As discussed in class, equivalent state$-$space representation can be represented in several

forms, and are, in general, not unique. Show that the systems $($a$)-($c$)$ will result in the same

transfer function.

$($a$)$

$x^{}=-5x+3u$

$y=7x$

$($b$)$

$\left[\begin{array}{c}{x}_1^{}\end{array}\right]$

$x_2^{}=\left[\begin{array}{ccc}-5& 0& -1\end{array}\right]$

$0\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2+\left[\begin{array}{c}3\end{array}\right]$

$1u$

$y=\left[\begin{array}{cc}7& 0\end{array}\right]\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2$

$($c$)$

$\left[\begin{array}{c}{x}_1^{}\end{array}\right]$

$x_2^{}=\left[\begin{array}{ccc}-5& 0& -1\end{array}\right]$

$0\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2+\left[\begin{array}{c}3\end{array}\right]$

$0u$

$y=\left[\begin{array}{cc}7& 3\end{array}\right]\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2$