EXERCISE:

$1)$ 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

$2)$ Using the diagram in Figure $1,$

a$)$ Using MATLAB, $\backslash ($ G$\_\{1\}($s$)=\backslash$frac$\{$K$\}\{$s$($s$+1)\}\backslash )$ sampling period of $\backslash ($ T$=0\mathrm{.}1\backslash )$ seconds, derive the loop transfer function $\backslash ($ G$($z$)\backslash ).$

b$)$ Using the bilinear transform of $\backslash ($ z $\backslash )$ to help you, determine the range of values for $\backslash ($ K $\backslash )$ for stability.

c$)$ Using MATLAB and critical gain $($i$.$e$.\backslash ($ K$=$K$\_\{$c r$\}\backslash )),$ derive the closed loop transfer function

d$)$ Repeat part a$)$ when $\backslash ($ T$=1\backslash )$ seconds.

e$)$ Using the bilinear transform of $\backslash ($ z $\backslash )$ to help you, determine the range of values for $\backslash ($ K $\backslash )$ for stability. NB$.$ Use the closed loop transfer function produced in part e$)$