$2.$ Consider the following closed loop system where the plant has a delay $4$ second. $(50$ points$)$

$-$ Design digital controllers for different values of sampling times $\backslash ($ T$=1,$ T$=2\backslash )$ and $\backslash ($ T$=4\backslash ),$ such that the step response of the closed loop system has no overshoot and reaches to its steady state value at $\backslash ($ t$=12\backslash )$ and remain at the constant value with no inter sampling ripples.

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

$-$ Simulate the designed controllers in MATLAB