A digital control U $($z$)$ is defined by means of a digital compensator D$($z$)$ such that U $($z$)=$

D$($z$)$E$($z$).$ E$($z$)$ is the error signal.

Write down a finite difference equation for computing the control u$($k$)$ in discrete time domain

with the error e$($k$)$ as input for the following compensator.

D$($z$)=$ K

$1+$ rD$(1$ z$1)+$ rI

$1$ z$1$

$(1)$

Use the delay property of the z$-$transform: Z$[$u$($n $$ k$)]=$ z$$kU $($z$)($k $=0,1,2,).$