臺灣博碩士論文加值系統

本文主旨針對單片壓電懸臂樑（Piezoelectric Cantilever Beam）設計精密定位控制器。系統之動態模型由兩種方法獲得，第一種為考慮壓電懸臂樑本身之物理行為與壓電方程式，利用漢米頓原理（Hamilton’s principle）推導出完整的統御方程式，並利用有限元素法得到系統之轉移函數；另一方法為利用實驗鑑別出系統之轉移函數，比較並分析兩模型的精確性。並以壓電懸臂樑作為光碟機之致動器為例，利用輸入電壓控制使懸臂樑末端的位移達到所要求的高度。控制器設計方面，利用古典控制的 加上落後領先補償器作初步的設計，進而考慮光碟機因碟片旋轉時所產生的振動以及對於系統模型的不確定性設計具有強健性之H∞ 定位控制器。最後利用dSPACE即時控制之人機介面，將所設計之控制器以實作驗證。

This study performs precision position control of the piezoelectric cantilever beam with consideration of nonlinearity. Two different dynamic models were obtained. The first one is derived in the forms of linear systems through application of basic physic laws of piezoelectric material, Hamilton’s principle and modeling technique of finite elements. The second one is established as in through experimentally-obtained frequency responses. With theoretical models in hand, the controllers aimed to perform precision positioning of the piezoelectric cantilever beam are next designed to work for a pickup actuator in optical disc drives, which ought to suppress the vibratory disturbance caused by the rotation of the disc. Two types of controllers, PI-and-lag-lead compensator and H∞ controller, are synthesized herein to perform the precision positioning due to the simple structure of the PI-and-lag-lead one and the robustness achieved by the H∞ one. Note that the H∞ controller designed herein would able to work against plant uncertainty, sensor noise and the extraneous disturbance caused by the eccentric rotation of the disk. Simulations are performed to validate the performance expected by previously-designed controllers. Finally, experiments are conducted to verify the effectiveness foreseen by simulations.

摘要 Ⅰ
Abstract Ⅱ
誌謝 Ⅲ
Contents Ⅳ
Figure Captions Ⅶ
Table Title Ⅸ
Nomenclature Ⅹ
一、簡介 1
二、數學模型 3
三、控制器設計 9
四、數值模擬結果 17
五、實驗結果 20
六、結論與未來工作 22
1. Introduction 24
2. Modeling 26
2.1 Theoretical Analysis 26
2.1.1 Equations of Motion 26
2.1.2 Finite Element Method 30
2.2 System Identification 31
2.3 Discussion on various models 32
3. Control Design 34
3.1 PI-and-Lag-Lead Compensator 34
3.2 H∞ Controller 38
3.2.1 Design Structure for H∞ Control 38
3.2.2 Weighting Function 39
3.2.3 Modeling of Uncertainty 41
3.2.4 Representation of system in Linear Fractional Transformation 42
3.2.5 Design of H∞ Control 43
3.2.6 Examination on Robust Performance and Robust Stability 44
4. Numerical Results 45
4.1 PI-and- Lag-Lead Compensator 45
4.2 H∞ controller 46
4.2.1 Without Considering Plant Uncertainty 46
4.2.2 With Considering Plant Uncertainty 46
5. Experiment Validation 47
6. Conclusions and Future Works 51
Reference 53
Figure 56
Table 87
Appendix 88
簡歷 92

Figure Captions
Fig. 1 Schematic diagram of the cantilever beam with one piezoelectric layer bounded on the top and the input voltage V is applied across the thickness.
Fig. 2 (a) Experiment devices for system identification. (b) Schematic of experiment for system identification.
Fig. 3 Frequency response in bode plot of the piezoelectric cantilever beam by system identification.
Fig. 4 Frequency responses of the real model and identified model.
Fig. 5 Frequency responses of finite element model, real model and identified model.
Fig. 6 Frequency responses of real model, finite element model and finite element model with hysteresis.
Fig. 7 Bode plot of uncompensated system.
Fig. 8 Bode plot of -and-lag compensator.
Fig. 9 Bode plot of compensated system.
Fig. 10 control design structure.
Fig. 11 (a) Performance weight and desired ; (b) Control weight and desired .
Fig. 12 The block diagram of the control system with considering the plant uncertainty.
Fig. 13 and the bond .
Fig. 14 (a) Closed-loop interconnection of control without considering plant uncertainty; (b) Closed-loop interconnection of control with considering plant uncertainty.
Fig. 15 (a) LFT framework of control without considering plant uncertainty; (b) LFT framework of control with considering plant uncertainty.
Fig. 16 Step response of the -and-lag-lead compensator with P.M. 50.7deg.
Fig. 17 Bode plot of compensated system.
Fig. 18 Step response of the -and-lag-lead compensator with P.M. 24.4deg.
Fig. 19 Step response of the control without considering plant uncertainty.
Fig. 20 Step response of the control without considering plant uncertainty but with a sine-wave disturbance.
Fig. 21 Step response of the control with considering plant uncertainty.
Fig. 22 (a) Robust performance; (b) Robust performance; (c) Robust stability.
Fig. 23 Step response of control with considering plant uncertainty and a sine-wave disturbance.
Fig. 24 (a) Experiment loop and devices; (b) SIMULINK block diagram of the control loop; (c) Experimental piezoelectric cantilever beam.
Fig. 25 Experimental and numerical results of the -and-lag-lead compensator.
Fig. 26 Experimental and numerical results of the control without considering plant uncertainty.
Fig. 27 Experimental and numerical results of the control with considering plant uncertainty.
Fig. 28 Experimental and numerical results of the -and-lag-lead compensator with a sine-wave disturbance.
Fig. 29 Experimental and numerical results of the control without considering plant uncertainty and a sine-wave disturbance.
Fig. 30 Experimental and numerical results of the control with considering plant uncertainty and a sine-wave disturbance.
Fig. 31 The creep effect of the piezoelectric cantilever beam.

TABLE TITLES
Table 1 Material properties and geometric dimensions of the piezoelectric cantilever beam.

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