Test $2(25$ marks$)$

A state$-$space r representation of a system in the controllable canonical form is given by:

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

$x_2^{}=\left[\begin{array}{ccc}0& 1& -1\mathrm{.}3\end{array}\right]$

$-0\mathrm{.}4\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2+\left[\begin{array}{c}0\end{array}\right]$

$1[u]$

$y=\left[\begin{array}{cc}0\mathrm{.}8& 1\end{array}\right]\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2$

The same system may be represented by the following state$-$space equation, which is in the observable canonical form:

$\left[\begin{array}{c}{x}_1^{}\\ x_2^{}\end{array}\right]=\left[\begin{array}{cc}0& -0\mathrm{.}4\\ 1& -1\mathrm{.}3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\mathrm{.}8\\ 1\end{array}\right][u]$

$y=\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

Show that the state$-$space representation given in Eq$.(1$ and $2)$ give a system that is state controllable but not observable, and the state$-$space representation given in Eq$.(3$ and $4)$ give a system not completely state controllable but observable. Explain what causes the apparent difference in the controllability and observability of the same system.