A certain linear, time-invariant system has the state and output representation shown below:
\[
\left(\begin{array}{l}
\dot{x}_{1} \\
\dot{x}_{2}
\end{array}\right)=\left(\begin{array}{rr}
-2 & 1 \\
0 & -3
\end{array}\right)\left(\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right)+\left(\begin{array}{l}
1 \\
0
\end{array}\right) u
\]
\[
y=\left(\begin{array}{ll}
1 & 1
\end{array}\right)\left(\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right)
\]
- Find the eigenvalues (natural frequencies) of the system.
- If $u(t)=\delta(t)$ and $x_{1}\left(0_{+}\right)=x_{2}\left(0_{+}\right)=0$, find $x_{1}(t)$, $x_{2}(t)$ and $y(t)$, for $t>0$.
- When the input is zero, choose initial conditions $x_{1}\left(0_{+}\right)$and $x_{2}\left(0_{+}\right)$such that $y(t)=A e^{-2 t}$ for $t>0$.