$(3$ marks$)$ The transfer function of a process having dead time is given as:

G$\_($p$)($s$)=$G$($s$)$e$^(-\backslash$tau s$)$

where and $\backslash$tau represent the actual process $($dead time free part$)$ and actual dead

time. If the estimated dead time is $\backslash$tau $^(\text{'}),$ derive the open$-$loop transfer function G$\_($OL$)$

having uncompensated dead time with P$-$only controller as:

G$\_($OL$)($s$)=$K$\_($c$)$G$($s$)$e$^(-(\backslash$tau $-\backslash$tau $^(\text{'}))$s$)$

in which K$\_($c$)$ is proportional gain and G$\_($f$)=$G$\_($m$)=1.$ please give the answer in a detailed way