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研究生:劉子鳴
研究生(外文):Tzu-Min Liu
論文名稱:以李群打靶法求解杜芬非線性振子的最佳化控制問題
論文名稱(外文):By Using the Lie-group Shooting Method to Solve the Optimal Control Problems of Nonlinear Duffing Oscillators
指導教授:劉進賢
指導教授(外文):Chein-Shan Liu
口試委員:陳永為郭仲倫
口試日期:2015-06-30
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:中文
論文頁數:87
中文關鍵詞:杜芬振子最佳化控制問題哈密頓函數保群算法李群打靶法李群微分代數方程法
外文關鍵詞:Duffing oscillatorOptimal control problemHamiltonian formulationLie-group methodLie-group shooting methodLie-group differential algebraic equations method
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在最佳化控制理論中,通常使用哈密頓函數,利用其方便找尋控制力函數的特點來設計控制力。然而,當狀態函數形式較為複雜時,哈密頓函數將構成一非線性微分代數方程組的兩點邊界值問題而難以找出封閉解,因此需使用其他數值方法輔助求解。
本篇論文將杜芬非線性振子代入兩點邊界值問題模擬非線性微分代數方程組,藉此探討上述議題,並建立一套數值方法利用李群 及 打靶法配合李群微分代數方程法對杜芬非線性振子的最佳化控制問題求出數值近似解。在論文中將演示如何使用上述方法求解六個單自由度以及一個雙自由度的杜芬非線性振子最佳化控制問題,並分析其數值結果。


In the optimal control theory, the Hamiltonian formulation is a famous one which is convenient to find an optimally designed control force. However, when the performance index is a complicated function of control force, the Hamiltonian method is not easy to find the optimal closed-form solution, because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations (DAEs).
In this thesis, we address this issue via an novel approach, of which the optimal vibration control problem of Duffing oscillator is recast into a two-point nonlinear DAEs. We develop the corresponding and shooting methods, as well as a Lie-group differential algebraic equations (LGDAE) method to numerically solve the optimal control problems of nonlinear Duffing oscillators. Seven examples of a single Duffing oscillator and one coupled Duffing oscillators are used to test the performance of the present method.


口試委員審定書 ii
誌謝 iii
摘要 v
ABSTRACT vi
目錄 vii
圖目錄 x
第 一 章 緒論 1
1.1 前言 1
1.2 文獻回顧 2
1.3 研究動機與目的 3
1.4 論文架構 4
第 二 章 數值分析方法 6
2.1 四階龍格-庫塔法 6
2.2 保群算法 9
2.2.1 群 9
2.2.2 李群(Lie Group) 11
2.2.3 增廣動態系統 13
2.2.4 李代數 16
2.2.5 光錐構造 17
2.2.6 凱萊轉換(Cayley Transformation) 20
2.2.7 指數映射 24
2.2.8 一步保群算法 28
2.3 李群微分代數方程法(Lie-Group Differential Algebraic Equations method, LGDAE) 30
2.3.1 微分方程系統中的 結構 30
2.3.2 李群微分代數方程法(LGDAE) 31
2.3.3 數值分析流程 33
第 三 章 杜芬非線性振子的最佳化控制問題 35
3.1 最佳化控制 35
3.2 打靶法 35
3.3 杜芬非線性振子與動態系統 36
3.4 哈密頓函數(Hamiltonian formulation) 37
3.5 李群打靶法(Lie-group shooting method) 38
3.5.1 李群迭代法(Lie-group scheme) 38
3.5.2 李群 打靶法 42
3.5.3 李群 打靶法 43
3.6 數值分析流程 45
3.6.1 李群打靶法配合四階龍格-庫塔法 45
3.6.2 李群打靶法配合李群微分代數方程法及四階龍格-庫塔法 47
第 四 章 數值模擬計算與結果分析 49
4.1 數值算例一 49
4.2 數值算例二 54
4.3 數值算例三 57
4.4 數值算例四 60
4.5 數值算例五 69
4.6 數值算例六 74
4.7 數值算例七 77
第 五 章 結論與未來展望 81
參考文獻 84


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