Q$-10$ For the system shown in Figure $Q-2,G_P=\frac{1}{5s+1},G_f=\frac{1}{2s+1},G_m=\frac{1}{s+a},$ and $G_c=K_c$ are given.

$($a$)$ Determine the transfer function, $Y\frac{s}{R}(s)$ and the characteristic equation, writing it in the form, $(s)=0.$

$[12$ pts$.\}$

$($b$)$ Determine the values of the controller gain, Kc by using Routh criteria, so that the feedback control system becomes stable.

$[18$ pts$.]$