The following systems are targeted.

$x^{}(t)=Ax(t)+bu(t),y(t)=cx(t)$

$x(t)=\left[\begin{array}{c}{x}_1(t)\end{array}\right]$

$x_2(t),A=\left[\begin{array}{ccc}-5& 6& 0\end{array}\right]$

$-1,b=\left[\begin{array}{c}1\end{array}\right]$

$0,c=\left[\begin{array}{cc}4& 7\end{array}\right]$

Determine the stability of the system $($state reasons for stability$/$unstability$).$

Determine the controllability and observability of the system $($give reasons for your determination$).$

Find the transfer function from input $u(t)$ to output $y(t).$

We want to design a state feedback control $u(t)=-fx(t)+r(t).$ Here, $r(t)$ is the target value $($it is fine to assume it is $0).$ Find the feedback gain $f=[f_1{f}_2]$ that sets the eigenvalues $($poles$)$ of the closed loop system to $-6$ and $-7.$ Also, confirm that the poles of the closed loop system are the specified poles.

$($Continuation of question $4)$ Find the transfer function from the target value $r(t)$ to the output $y(t).$ Also, consider the difference with the transfer function found in question $3($the transfer function of the controlled object before control is implemented$).$

I want to design the following observer. Find the observer gain $h=[h1h2]$ that makes the observer's eigenvalues $($poles$)-8,-9.$ Also, make sure that the observer poles are the specified poles $($Hint: The observer gain $h=[h_{1}h2]$ is not an integer$).$

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$(t))$

$((t$

do all parts specialspecialty $4,5,6$