From the main problem, the dynamic equations in state$-$space form are the following where $Y1=x1-x2.$

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

$x_1^{}$

$Y_1^{}$

$Y_1^{}=\left[\begin{array}{ccccccccccccc}0& 1& 0& 0& 0& \frac{b_n}{M_1}(\frac{b_n}{M_1}+\frac{b_n}{M_2}+\frac{b_2}{M_2})-\frac{K_1}{M_1}& \frac{-b_1}{M_1}& 0& -(\frac{b_1}{M_1}+\frac{b_1}{M_2}+\frac{b_2}{M_2})& 1& 0& -(\frac{K_1}{M_1}+\frac{K_1}{M_2}+\frac{K_2}{M_2})& 0\end{array}\right]$

$\frac{K_2}{M_2}$

$\frac{b_n}{M_2}$

$\frac{-b_b}{M_1{M}_2}\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_1^{}$

$Y_1$

$Y_1^{}+$

$\left[\begin{array}{ccccc}0& 0& \frac{b,b_2}{M_1{W}_2}& \frac{b_1}{M_2}& \frac{-K_2}{M_2}\end{array}\right]$

$(\frac{1}{M_2}+\frac{1}{M_2})$

$0$

$\frac{1}{M_1}\left[\begin{array}{c}U\end{array}\right]$

$W$

$Y=\left[\begin{array}{cccc}0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_1^{}$

$Y_1$

$Y_1^{}+\left[\begin{array}{cc}0& 0\end{array}\right]\left[\begin{array}{c}U\end{array}\right]$

$W$

$6-$ Is the open$-$loop system stable? Justify your answer.

$7-$ Check the system controllability and observability of the state space model derived in part $(5)($Given in the photo$).$ Comment on the results.