The transfer function for a SISO system is given by

$\frac{x(s)}{U(s)}=\frac{5(s+4)}{s^2+6s+25}$

write the equations in state$-$space form. These equations should be in the form:

$x^{}=Ax+Bu$

$y=Cx+Du$

For the system shown in state$-$space form

$x^{}=\left[\begin{array}{ccccccc}-2& 3& 0& -2& 0& 0& -1\end{array}\right]$

$0$

$0$

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

$1$

$1u$

$y=\left[\begin{array}{ccccccc}1& 0& 0& 1& 0& 1& 0\end{array}\right]$

$0$

$-1x$

determine:

a$.$ whether the system is state controllable.

b$.$ whether the system is output controllable.

$3.$ For the system shown in state$-$space form

$x^{}=\left[\begin{array}{cc}-3& 2\\ 0& -1\end{array}\right]x+\left[\begin{array}{c}-1\\ 1\end{array}\right]u$

determine the feedback gain $K=\left[\begin{array}{cc}{k}_1& k_2\end{array}\right]$ where $u=-Kx$ such that the system has closed$-$loop poles at $s=-5,-6.$