(a) Find an expression for the amount of derivative kick that wilt bc applied to the process when using the position form of the PID digital algorithm (Eq. 8-26) if a step set-point change of magnitude is made between the k ? 1 and k sampling instants.

(b) Repeat for the proportional kick, that is, the sudden change caused by the proportional mode.

(c) Plot the sequence of controller outputs at, the k ? 1, k, sampling times for the case of a step set-point change of magnitude made just after the k ? 1 sampling time if the controller receives a constant measurement j, and the initial set point is y_sp, = y_m. Assume that the controller output initially is.

(d) How can Eq. 8-26 be modified to eliminate derivative kick?

(ex Pk =p + Ke ek + ek- (8-26) LT LC LT %23 LC (a) (b)