PID controllers
FIR system relationship

Control variable increment is a linear combination of three samples of the error – current ek, previous ek-1 and even one before ek-2. The increment is a linear combination of values in three element delay line. Such system is an example of a FIR system. The algorithm contains only three multiplications and two additions, coefficients q0, q1, q2 can be precomputed once from given coefficients P, I, D (gains).

Fig. 8.19: Incremental PID controller as a FIR system