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研究生：	黃耀平 Yao-Ping Huang
論文名稱：	A Nonlinearly Preconditioned Full-space Lagrange-Newton Method for Low Thrust Orbit Transfer Optimization Problems
指導教授：	黃楓南 Feng-Nan Hwang
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2019
畢業學年度：	107
語文別：	英文
論文頁數：	46
中文關鍵詞：	非線性
外文關鍵詞：	Nonlinearly
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在太空任務中電力推進系統被稱為低推力推進系統。最近，在大多數的太空
任務中傳統化學推進系統都改變為電力推進系統，因此低推力問題變得更加普
遍。而低推力軌道轉移的優化和設計一直是太空探索任務中的難題。針對這些問
題, 我們提出了A Nonlinearly Preconditioned Full-space Lagrange-Newton Method
該方法一種基於右非線性預處理技術分別通過替換非線性函數或改變未知數來
處理非線性，但它需要在原始系統的子集上進行內部迭代，這導致每步的額外成
本。因此，為了有效地提高效率，不需要在每次牛頓迭代上調用非線性預處理，
尤其是當近似解接近收斂時。數值計算驗證了該方法的有效性。
iv

In space missions, the low-thrust propulsion system is another name for the
electrically-powered spacecraft propulsion system. Recently, traditional chemical
propulsion system change to the electrically-powered spacecraft propulsion system
in the most space missions, so the low-thrust problems become more common. The
optimization and design of low-thrust orbit transfer always have been a difficult
problem in space exploration missions. For these problems, we propose a nonlinearly
preconditioned Full-space Lagrange-Newton Method. The method is kind of
the right nonlinear preconditioning techniques deals with nonlinearities by changing
unknowns or replacing nonlinear functions, respectively. Owing to it needs inner
iterations working on subsets of the original system, which lead to additional cost
per step. Therefore, for the purpose of effectively improve efficiency nonlinear preconditioner
require not to be invoked on every Newton iteration, especially when
the approximate solution close to the typical solution. The numerical result verifies
the effectiveness of the method.

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Low thrust optimal orbit transfer problems . . . . . . . . . . . . . . 3
1 Mathematical model of low thrust optimal orbit transfer problems . . 3
2 Indirect methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Indirect method formulation . . . . . . . . . . . . . . . . . . . 6
2.2 Applications: lunar launch problem . . . . . . . . . . . . . . . 8
3 Constrained optimization . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 The quadratic penalty method . . . . . . . . . . . . . . . . . 11
3.2 The method of Lagrange multipliers . . . . . . . . . . . . . . 12
Nonlinear preconditioned full-space Lagrange-Newton method . . 13
1 Description of Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Karush-Kuhn-Tucker Condition . . . . . . . . . . . . . . . . . . . . . 13
3 Nonlinear precondition . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Nonlinearly preconditioned full-space Lagrange Newton Algorithm . . 16
5 Lagrange-Newton method . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Newton step computation . . . . . . . . . . . . . . . . . . . . . . . . 19
6.1 Merit Function . . . . . . . . . . . . . . . . . . . . . . . . . . 19
vii
6.2 Line-search methods . . . . . . . . . . . . . . . . . . . . . . . 20
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1 Test problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Initial guess and typical solution . . . . . . . . . . . . . . . . . . . . 24
3 The Full-space Lagrange-Newton . . . . . . . . . . . . . . . . . . . . 27
4 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Compare with Matlab optimization toolbox . . . . . . . . . . . . . . 31
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
                                

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