本論文研究撓性臂的強韌控制之設計。首先採用有限元素方法 架構出系統的常微分模型，接著 再應用H-inf 控制理論來設計控制器，並由電腦模擬顯示所設計的H-inf 控制器確實使得系統具有良好的定位控制、降低外在擾動與強韌穩定性。

更進一步地，對於撓性臂的控制，再將一維系統H-inf 控制推廣到二維系統H-inf 控制。藉著對系統模型時間與空間予以離散化，建立出受控系統的二維Roesser數學模型。並探討關於二維系統 H-inf 控制的應用，再以線性矩陣不等式(LMIs)的運算方法，採用反覆疊代的方式來計算求解二維系統H-inf 控制器。

The purpose of this paper is to study the design of robust control for a flexible beam. First, the finite element method (FEM) is used to formulate the ordinary differential equations (ODEs) of the dynamic system of the beam model, and a controller is designed via the H-inf control theory. From the computer simulation results, the following conclusions could be drawn.
1. The developed controller is capable of precision positioning and tolerating external disturbance.
2. The robust stability for the system is assured.

Furthermore, robust control of linear one-dimensional systems is extended here to two-dimensional systems. The finite difference method (FDM) is applied to partial differential equations (PDEs) to obtain the so-called Roesser discrete state-space model. We discussed the application of H∞ control for two-dimensional systems and the design of H-inf controller is formulated into a convex optimization problem characterized by linear matrix inequalities (LMIs). Finally, the solutions of the LMIs are then used to construct a two-dimensional H∞ controller.

文目錄..………………………………………………………………I
符號表…………………………………………………………………iv
圖表索引………………………………………………………………vi
論文摘要(中文)………………………………………………………ix
論文摘要(英文)………………………………………………………x
第1章 緒論
1.1 前言……………………………………………………1-1
1.2 研究動機………………………………………………1-2
1.3 文獻回顧………………………………………………1-3
1.4 內容架構………………………………………………1-6
第2章 系統架構與分析
2.1 撓性臂與壓電致動器系統……………………………2-2
2.1.1 PZT壓電材料之簡介……………………………2-2
2.1.2 系統結構…………………………………………2-7
2.2 有限元素法……………………………………………2-9
2.2.1 形狀函數與運算矩陣的定義……………………2-9
2.2.2 動態分析狀態空間表示法………………………2-12
2.3 有限元素法模擬與分析………………………………2-14
2.3.1 數值模擬…………………………………………2-16
第3章 強韌控制
3.1 基本定理……………………………………………3-2
3.1.1 奇異值…………………………………………3-2
3.1.2 範數…………………………………………3-2
3.1.3 線性分式轉換…………………………………3-3
3.1.4 適定與內部穩定……………………………3-4
3.1.5 小增益理論…………………………………3-5

3.2 強韌控制的基本應用……………………………3-6
3.2.1 外在干擾抑制…………………………………3-7
3.2.2 強韌穩定性……………………………………3-9
3.3 控制器的求解方法………………………………3-11
第4章 控制器設計與模擬
4.1 控制器設計…………………………………………4-1
4.1.1 命令追隨與參考函數成型……………………4-1
4.1.2 輸出項與輸入項的選取與量測………………4-4
4.1.3 參考函數與權重函數的選取…………………4-5
4.2 數值模擬……………………………………………4-11
4.3 結論…………………………………………………4-20
第5章 二維模型架構
5.1 二維系統模型的簡介………………………………5-1
5.1.1 二維模型的描述………………………………5-2
5.1.2 各模型之間的關聯……………………………5-3
5.2 系統的轉移函數……………………………………5-5
5.2.1 數學表示式與無因次化………………………5-6
5.2.2 有限差分法與z轉換…………………………5-7
5.3 二維系統Roesser模型的實現………………………5-11
5.3.1 Roesser模式的實現…………………………5-11
第6章 二維系統 強韌控制
6.1 線性矩陣不等式(LMI)簡介………………………6-1
6.1.1 線性矩陣不等式的表示式…………………6-2
6.1.2 求解線性矩陣不等式算法…………………6-4
6.2 Roesser模型系統的穩定度與實界引理…………6-6
6.2.1 穩定度………………………………………6-6
6.2.2 實界引理……………………………………6-7
6.3 Roesser系統的 強韌控制………………………6-10

第7章 二維系統控制器的設計
7.1二維系統靈敏度極小化概念……………………7-1
7.2 二維系統控制器設計……………………………7-4
7.2.1二維狀態回授控制器設計……………………7-4
7.2.2二維系統控制器設計………………………7-8
7.3 結論……………………………………………7-11
第8章 總結與未來展望
8.1 本文總結………………………………………8-1
8.2 未來展望………………………………………8-2

參考文獻
附錄A. 有限元素的假設模態分析
附錄B. Riccati方程式與實界引理
附錄C. Matlab LMI Toolbox相關指令介紹
附錄D. 基本名詞解釋與定義

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