In a chemical reactor it is desired to control the reaction temperature $(T)$ at $90C$ with PD $($Proportional $+$ derivative$)$ controller, by manipulating the flow rate of cooling water $($Fc$).$ In the PD controller, the proportional band value is $50\%$ and the derivative time constant $($ D$)$ is $5$ min. The transfer function of the process, the transfer function of the disturbance, the transfer function of the thermocouple and the transfer function of the valve used are as follows;

$G_p(s)=\frac{-0\mathrm{.}2}{(4^s+1)(5s+1)}$

$G_d(s)=\frac{0\mathrm{.}4}{(4s+1)(5s+1)}\frac{C}{}C,$

$G_s(s)=\frac{\frac{16}{200}}{(s+1)}$

$G_V(s)=\frac{-500}{16}\frac{t}{m}i\frac{n}{m}A$

a$)$ Is this system stable?

b$)$ Is it oscillating?

c$)$ If feed temperature increases by $30C,$ does the response of the system fit to the set point?

d$)$ How much offset value is seen?

e$)$ What is the value of the temperature when the system becomes stable?