Q$1.($a$)$ What is derivative kick? Is there any way to overcome this problem in the feedback controller?

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$[2+2+2+(2+2+2)=12]$

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$($b$)$ Find the amplitude ratio $($AR$)$ and phase angle $\backslash ((\backslash$phi$)\backslash )$ for the PID controller.

$($c$)$ What are the major limitations of Ziegler$-$Nichols tuning method?

$($d$)$ Consider a closed$-$loop process shown in Fig. $1,$ in which a PID controller is used to control the system.

Fig. $1.$ A closed$-$loop process with PID controller.

$($i$)$ Find the ultimate gain $(\backslash ($ K$\_\{$u$\}\backslash ))$ and ultimate period $(\backslash ($ P$\_\{$u$\}\backslash )).$

$($ii$)$ Use the Tyreus $-$ Luyben recommended settings and tune the PID controller.

$($iii$)$ Find the pole$($s$)$ and zero$($s$)$ of the PID controller.