The model of a continuous-time system with algebraic loops is given as

\[
\begin{aligned}
\dot{x}_{1}(t) & =-x_{1}(t)+2 \dot{x}_{2}(t)+u_{1}(t) \\
\dot{x}_{2}(t) & =-\dot{x}_{2}(t)-x_{2}(t)-\dot{x}_{1}(t)+x_{1}(t)+u_{2}(t) \\
\dot{x}_{3}(t) & =-\dot{x}_{3}(t)-x_{3}(t)-\dot{x}_{2}(t)+x_{1}(t)+u_{2}(t) \\
y(t) & =\dot{x}_{3}(t)
\end{aligned}
\]

Derive the state equations for this system. Use the matrix technique of Section 4.9.

Verify the results by solving the given equations for \(\dot{x}_{1}(t), \dot{x}_{2}(t)\), and \(\dot{x}_{3}(t)\) in the standard state-variable format.