Similar realizations.

$($a$)(3$ points$)$ Are the following two state$-$space models

$x^{}=\left[\begin{array}{ccc}2& 1& 2\\ 0& 2& 2\\ 0& 0& 1\end{array}\right]x+\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]u,y=\left[\begin{array}{ccc}1& -1& 0\end{array}\right]x$

and

$x^{}=\left[\begin{array}{ccc}2& 1& 1\\ 0& 2& 1\\ 0& 0& -1\end{array}\right]x+\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]u,y=\left[\begin{array}{ccc}1& -1& 0\end{array}\right]x$

equivalent when both $x(0)$ and $u(t)$ are non$-$zero? Are they equivalent when $x(0)=0?$

$($b$)(3$ points$)$ Verify that

$=\left[\begin{array}{cccc}-a_2& 1& 0& b_2\\ -a_1& 0& 1& b_1\\ -a_0& 0& 0& b_0\\ 1& 0& 0& d\end{array}\right]$

is similar to

${}^{**}=\left[\begin{array}{cccc}0& 0& -a_0& b_0\\ 1& 0& -a_1& b_1\\ 0& 1& -a_2& b_2\\ 0& 0& 1& d\end{array}\right]$