Problem $2$: Consider two different implementations $($Systems B and C$)$ for System A below where $K$ is positive and $T_N=1$ :

Derive the $z-$transform of the plant, $G(z),$ for the two different types of implementations:

a$)$ System B without a zero$-$order$-$hold $(ZOH),Z[G_p((s))].$

b$)$ System C with a ZOH,$Z[G_{ho}(s)G_p(s)].$

Analyze the stability conditions of the closed$-$loop system $($without and with ZOH $)$ where $D(z)=K$ in terms of $T$ and $K.$

Assume $G(z)=\frac{0\mathrm{.}3678(z+0\mathrm{.}7189)}{(z-1)(z-0\mathrm{.}3678)},$ derive the closed$-$loop transfer function $Y\frac{z}{R}(z)$ and compare the unit$-$step response of two different controlled systems using MATLAB: $D(z)=1$ and $D(z)=\frac{1\mathrm{.}359(z-0\mathrm{.}3678)}{z+0\mathrm{.}24}.$

Given $G_p(s)=\frac{1}{s^2},$ compare the root locus for two different controlled systems; continuous$-$time and discrete$-$time with a ZOH and $T=\sqrt[\phantom{2}]{2}$ second.