A process stream is heated using a shell and tube heat exchanger. The exit temperature is controlled by adjusting the steam control valve shown in Fig. E$12\mathrm{.}5.$ During an open$-$loop experimental test, the steam pressure $P_s$ was suddenly changed from $18$ to $20$ psig and the temperature data shown below

were obtained. At the nominal conditions, the control valve and current$-$to$-$pressure transducers have gains of $K_v=0\mathrm{.}9\frac{}{}$ and $K_{IP}=0\mathrm{.}75\frac{}{m}A,$ respectively. Determine appropriate PID controller settings using the following approaches:

$($a$)$ Internal Model Control $($select a reasonable value of ${}_c)$

$($b$)$ AMIGO $($with $=1$ and $=0)$

$($c$)$ ITAE $($disturbance$)$

Which controller is the most aggressive?

$\backslash$table$[[t(min),T_{2m}(mA)$