This work deals with the LQG control problem of a flexible beam clamped at one end and with a point mass at the free end, where a boundary control force can be applied. A class of finite-dimensional control laws is proposed here, derived on the basis of the Euler-Bernoulli infinite-dimensional beam model. By means of this approach it is possible to take into account also the higher order modes that are indeed neglected in the more usual methods based on a finite-dimensional model of the beam. The main motivation for the approach followed here is that it naturally allows to overcome the phenomenon of spillover, occurring when unmodeled modes are excited by the control law itself. The finite-dimensional control law here proposed is derived by a Galerkin approximation of the solution of the LQG control problem, in a proper Hilbert space setting. In particular, the novelty of the approach is the definition of an implementable Galerkin approximation scheme based on generalized eigenfunctions of the Euler-Bernoulli model instead of the usual splines. It is here proved that, for any given finite time horizon, the evolution of the system state driven by the proposed control input converges, in $L_2$ norm, to the optimal LQG evolution, as the order of the approximation scheme increases. The strong stability of the closed loop system is guaranteed for any order of the approximation scheme. Moreover, it is proved that the proposed compensator guarantees modal stability of the closed loop system also in the presence of stiffness/inertia parameters uncertainties.
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