$G(s)=\frac{4}{s+2}$

$K(s)=K_P+\frac{K_{\text{I}}}{s}$

Consider the feedback control system with the PI controller shown in Fig. $3.$

$(3$a$)$ Find the transfer functions $Y\frac{s}{R}(s),E\frac{s}{R}(s),$ and $U\frac{s}{R}(s)$ in terms of the parameters $K_P$ and $K_{\text{I}}.$

$(3$b$)$ Find the poles and zeros of the closed$-$loop system transfer function $Y\frac{s}{R}(s)$ in terms of the controller parameters $K_P$ and $K_l.$

$(3$c$)$ Determine the values of $K_P$ and $K_{\text{I}}$ so that the damping ratio and the natural frequency of the closedloop system are $=0\mathrm{.}8$ and ${}_n=50ra\frac{d}{s},$ respectively.

$(3$d$)$ Determine another set values of $K_P$ and $K_{\text{I}}$ so that the damping ratio of the closed$-$loop system are $=0\mathrm{.}8$ and ${}_n=70ra\frac{d}{s},$ respectively.