Problem $1(30+5pt)$ Given the follow system

$z^{}=\left[\begin{array}{ccc}0& -83\mathrm{.}33& -10\end{array}\right]$

$500z+\left[\begin{array}{c}166\mathrm{.}67\end{array}\right]$

$0u(t)$

$y=\left[\begin{array}{cc}0& 1\end{array}\right]z$

with the corresponding transfer function

$G(s)=\frac{83335}{s^2+10s+41665}$

$($a$)$ Express the system's state equations in phase$-$variable form, $x^{}=Ax+Bu$ and $y=Cx.(10$ pts$)$

$($b$)$ Find a set of state$-$feedback gains to obtain $4\mathrm{.}5\%$ overshoot and a settling time of $0\mathrm{.}5$ second in the phase$-$variable system. $($Round off the result to $2$ decimal places.$)(15$ pts $)$

$($c$)$ The original system $(z)$ and the phase$-$variable form system $(x)$ can be linked by a matrix $P$ where $z=Px.$ Find $P$ and verify Eq$.(4)$ by using $P{P}^{-1}=$I. $(5$ pts$)$

$P^{-1}=\left[\begin{array}{cc}0& 1\mathrm{.}2{10}^{-5}\\ 6\mathrm{.}0{10}^{-3}& -1\mathrm{.}2{10}^{-4}\end{array}\right]$

$($d$)$ Obtain the corresponding set of state$-$feedback gains in the original system. $($Round off the result to $3$ decimal places.$)(+5$ pts$)$