The parallel form of PID controller has the transfer function given by Eq $8-14.$ Many commercial analog controllers can be described by the series form given by Eq $8-15.$

$($a$)$ For the simplest case, $0,$ find the relations between the settings for the parallel form $($K$$ c$,$$$ I,$$ D$)$ and the settings for the series form $($Kc$,$ I$,$ D$).$

$($b$)$ Does the series form make each controller setting $($Kc$,$ I$,$ or D$)$larger or smaller than would be expected for the parallel form?

$($c$)$ What are the magnitudes of these interaction effects for Kc $=4,$ I $=10$min, D $=2$min?

$($d$)$ What can you say about the effect of nonzero on these relations? $($Discuss only first$-$order effects.$)$