Stabilization Results for Well-Posed Potential Formulations of a Current-Controlled Piezoelectric Beam and Their Approximations

Abstract

Hysteresis is highly undesired for the vibration control of piezoelectric beams especially in high-precision applications. Current-controlled piezoelectric beams cope with hysteresis substantially in comparison to the voltage-controlled counterparts. However, the existing low fidelity current-controlled beam models are finite dimensional, and they are either heuristic or mathematically over-simplified differential equations. In this paper, novel infinite-dimensional models, by a thorough variational approach, are introduced to describe vibrations on a piezoelectric beam. Electro-magnetic effects due to Maxwell’s equations factor in the models via the electric and magnetic potentials. Both models are written in the standard state-space formulation (A, B, C), and are shown to be well-posed in the energy space by fixing the so-called Coulomb and Lorenz gauges. Different from the voltage-actuated counterparts, the control operator B is compact in the energy space, i.e. the exponential stabilizability is not possible. Considering the compact $$C=B^*-$$type state feedback controller (induced voltage), both models fail to be asymptotically stable if the material parameters satisfy certain conditions. To achieve at least asymptotic stability, we propose an additional controller. Finally, the stabilizability of infinite-dimensional electrostatic/quasi-static model (no magnetic effects) is analyzed for comparison. The biggest contrast is that the asymptotic stability is achieved by an electro-mechanical state feedback controller for all material parameters. Our findings are simulated by the filtered semi-discrete Finite Difference Method.