(a) For the digital velocity P and PD algorithms, show how the set point enters into calculation of \Delta p_{k} on the assumption that it is not changing, that is, y_{s p} is a constant.
(b) What do the results indicate about use of the velocity form of P and PD digital control algorithms?
(c) Are similar problems encountered if the integral mode is present, that is, with PI and PID forms of the velocity algorithm? Explain.
a) Let the constant set point be denoted by \bar{y}_{s p} . The digital velocity P algorithm is obtained by setting 1 / \tau_{I}=\tau_{D}=0 in Eq. 8-27:
\begin{aligned}\Delta p_{k}=p_{k}-p_{k-1}=& K_{c}\left[\left(e_{k}-e_{k-1}\right)+\frac{\Delta t}{\tau_{I}} e_{k}\right.\\&\left.+\frac{\tau_{D}}{\Delta t}\left(e_{k}-2 e_{k-1}+e_{k-2}\right)\right]\end{aligned} (8-27)
\begin{aligned}\Delta p_{k} &=K_{c}\left(e_{k}-e_{k-1}\right) \\&=K_{c}\left\lfloor\left(\bar{y}_{s p}-y_{k}\right)-\left(\bar{y}_{s p}-y_{k-1}\right)\right] \\&=K_{c}\left[y_{k-1}-y_{k}\right]\end{aligned}
The digital velocity PD algorithm is obtained by setting 1 / \tau_{I}=0 in Eq. 8 27:
\begin{aligned}\Delta p_{k} &=K_{c}\left[\left(e_{k}-e_{k-1}\right)+\frac{\tau_{D}}{\Delta t}\left(e_{k}-2 e_{k-1}+e_{k-2}\right)\right] \\&=K_{c}\left[\left(-y_{k}+y_{k-1}\right)+\frac{\tau_{D}}{\Delta t}\left(-y_{k}-2 y_{k-1}+y_{k-2}\right)\right]\end{aligned}
In both cases, \Delta p_{k} does not depend on \bar{y}_{s p}.
b) For both these algorithms \Delta p_{k}=0 if y_{k-2}=y_{k-1}=y_{k} . Thus a steady state is reached with a value of y that is independent of the value of \bar{y}_{s p} . Use of these control algorithms is inadvisable if offset is a concern.
c) If the integral mode is present, then \Delta p_{k} contains the term K_{c} \frac{\Delta t}{\tau_{I}}\left(\bar{y}_{s p}-y_{k}\right).
Thus, at steady state, \Delta p_{k}=0 and y_{k-2}=y_{k-1}=y_{k}, y_{k}=\bar{y}_{s p}, and the offset problem is eliminated.