First-Order Process Response with Delay

Method & assumptions

1. EU means engineering unit. Use consistent input/output units; gain is output-unit change divided by input-unit change. This is a stable linear first-order-plus-dead-time process initially at equilibrium, with one sustained step at time zero.
2. Final change = gain × input step; final output = initial output + final change. Negative gains and steps are allowed.
3. Before delay θ, output stays at its initial value. At and after θ: y = y_initial + final_change × [1 − exp(−(t − θ)/τ)], with positive time constant τ. At θ + τ, 63.212% of the final change is complete.
4. After the delay, output rate = (final_change/τ) × exp(−(t − θ)/τ). At the delay boundary the tool reports the right-hand derivative; the derivative jumps from zero for a nonzero step.
5. Time to fraction f is θ − τ ln(1 − f), measured from the input step. For this monotonic model, 90%, 95% and 98% correspond to remaining-error bands of 10%, 5% and 2% of the final change, not of the absolute output.
6. If the final change is zero, output stays constant, rate and response times are zero, and percent completion is not applicable. Exact final equilibrium is approached asymptotically for a nonzero change; displayed numbers eventually round to it.
7. This does not fit parameters, tune a controller or simulate a closed loop, oscillation, actuator saturation, multiple steps, nonlinear behavior or an unstable/integrating process. Use identified model parameters valid around the operating point.

Sources

• APMonitor Process Dynamics and Control: First Order Plus Dead Time ↗