Problem $3(20$ pts$)$:

The following EOMs describe the behavior of a dynamic system.

$z^{}+z+v_b=u(t)$

${}^{}+{}^{}+-z=$

The equations are coupled by the fact $=z^{}$ and $v_b=-3.$

A$.$ Write the state space form of the system as $x^{}=Ax+Bu$ by substituting the coupled terms into the EOMs. Clearly identify the matrices. You can take the state variables as $x_1=z,x_2=,$ and $x_3={}^{}.$ Hint: You may have to substitute from one EOM to the other to get things in the right form. $(7$ pts $)$

B$.$ Assume the desired output of the system is the vector $y$ that consists of $z$ and $$ as individual elements. Write the output equation in the form of $y=Cx+Du.$ Clearly identify the matrices. $(6$ pts $)$

C$.$ Find all eigenvalues and corresponding eigenvectors for the matrix A$.(7$ pts$)$