You're right, it's a second order plant representing the mass-spring-damper system $P\left(s\right)=1/\left(s^2-bs -k\right)$ (stiffness $k$ and damping $b$ should be thus negative) under the closed-loop control action of the compensator $C\left(s\right)$, whose transfer function is:
The negative feedback is closed over the position of the mass, while the set-point is given by $u$.
We can easily derive these results by applying the Laplace transform to the system of equations and then substituting $x_1$:
sx_3&=\alpha (u-x_2)-\beta x_3
from which we obtain the following transfer functions:
corresponding to the closed-loop system here below: