PID controllers
Incremental algorithm

The incremental algorithm is given by formal modification of the absolute one, according to following expression:

Δuk = P·Δek + I·ek + D·Δ2ek

Output of the algorithm is a control variable increment, Δuk. Integral component is proportional to current error ek , proportional component is proportional to its first difference Δek = ek – ek-1 and a derivative component is proportional to a second difference, Δ2ek = Δek – Δek-1 = ek – 2ek-1 + ek-2. The increment Δuk represents a velocity of control variable uk change. Thus, the algorithm is often called a velocity or incremental. Absolute value of uk can be computed by cumulative sum of the increments, uk = uk-1 + Δuk). Lettres P, I, D represents particular gains (amplifications) of respective components.

After formal modification, the incremental algorithm can be simplified to:

Δuk = q0·ek + q1·ek-1 + q2·ek-2

Fig. 8.18: Principial schematics of velocity (incremental) PID algorithm