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http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/54293完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 劉進賢(Chein-Shan Liu) | |
| dc.contributor.author | Tzu-Min Liu | en |
| dc.contributor.author | 劉子鳴 | zh_TW |
| dc.date.accessioned | 2021-06-16T02:48:56Z | - |
| dc.date.available | 2015-07-22 | |
| dc.date.copyright | 2015-07-22 | |
| dc.date.issued | 2015 | |
| dc.date.submitted | 2015-07-15 | |
| dc.identifier.citation | [1] M. Kubicek, V. Hlavacek, Numerical solution of nonlinear boundary value problems with applications. Prentice-Hall, New York (1983).
[2] H.-B. Keller, Numerical methods for two-point boundary value problems. Dover, New York (1992). [3] U. Ascher, R. Mattheij, R. Russell, Numerical solution of boundary value problems for ordinary differential equations. SIAM, Philadelphia (1995). [4] D.-D. Morrison, J.-D. Riley, J.-F. Zancanaro, Multiple shooting method for two-point boundary value problems. Communications of the ACM, vol. 5, no. 12, pp. 613-614 (1962). [5] R. Holsapple, R. Venkataraman, D. Doman, A modified simple method for solving two point boundary value problems. In: Proceedings of the IEEE Aerospace Conference, vol. 6 (2003). [6] C.-S. Liu, Cone of non-linear dynamical system and group preserving schemes. International Journal of Non-linear Mechanics, vol. 36, no. 7, pp. 1047-1068 (2001). [7] C.-S. Liu, The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions. CMES: Computer Modeling in Engineering & Sciences, vol. 13, pp. 149-163 (2006). [8] F. Stetter, On a generalization of the midpoint rule. Mathematics of Computation, vol. 22, pp. 661-663 (1968). [9] C.-S. Liu, Efficient shooting methods for the second order ordinary differential equations. CMES: Computer Modeling in Engineering & Sciences, vol. 15, pp. 69-86 (2006). [10] C.-S. Liu, The Lie-group shooting method for singularly perturbed two-point boundary value problems. CMES: Computer Modeling in Engineering & Sciences, vol. 15, no. 179-196 (2006). [11] C.-S. Liu, An efficient simultaneous estimation of temperature-dependent thermophysical properties. CMES: Computer Modeling in Engineering & Sciences, vol. 14, no. 77-90 (2006). [12] C.-S. Liu, L. W. Liu, and H. K. Hong, Highly accurate computation of spatial dependent heat conductivity and heat capacity in inverse thermal problem. CMES: Computer Modeling in Engineering & Sciences, vol. 17, pp. 1-18 (2007). [13] C.-S. Liu, The Lie-group shooting method for thermal stress evaluation through an internal temperature measurement. CMC: Computers, Materials & Continua, vol. 8, pp. 1-16 (2008). [14] C.-S. Liu, A Lie-group shooting method for computing eigenvalues and eigenfunctions of Sturm-Liouville problems. CMES: Computer Modeling in Engineering & Sciences, vol. 26, no. 157-168 (2008). [15] C.-S. Liu, A Lie-group shooting method for post buckling calculations of elastica. CMES: Computer Modeling in Engineering & Sciences, vol. 30, pp. 1-16 (2008). [16] C.-S. Liu, A Lie-group shooting method for simultaneously estimating the time dependent damping and stiffness coefficients. CMES: Computer Modeling in Engineering & Sciences, vol. 27, pp. 137-149 (2008). [17] C.-S. Liu, Computing the eigenvalues of the generalized Sturm-Liouville problems based on the Lie-group SL(2,R). Journal of Computational and Applied Mathematics, vol. 236, pp. 4547-4560 (2012). [18] C.-S. Liu, The Lie-group shooting method for solving the Bratu equation. Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4238-4249 (2012). [19] C.-S. Liu, Developing an SL(2,R) Lie-group shooting method for a singular ϕ-Laplacian in a nonlinear ODE. Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 9, pp. 2327-2339 (2012). [20] C.-S. Liu, The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems. Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1506-1521 (2012). [21] C.-S. Liu, The Lie-group shooting method for solving multi-dimensional nonlinear boundary value problems. Journal of Optimization Theory and Applications, vol. 152, no. 2, pp. 468-495 (2012). [22] L. Cveti´canin, Ninety years of Duffing’s equation. Theoretical and Applied Mechanics, vol. 40, no. 1, pp. 49-63 (2013) [23] J. Suhardjo, B.