$1.$ Consider the following closed loop system where the plant has a delay $3$ second. $(50$ points$)$

$-$ Design PI controllers for different values of sampling times $\backslash ($ T$=1,$ T$=2\backslash )$ and $\backslash ($ T$=3\backslash ),$ such that dominant closed$-$loop poles have a damping ration $\backslash (\backslash$zeta$=0\mathrm{.}5\backslash )$ and the number of samples per cycle of damped sinusoidal oscillation is $10.$

$-$ Obtain the sample values of the step response of the system and also find the steady state error of the unite ramp input.

$-$ Study the effect of the sampling time on the magnitude of the control input $\backslash ($ u$($t$)\backslash ).$

$-$ Simulate the designed controllers in MATLAB