Consider the feedback temperature control loop with a stirred heater tank given in the block diagram

below. The transfer functions of the loop elements except for the controller are provided as follows.

Figure $1.$ Feedback temperature control loop $($block diagram$)$ for the stirred heater tank system.

$G_s(s)=\frac{1\mathrm{.}2}{(8\mathrm{.}3s+1)(0\mathrm{.}5s+1)},G_F(s)=\frac{-3\mathrm{.}3(0\mathrm{.}52s+1)}{(8\mathrm{.}3s+1)(0\mathrm{.}5s+1)},G_v(s)=\frac{1\mathrm{.}65}{0\mathrm{.}2s+1},H(s)=\frac{e^{-1\mathrm{.}5s}}{(0\mathrm{.}75s+1)}$

a$.$ Find the transfer function related to the output current temperature $($including setpoint changes

and disturbances$).$

b$.$ Create a MATLAB SIMULINK model for the given block diagram.

c$.$ Consider separately the conditions of having P$,$ PI$,$ and PID controllers in this loop. Using MATLAB

SIMULINK, find and plot the dynamic response of the system to disturbance of $2$ units occurring at

$t=0,$ in addition to a setpoint change of $1$ unit at $t=5.$

The controller parameters are listed in the table below.