Q$2.$ Consider the system

$x^{}=Ax+Bu$

$y=Cx$

where

$($a$)A=\left[\begin{array}{ccc}0& 2& 0\\ 1& 2& 0\\ -1& 0& 1\end{array}\right],B=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right],C=\left[\begin{array}{ccc}1& 0& 1\end{array}\right]$

$($b$)A=\left[\begin{array}{ccc}-6& 1& 0\\ -9& 0& 1\\ 0& 0& 0\end{array}\right],B=\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right],C=\left[\begin{array}{ccc}3& -2& 1\end{array}\right]$

$($c$)A=\left[\begin{array}{cccc}-5& \frac{1}{6}& -\frac{1}{6}& 0\\ -\frac{1}{2}& -\frac{16}{3}& \frac{1}{3}& \frac{1}{2}\\ -\frac{1}{2}& -\frac{1}{3}& -\frac{14}{3}& \frac{1}{2}\\ 0& -\frac{1}{6}& \frac{1}{6}& -5\end{array}\right]$

Find the transformation $x=P\stackrel{}{x}$ that transform the state equations into the DCF or JCF$.$ Give the

transformed state equations. $($In $($c$),$ you only need to find $\frac{?}{\text{b}\text{a}\text{r}}(A)).$