A Full Space Lagrange-Newton-Krylov Algorithm for Minimum Time Trajectory Optimization

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姓名

駱易俗(Yi-Su Lo) 查詢紙本館藏

畢業系所

數學系

論文名稱

A Full Space Lagrange-Newton-Krylov Algorithm for Minimum Time Trajectory Optimization
(A Full Space Lagrange-Newton-Krylov Algorithm for Minimum Time Trajectory Optimization)

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摘要(中)

軌道最佳化問題的主要目的在於設計一最佳軌道，其必須滿足問題給定的特定條件，並達成某項衡量標準的最大或最小化。因為這樣的特性，這類問題通常被描述成一最佳化控制問題，也因此屬於數學上延伸自「變分法」領域並作為其應用之一的「最佳化控制理論」範疇。近來，隨著電子計算機效能的提昇，各種數值計算技術越來越廣泛地被應用在最佳化問題的求解上。其中兩種主要的方式稱為間接法和直接法，前者將最佳化控制問題轉換成雙點邊界值問題，後者則是轉換成一非線性規劃問題，然後再嘗試以各種數值方法求解。在這篇論文的工作中，我們將重心放在某一類型的直接法，並針對非線性規劃問題的求解提出一全空間 Lagrange-Newton-Krylov 演算法。這個演算法建立在全空間序列二次規劃的架構上，並結合全域化策略和產生初始值的程序。透過這個演算法的執行，我們試著求解數個最小時間軌道最佳化問題，而從其產生的數值結果中可以看出，這個演算法是可行並具有發展潛力的。

摘要(英)

Trajectory optimization problem is concerned with the design of an optimal trajectory that maximizes or minimizes some measurement and satisfies prescribed conditions. Because of this characteristic, it is in general formulated as an optimal control problem and hence is related to the optimal control theory, a branch of mathematics as an application of the calculus of variations. Recently, with an improvement of computer powers, computational techniques become more widely used in solving optimal control problems. Two main approaches, namely direct and indirect methods, reformulated an optimal control problem as a boundary value problem and a nonlinear programming problem respectively and then numerical methods can be employed. In this work, we focus on a class of direct methods and purposed a full space Lagrange-Newton-Krylov algorithm for the nonlinear programming problems. This algorithm is based on the full space sequential quadratic programming framework and associated with particular globalization strategy and process to generate the initial guess. With the implementation of this algorithm, we try to solve several minimum time trajectory optimization problems and the numerical results exhibit the practicability and potentiality of this algorithm.

關鍵字(中)

★ 全空間二次序列規劃
★ 非線性規劃
★ 最佳化控制
★ 軌道最佳化

關鍵字(英)

★ nonlinear programming
★ optimal control
★ trajectory optimization
★ full space sequential quadratic programming

論文目次

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Optimal control problem and direct transcription . . . . . . . . . . . . 3
2.1 An introduction to optimal control problems . . . . . . . . . . . . . 3
2.1.1 Definition of variables . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2 A baseline optimal control problem . . . . . . . . . . . . . . . . . 4
2.1.3 The other forms . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.4 Multi-phase problem . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.5 Indirect and direct methods . . . . . . . . . . . . . . . . . . . . 7
2.2 Indirect method formulation . . . . . . . . . . . . . . . . . . . . . 8
2.3 Direct transcription employing collocation method . . . . . . . . . . 10
2.3.1 Discretization and transformation for free final time phase . . . . 10
2.3.2 Numerical integration of state equations . . . . . . . . . . . . . . 11
2.3.3 Collocation methods . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4 Resulting nonlinear programming problem . . . . . . . . . . . . . . 16
3 A full space Lagrange-Newton-Krylov algorithm for nonlinear programming 17
3.1 An introduction to parameter optimization problems . . . . . . . . . . 17
3.1.1 General equality-constrained optimization problem . . . . . . . . . 17
3.1.2 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Sequential quadratic programming . . . . . . . . . . . . . . . . . . . 21
3.2.1 Lagrange-Newton method . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Concept of sequential quadratic programming . . . . . . . . . . . . 23
3.3 Step computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Globalization strategy . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.1 Merit function method . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.2 Globalization mechanism . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Generating an initial guess . . . . . . . . . . . . . . . . . . . . . 27
3.6 A full space Lagrange-Newton-Krylov algorithm . . . . . . . . . . . . 28
4 Applications: minimum time trajectory optimization . . . . . . . . . . . 30
4.1 Aircraft navigation problem . . . . . . . . . . . . . . . . . . . . . 30
4.1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.2 Indirect method formulation . . . . . . . . . . . . . . . . . . . . 32
4.1.3 Direct transcribing formulation . . . . . . . . . . . . . . . . . . 35
4.2 Lunar launch problem . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.2 Indirect method formulation . . . . . . . . . . . . . . . . . . . . 38
4.2.3 Direct transcribing formulation . . . . . . . . . . . . . . . . . . 40
Satellite launch vehicle problem . . . . . . . . . . . . . . . . . . . . . 42
4.3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.2 Direct transcription formulation . . . . . . . . . . . . . . . . . . 46
5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1 Grid tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.1 Lunar launch problem . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.2 Satellite launch vehicle problem . . . . . . . . . . . . . . . . . . 51
5.2 Typical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2.1 Aircraft navigation problem . . . . . . . . . . . . . . . . . . . . 52
5.2.2 Lunar launch problem . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.3 Satellite launch vehicle problem . . . . . . . . . . . . . . . . . . 60
5.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3.1 Lunar launch problem . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3.2 Satellite launch vehicle problem . . . . . . . . . . . . . . . . . . 66
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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指導教授

黃楓南(Feng-Nan Hwang)

審核日期

2012-8-1

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