臺灣博碩士論文加值系統

　　在土木領域裡，對結構的保護及控制來說，能夠即時地重建施於系統的外力向來是個不可忽視的研究議題。
　　過去雖有許多重建外力的方案提出，但因運算求解費時，故欲達到「即時」重建的方法仍舊有限。本論文中，將過去已廣泛應用的強健微分器及追蹤微分器，自常微分方程轉換為一微分代數方程組，透過控制力加以調配控制，並利用結合隱格式李群法和牛頓迭代法的新數值方法——李群微分代數方程法進行運算求解，內外迴圈雙重迭代增加穩定性。此方法可利用於設計線上偵測器，意為可在僅有已受噪音干擾的位移訊號資料下，即時重建施於系統的未知外力。隨著線性結構、杜芬方程式、范德波爾方程式及地震力作用下的線性結構等四個數值算例顯示，此方法呈現的效果有相當不錯的精度與效率，且簡單易於使用，未來發展應用的機會極大。

For structure protection and control, it is utmost to immediately detect the external force being imposed on the structure currently in civil engineering. In this thesis, we remodel the famous robust exact differentiator and tracking differentiator into a type of differential algebraic equations (DAEs), and then we solve the resultant DAEs by a Lie-group method, which can be used as an on-line estimator to detect unknown external force by using only a real-time measurement of the structural displacement under random noise in time. The estimated results obtained by the novel methods are quite promising.

口試委員會審定書 i
誌謝 ii
摘要 iii
ABSTRACT iv
目錄 v
圖目錄 vii
1第一章 緒論 1
1.1 前言 1
1.2 文獻回顧 2
1.3 論文架構 3
2第二章 數值分析方法 5
2.1 群 5
2.1.1 群的歷史 5
2.1.2 群的定義 6
2.1.3 李群與李代數 6
2.2 數值分析方法 8
2.2.1 微分方程中的 結構 9
2.2.2 隱格式李群 法 11
2.2.3 牛頓法求解非線性代數方程 14
2.2.4 數值計算流程圖 15
3第三章 現有偵測器之改良 16
3.1 方法一 16
3.1.1 強健微分器（Robust Exact Differentiator, RED）之改良 16
3.1.2 LGDAE於強健微分器的解析與應用 18
3.1.3 方法一數值計算流程圖 21
3.2 方法二 22
3.2.1 追蹤微分器（Tracking Differentiator, TD）之改良 22
3.2.2 LGDAE於追蹤微分器的解析與應用 23
3.2.3 方法二數值計算流程圖 26
4第四章 數值算例 27
4.1 算例一 線性結構 27
4.1.1 計算量測位移無噪音影響下之外力 27
4.1.2 計算當量測位移具有噪音影響下之外力 35
4.2 算例二 Duffing Oscillator 47
4.2.1 計算量測位移無噪音影響下之外力 47
4.2.2 計算當量測位移具有噪音影響下之外力 55
4.3 算例三 Van der Pol Oscillator 67
4.3.1 計算量測位移無噪音影響下之外力 67
4.3.2 計算當量測位移具有噪音影響下之外力 75
4.4 算例四　線性結構下地震力重建 87
5第五章　結論與未來展望 98
參考文獻 99

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