臺灣博碩士論文加值系統

有界不確定量之分數階系統之狀態方程式強健穩定化問題為本文探討之主題。文中考慮的不確定量假設為多邊形態之有界不確定量，當未確定量有過量巨大的變化時，使用文獻中的方法，吾人不能得到解。根據模擬的結果，能容許的多邊形態不確定量比範數有界形態不確定量要小的多。探討強健穩定化問題的目的在於給出有界不確定狀態分數階系統為穩定的條件，同時設計使用分數階之狀態反饋，使得閉迴路系統為強健穩定。文中推導的五個定理使用嚴格線性矩陣不等式，得到分數階系統之狀態反饋的公式。一系列的數值例題用以驗證設計的結果。當系統瀕臨故障或重大災害時，未確定量將有巨量的變化，而容錯控制因應的不確定量，正是多邊形態不確定量的邊緣。所以，最後將本文中設計控制器的方法應用到分數階系統的容錯控制上，以顯示所提議之方法的有效性。


關鍵詞：分數階系統、線性矩陣不等式(LMI)、強健穩定、容錯控制。

The problems of robust stability and stabilization for continuous time uncertain fractional order systems are solved in this thesis. The parametric uncertainty considered is of polytopic type. The purpose of the robust stability problem is to give condition such that the uncertain fractional order system is stable for all admissible uncertainties, while the purpose of robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. A strict linear matrix inequality (LMI) design approach is derived, and an explicit expression for the desired robust state feedback control law is also given. Numerical examples are provided to demonstrate the application of the proposed method, and finally the controller design method is applied to the fault-tolerant control to show the applicability of the design method.


Keywords: Fractional Order PID Controller, Linear Matrix Inequality (LMI), Robust Stability, Fault-Tolerant Control.

目錄
摘要 I
Abstract II
致謝 III
圖目錄 VI
符號說明 VII
第一章 緒論 1
1.1 文獻回顧與研究動機 1
1.2 論文綱要 3
第二章 控制系統設計與分數階控制系統概論 4
2.1 控制的重點觀念 4
2.1.1 狀態反饋控制器之設計 4
2.1.2 不確定整數階系統數學模型描述 5
2.1.3 整數階系統穩定之相關的基本定義 6
2.1.4 不確定整數階系統之強健穩定化 8
2.2 容錯控制 12
2.3 分數階微積分 15
2.3.1 分數階微積分定義Grünwald-Letnikov(G-L) 16
2.3.2 分數階微積分定義 Riemann-Liouville (R-L) 17
2.3.3 分數階微積分定義 Caputo 18
2.4 分數階系統 19
2.4.1 分數階拉普拉斯轉換 20
2.4.2 分數階微分方程 22
2.5 分數階系統 (Fractional order system) 的含意 23
2.6 分數階系統狀態方程式的穩定性 25
2.7 不具不確定量之分數階系統 26
第三章 具不確定量分數階系統強健穩定化 29
3.1 分數階系統數學模型描述 29
第四章 數值模擬 33
第五章 結論與未來研究方向 49
5.1 結論 49
5.2 未來研究方向 49
參考文獻 50

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