\thetaθ (SIMC rules). Ten minutes of testing and two lines of arithmetic.

The dead time limits the loop. It sits in the denominator of every tuning rule, because delay eats phase and caps the achievable gain; a process with θ\thetaθ comparable to T1T_1T1​ is hard to control tightly whatever the controller, and calls for a Smith predictor or an MPC.

A step test is an experiment, and its conditions matter: the plant should be at steady state before the step, the step large enough to stand above noise and small enough to stay linear and safe, and a step in each direction reveals asymmetry such as valve stiction (experiment design).

It is the flyswatter of modelling. A 40-state model pays only when something needs it, such as a multivariable predictive controller; for single loops, three numbers and a rule beat a sophisticated model that nobody on the night shift can check (system identification).

For an integrating process (a tank level with an outlet pump) the step never settles, and the right model is an integrator plus dead time; forcing a FOPDT onto it gives a huge, meaningless T1T_1T1​.