Problem $5.(40$ pts$)$

Given the plant

$\backslash [$

$\backslash$mathrm$\{$G$\}($s$)=\backslash$frac$\{8($s$+6)\}\{$s$($s$+1)($s$+3)\}$

$\backslash ]$

Design the state feedback gains to yield approximately a $10\backslash \%$ overshoot and a settling time of $0\mathrm{.}8$ second.

$(1)$ Please draw the allowable region for the closed$-$loop poles on the s$-$plane using the transient response requirement. In addition, give the desired closed$-$loop poles values based on this region. $(10$pts$)$

$(2)$ Write the state$-$space equations of $\backslash (\backslash$mathrm$\{$G$\}(\backslash$mathrm$\{$S$\})\backslash )$ in controllable canonical from. $(10$pts$)$

$(3)$ Find the state feedback gains $\backslash ($ K $\backslash )$ if $\backslash ($ u$($t$)=-$K B $\backslash )$ such that the closed$-$loop poles are on the locations you specified in $(1).(10$pts$)$

$(4)$ What is the steady state error of the designed state feedback system to an unit step input? $(10$pts$)$

the question is just that no more additional information