Obtain a state-space representation for a system whose differential equation is given by

\[\dddot{x}+3 \ddot{x}+3 \dot{x}+x=\dot{u}+u \text {, }\]

where the output is $y=x$.

(a) Use this result to determine the system transition matrix $\varphi \left(t\right)$ and $\varphi \left(s\right)$.

(b) Use Ackermann's formula to determine the controller K that places the roots of this system at $-1,-2\pm 2j$.