$(25$ pt$)$ Consider the system shown in Figure $P16\mathrm{.}10,$ where ${}_1,{}_2,{}_3,{}_{\text{I}}$ and $k_{c}k>0.$

Figure P$16\mathrm{.}10$

$($a$)$ Derive the necessary and sufficient conditions for closed$-$loop stability.

$($b$)$ Derive an analytic formula for the ISE criterion as a function of the controller gain and integral

time for a unit step change in the set point.

$($c$)$ Determine the optimal values of the controller parameters and simulate the corresponding closed$-$

loop response.

please solve it fully