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Modeling with uncertainty and robust control of smart beams

Pouliezos Anastasios, Stavroulakis Georgios, Μουτσοπούλου Αμαλία

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A mathematical formulation and finite element model for the vibration suppression of laminated smart beams with embedded piezoelectric material is considered in this paper. Cubic and quadratic Lagrangian polynomials are used for the transverse and rotational displacements, respectively, and the differential equations of the beam are based on the Timoshenko beam theory.The design of the piezoelectric active control using LQG and Hinfinity control theory has been studied. The fact that the system is influenced by disturbances, such as the external loading which may be caused from a wind power or the unavoidable noise of measurements or even some damage of the structure, is taken into account. At the beginning the classical optimal linear quadratic regulator is used. In order to take into account our incomplete information about the damage and the external disturbances the theory of a robust Hinfinity feedback controller is also used. The following three steps are followed in the robustness analysis:Expression of the uncertainty set by a suitable mathematical model.Robust stability (RS): check if the system remains stable for all plants within the uncertainty set.Robust performance (RP): if the system is robustly stable, check whether the performance specifications are met for all plants within the uncertainty set.After the analysis of the system we check the robust stability and performance of the system. We encounter the solution to robust stability but not to the problem of robust performance, event though the criteria of nominal performance is very satisfactory and the beam keeps in equilibrium with controls within limits. The Hinfinity feedback controller has been determined by using a nonconvex, nondifferentiable optimization approach with the use of HIFOO software within MATLAB.The numerical results show that the proposed method is effective and that the control behavior of the beam permits us to keep the structure in service up to the given limits of uncertainties. The control behavior of the beam achieves the predicted characteristics. Simulations show that the control strategy produces very good results by reducing the disturbances within the limits of the piezoelectric effect's resistance.

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