壓電懸臂樑磁滯效應建模與強健控制器設計及實驗驗證

Dynamic Modeling on Hysteresis Effects in Piezoelectric Structures via Preisach Model and Subsequent Robust Control Designs

廖本諺(Pen-Yen Liao)

指導教授 ： 趙昌博

中原大學/工學院/機械工程研究所/碩士(2005年)

https://doi.org/10.6840/cycu200500671

摘要

本文主旨是對於考慮磁滯效應的壓電材料進行微觀建模，在此利用雙層壓電片懸臂樑做為建模應用，且同時以推導出的動態模型設計強健控制器，以消除磁滯效應在定位上造成的誤差。首先在壓電材料的微觀建模方面，利用壓電方程式（constitutive equations）中加入磁滯效應的極化項P（polarization term），並且以Preisach model 來描述磁滯效應。以此修正後的壓電方程式，透過有限元素法推導出壓電懸臂樑之運動方程式。因此可利用包含磁滯效應的運動方程式設計控制器，再以控制方式消除實際磁滯對於系統定位不佳的影響。在本文控制分為兩部分，第一是消除在運動方程式中包含之磁滯項次，第二利用強健控制理論，考慮Preisach model建模的微小誤差值，即以實際磁滯與建模後磁滯之間誤差值考慮為系統之不確定項，再用強健控制設計步驟設計出控制器。同時設計一PI雙級領先控制器，比較強健控制器對於消除磁滯效應的能力。最後，透過模擬與實驗結果，顯示經由Preisach model結合有限元素法能有效建立壓電材料的磁滯效應之微觀模型。

關鍵字

磁滯 ； Preisach model ； 強健控制

並列摘要

This study is aimed to conduct the microscopic modeling of hysteresis effects in the piezoelectric materials via finite elements. A robust controller is subsequently designed to demonstrate the effectiveness of the microscopic modeling in terms of minimizing positioning error for a simple bimorph piezoelectric beam system. To perform the microscopic modeling on hysteresis, the constitutive equations of a general piezoelectric material are modified to include the hysteresis effect by adding a polarization term in one of the constitutive equations. The well-known Perisach model is employed to prescribe the hysteresis effects through polarization. The equations of motion of the piezoelectric beam are next derived through the utilization of the finite element method based on modified constitutive equations. With the system equations in hand, a control is designed with the two components: the first one is forged to cancel the micro-hysteresis term and the second one is synthesized via the theory of robust H∞ control design in the aim of overcoming unavoidable small level of modeling error by the Preisach model. In the process of the robust H∞ control design, the difference between the predicted hystersis by the Preisach model and the realistic measurements is considered as the model uncertainty for later control design. Finally, PI and double lead compensators are also designed via conventional processes to provide the basis that demonstrates the superiority of the previously-designed controller. Both simulations and experimental results show the effectiveness and efficiency of the microscopic hysteresis model established via the Preisach modeling.

並列關鍵字

Preisach model ； hysteresis ； robust H∞ control

參考文獻

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