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The paper is devoted to the observability study of a dynamic system, whichdescribes the vibrations of an elastic beam with an attached rigid body anddistributed control actions. The mathematical model is derived using Hamilton'sprinciple in the form of the Euler-Bernoulli beam equation with hinged boundaryconditions and interface condition at the point of attachment of the rigidbody. It is assumed that the sensors distributed along the beam provide outputinformation about the deformation in neighborhoods of the specified points ofthe beam. Based on the variational form of the equations of motion, thespectral problem for defining the eigenfrequencies and eigenfunctions of thebeam oscillations is obtained. Some properties of the eigenvalues andeigenfunctions of the spectral problem are investigated. Finite-dimensionalapproximations of the dynamic equations are constructed in the linear manifoldspanned by the system of eigenfunctions. For these Galerkin approximations,observability conditions for the control system with incomplete informationabout the state are derived. An algorithm for observer design with an arbitrarynumber of modal coordinates is proposed for the differential equation on afinite-dimensional manifold. Based on a quadratic Lyapunov function withrespect to the coordinates of the finite-dimensional state vector, theexponential convergence of the observer dynamics is proved. The proposed methodof constructing a dynamic observer makes it possible to estimate the fullsystem state by the output signals characterizing the motion of particularpoint only. Numerical simulations illustrate the exponential decay of the normof solutions of the system of ordinary differential equations that describesthe observation error.

Observer Design for a Flexible Structure with Distributed and Point Sensors