Problem $5$

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.

$D(z)=K(1+r_D(1-z^{-1})+\frac{r_{\text{I}}}{1-z^{-1}})$

Use the delay property of the $z-$transform: