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.

中文翻译：

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电流控制压电梁的适当位置势公式的稳定化结果及其近似值

摘要

对于压电梁的振动控制，特别是在高精度应用中，磁滞非常不理想。与电压控制的对应物相比，电流控制的压电束基本上可以应付滞后现象。但是，现有的低保真度电流控制射束模型是有限维的，它们要么是启发式方法，要么是数学上过于简化的微分方程。在本文中，通过彻底的变分方法，介绍了新颖的无穷维模型来描述压电梁上的振动。麦克斯韦方程组产生的电磁效应通过电势和磁势在模型中进行分解。两种模型均以标准状态空间公式（A，  B，  C），并通过固定所谓的Coulomb和Lorenz量规显示出在能量空间中的适当位置。与电压致动的对应物不同，控制操作员B在能量空间中紧凑，即不可能实现指数稳定。考虑紧凑\（C = B ^ *-\）类型的状态反馈控制器（感应电压），如果材料参数满足某些条件，则两个模型都无法渐近稳定。为了至少达到渐近稳定性，我们提出了一个附加控制器。最后，分析了无限维静电/准静态模型（无磁效应）的稳定性，以进行比较。最大的对比是，对于所有材料参数，通过机电状态反馈控制器可实现渐近稳定性。我们的发现是通过滤波的半离散有限差分法进行模拟的。