A linear time-invariant system is described by the differential equation

\[\frac{d^{3} y(t)}{d t^{3}}+3 \frac{d^{2} y(t)}{d t^{2}}+3 \frac{d y(t)}{d t}+y(t)=r(t)\]

(a) Let the state variables be defined as \(x_{1}=y, x_{2}=d y / d t, x_{3}=d^{2} y / d t^{2}\). Write the state equations of the system in vector-matrix form.

(b) Find the State-Transition Matrix \(\phi(t)\) of \(\mathbf{A}\).

(c) Let \(y(0)=1, \quad d y(0) / d t=0, \quad d^{2} y(0) / d t^{2}=0\), and \(r(t)=u_{s}(t)\). Find the state transition equation of the system.

(d) Find the characteristic equation and the eigenvalues of the system.