-F. Spencer. Jr, M.-K. Sain, Non-linear optimal control of a Duffing system. International Journal of Non-Linear Mechanics, Vol. 27, no. 2, pp. 157–172 (1992). [24] M.-J. Davies, Time optimal control and the Duffing oscillator. Journal of Institute of Mathematics and its Applications, Vol. 9, pp. 357-369 (1972). [25] R. Van Dooren, J. Vlassenbroeck, Chebyshev series solution of the controlled Duffing oscillator. Journal of Computational Physics, Vol. 47, pp. 321-329 (1982). [26] M. El-Kady, E.-M.-E. Elbarbary, A Chebyshev expansion method for solving nonlinear optimal control problems. Applied Mathematics and Computation, Vol. 129, pp. 171-182 (2002). [27] M. Razzaghi, G. Elnagar, Numerical solution of the controlled Duffing oscillator by the pseudospectral method. Journal of Computational and Applied Mathematics, Vol. 56, pp. 253-261 (1994). [28] C.-S. Liu, The optimal control problem of nonlinear Duffing oscillator solved by the Lie-group adaptive method. CMES: Computer Modeling in Engineering & Sciences, vol. 86, pp. 171-197 (2012). [29] C.-S. Liu, A method of Lie-symmetry GL(n;R) for solving non-linear dynamical systems. International Journal of Non-Linear Mechanics, Vol. 52, pp. 85-95 (2013). [30] C.-S. Liu, Solving nonlinear differential algebraic equations by an implicit Lie-group method. Journal of Applied Mathematics, Article ID 987905, 8 pages (2013). [31] C.-S. Liu, A new sliding control strategy for nonlinear system solved by the Lie-group differential algebraic equation method. Communications in Nonlinear Science and Numerical Simulation, Vol. 19, pp. 2012-38 (2014). | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/54293 | - |
| dc.description.abstract | 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。 | zh_TW |
| dc.description.abstract | 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. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T02:48:56Z (GMT). No. of bitstreams: 1 ntu-104-R02521241-1.pdf: 2367064 bytes, checksum: 548c8075be3fe19877d84fa8533df887 (MD5) Previous issue date: 2015 | en |
| dc.description.tableofcontents | 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 | |
| dc.language.iso | zh-TW | |
| dc.subject | 保群算法 | zh_TW |
| dc.subject | 李群打靶法 | zh_TW |
| dc.subject | 杜芬振子 | zh_TW |
| dc.subject | 李群微分代數方程法 | zh_TW |
| dc.subject | 最佳化控制問題 | zh_TW |
| dc.subject | 李群微分代數方程法 | zh_TW |
| dc.subject | 李群打靶法 | zh_TW |
| dc.subject | 保群算法 | zh_TW |
| dc.subject | 哈密頓函數 | zh_TW |
| dc.subject | 哈密頓函數 | zh_TW |
| dc.subject | 最佳化控制問題 | zh_TW |
| dc.subject | 杜芬振子 | zh_TW |
| dc.subject | Hamiltonian formulation | en |
| dc.subject | Duffing oscillator | en |
| dc.subject | Optimal control problem | en |
| dc.subject | Lie-group method | en |
| dc.subject | Lie-group shooting method | en |
| dc.subject | Lie-group differential algebraic equations method | en |
| dc.subject | Duffing oscillator | en |
| dc.subject | Optimal control problem | en |
| dc.subject | Hamiltonian formulation | en |
| dc.subject | Lie-group method | en |
| dc.subject | Lie-group shooting method | en |
| dc.subject | Lie-group differential algebraic equations method | en |
| dc.title | 以李群打靶法求解杜芬非線性振子的最佳化控制問題 | zh_TW |
| dc.title | By Using the Lie-group Shooting Method to Solve the Optimal Control Problems of Nonlinear Duffing Oscillators | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 103-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 陳永為,郭仲倫 | |
| dc.subject.keyword | 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, | zh_TW |
| dc.subject.keyword | Duffing oscillator,Optimal control problem,Hamiltonian formulation,Lie-group method,Lie-group shooting method,Lie-group differential algebraic equations method, | en |
| dc.relation.page | 87 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2015-07-15 | |
| dc.contributor.author-college | 工學院 | zh_TW |
| dc.contributor.author-dept | 土木工程學研究所 | zh_TW |
| 顯示於系所單位： | 土木工程學系 | |
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