  The paper is devoted to the observability study of a dynamic system, which
describes the vibrations of an elastic beam with an attached rigid body and
distributed control actions. The mathematical model is derived using Hamilton's
principle in the form of the Euler-Bernoulli beam equation with hinged boundary
conditions and interface condition at the point of attachment of the rigid
body. It is assumed that the sensors distributed along the beam provide output
information about the deformation in neighborhoods of the specified points of
the beam. Based on the variational form of the equations of motion, the
spectral problem for defining the eigenfrequencies and eigenfunctions of the
beam oscillations is obtained. Some properties of the eigenvalues and
eigenfunctions of the spectral problem are investigated. Finite-dimensional
approximations of the dynamic equations are constructed in the linear manifold
spanned by the system of eigenfunctions. For these Galerkin approximations,
observability conditions for the control system with incomplete information
about the state are derived. An algorithm for observer design with an arbitrary
number of modal coordinates is proposed for the differential equation on a
finite-dimensional manifold. Based on a quadratic Lyapunov function with
respect to the coordinates of the finite-dimensional state vector, the
exponential convergence of the observer dynamics is proved. The proposed method
of constructing a dynamic observer makes it possible to estimate the full
system state by the output signals characterizing the motion of particular
point only. Numerical simulations illustrate the exponential decay of the norm
of solutions of the system of ordinary differential equations that describes
the observation error.
