$4.$ Consider a closed$-$loop process shown in Fig. below, in which a PI controller is used to control the system.

$($a$)(3$ marks$)$ Find the ultimate gain $\backslash (\backslash$left$($K$\_\{$u$\}^\{\backslash$prime$\}\backslash$right$)\backslash )$

$($b$)(2$ marks$)$ and ultimate period $(\backslash ($ P$\_\{$u$\}\backslash )).$

$($c$)(2$ marks$)$ Use the Tyreus $-$ Liyben recommended settings and tume the PI controller.