PROBLEM: Design a linear state$-$feedback controller to yield $20\%$ overshoot and a settling time of $2$ seconds for a plant

$G(s)=\frac{(s+6)}{(s+9)(s+8)(s+7)}$

that is represented in state space in cascade form by

$z^{}=Az+Bu=\left[\begin{array}{ccccccc}-7& 1& 0& -8& 1& 0& -9\end{array}\right]$

$0$

$0$

$0z+\left[\begin{array}{c}0\end{array}\right]$

$0$

$1u$

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

ANSWER: $K_z=\left[\begin{array}{ccc}-40\mathrm{.}23& 62\mathrm{.}24& -14\end{array}\right]$