$12\mathrm{.}7.($a$)$ For the control system shown in Fig. P$12-7,$ obtain the closed$-$loop transfer function $\frac{C}{U}.$

$($b$)$ Find the value of $K_c$ for which the closed$-$loop response has a $$ of $2\mathrm{.}3.$

$($c$)$ Find the offset for a unit$-$step change in $U$ if $K_c=4.$

FIGURE P$12-7$