$1)$ Using the diagram in Figure $1,$ study the stability of the closed$-$loop system where the values $\backslash ($ G$\_\{1\}($s$)=\backslash$frac$\{1\}\{$s$+2\}\backslash )$ and sampling period $\backslash ($ T$=0\mathrm{.}5\backslash )$ seconds. Note, by study, the following exercise should be carried out:

a$)$ Manually determine the closed loop transfer function in the Z domain and determine stability.

b$)$ Using MATLAB programming determine the closed loop transfer function in the Z domain and determine stability.

c$)$ Check the stability with respect to the sampling period $\backslash ($ T $\backslash ).$

d$)$ Routh$-$Hurwitz criterion

Figure $1-$ Closed Loop System