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 $1.$ reedback temperature control loop $($block diagram$)$ tor 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 a 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.

$\backslash$table$[[,$Proportional $($P$),$Proportional$-$Integral $($PI$),\backslash$table$[[$Proportional$-$Integral$-],[$Derivative $($PID$)]],,],[K_c,9,12,20,8\mathrm{.}5,11,5,14],[{}_{\text{I}},-,-,-,0\mathrm{.}4,0\mathrm{.}8,0\mathrm{.}4,0\mathrm{.}8,16,0\mathrm{.}3],[{}_D,-,-,-,-,-,,4\mathrm{.}6,0\mathrm{.}05]]$