Q$3.(20$ P$)$ 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;

$G_p(s),=\frac{-0\mathrm{.}2}{((4s+1)(5s+1){)}^0}\frac{C}{L}\frac{?}{\text{m}\text{i}\text{n}},G_d(s)=\frac{0\mathrm{.}4}{(4s+1)(5s+1)},{?}^0\frac{C}{}C$

$G_s(s),=\frac{\frac{16}{200}}{(s+1)}m\frac{A}{}C,G_V(s)=\frac{-500}{16}\frac{L}{m}i\frac{n}{m}A$

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? How

much offset value is seen?

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