Problem #$2$: To convert the continuous time system to a discrete time system, a ZOH $($zero order hold$)$ and a sampler are introduced. By setting the sampling period $T=0\mathrm{.}005$ second, develop the transfer function of the discrete time closed loop system.

You should:

$(2)$ Provide a detailed mathematical derivation of the transfer function of the discrete time closed loop system shown above.

$(3)$ Plot the output response when the input response is the sampled sinusoidal signal $x(t)=sin(t),$ where $=\frac{40}{13}ra\frac{d}{?}$ second, and the sampling period $T=0\mathrm{.}005$ second

$(4)$ Compare the output response signal with the input signal, do they match?