This is the PID CPV discrete iterative control algorithm. CPV = Composite Past Value The single line diagram helps visualising what's going on. The field input is converted from Analog/Digital, then passed into the PID CPV, then converted back to Analog. The D/A to field may be optional, some field controlling devices are fit to discrete signal.

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The reaction curve is the open loop response from in-situ [ideally] at the commissioning. It is a robust method to set the P,I,D based on the Ziegler-Nichols method. It can be predicted from the two dominant time constants of the system. It closeley relates to the XFR which is otherwise undefinable in real process systems. ---------------------------------------------------------- In the example sheets, we simulate different long perturbations. For each of these long trailing input information, we apply a discrete control algorithm and inspect the response over evolution. From reverse thinking: those model input can be interpreted ... representing pieces of Laplace not to avail the maths-effort. Continuous analog control specific applications aren't considered... ... analog servomecanism [RADAR antenna, cannon turret ...]

There are many different styles of derivatives from accumulated data Read more in literature [F.B. Hildebrand]. Here, we give two common styles. The most common style: T-bar