Consider the following Dual$-$Loop feedback control system

Fig. $4$

$(4$a$)$ Find the transfer functions of the closed$-$loop system, $Y\frac{s}{R}(s),E\frac{s}{R}(s)$ and $U\frac{s}{R}(s),$ in terms of the controller parameters $K_1$ and $K_2.$

$(4$b$)$ Find the poles and zeros of the closed$-$loop system transfer function $Y\frac{s}{R}(s)$ in terms of the controller parameters $K_1$ and $K_2.$

$(4$c$)$ Determine the values of $K_1$ and $K_2$ so that the damping ratio and the natural frequency of the closed$-$loop system are $=0\mathrm{.}8$ and ${}_n=50ra\frac{d}{s},$ respectively.

$(4$d$)$ Determine another set of $K_1$ and $K_2$ so that the damping ratio and the natural frequency of the closedloop system are $=0\mathrm{.}8$ and ${}_n=70ra\frac{d}{s},$ respectively.