Editing Proportional–integral–derivative controller (section)

=== First-order model with dead time ===
The transfer function for a first-order process with dead time is

: <math>y(s) = \frac{k_\text{p} e^{-\theta s}}{\tau_\text{p} s + 1} u(s),</math>

where ''k''<sub>p</sub> is the process gain, ''τ''<sub>p</sub> is the time constant, ''θ'' is the dead time, and ''u''(''s'') is a step change input. Converting this transfer function to the time domain results in

: <math>y(t) = k_\text{p} \Delta u \left(1 - e^{\frac{-t - \theta}{\tau_\text{p}}}\right),</math>

using the same parameters found above.

It is important when using this method to apply a large enough step-change input that the output can be measured; however, too large of a step change can affect the process stability. Additionally, a larger step change ensures that the output does not change due to a disturbance (for best results, try to minimize disturbances when performing the step test).

One way to determine the parameters for the first-order process is using the 63.2% method. In this method, the process gain (''k''<sub>p</sub>) is equal to the change in output divided by the change in input. The dead time ''θ'' is the amount of time between when the step change occurred and when the output first changed. The time constant (''τ''<sub>p</sub>) is the amount of time it takes for the output to reach 63.2% of the new steady-state value after the step change. One downside to using this method is that it can take a while to reach a new steady-state value if the process has large time constants.<ref>{{Cite book |title=Process Control: Modeling, Design, and Simulation |last=Bequette |first=B. Wayne |publisher=Prentice Hall |year=2003 |isbn=978-0-13-353640-9 |location=Upper Saddle River, New Jersey |pages=129}}</ref>