A full-space quasi Lagrange-Newton-Krylov algorithm for  trajectory optimization problems with space mission

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王璿豪(Hsuan-Hao Wang) 查詢紙本館藏

畢業系所

數學系

論文名稱

(A full-space quasi Lagrange-Newton-Krylov algorithm for trajectory optimization problems with space mission)

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摘要(中)

軌跡最佳化是航太工業經常使用的技術，例如找出最佳軌跡使得酬載重量最大化或是縮短衛星到達目標軌道的時間等等，這種類型的問題可以用數學建模成連續時間的最佳化控制問題。本篇論文主要是研究full-space quasi Lagrange-Newton-Krylov method作為數值求解器，首先會引入拉格朗日乘子將有約束的最佳化問題轉為無約束的最佳化問題，並求解一階必要條件(KKT condition)，透過牛頓法結合回朔技術，在每一次牛頓迭代中，所有KKT系統的變數(狀態變數, 控制變數, 設計變數, 拉格朗日乘子) 會以 Krylov-subspace method 配合預處理技術同時被解出。本文會針對KKT系統中的 Hessian matrix 的建造找出提高計算效能的方法，並透過引入鬆弛變量來解決不等式限制條件問題。本文以三維座標的三節火箭問題和二維座標的金牛座火箭作為數值範例，相關的物理背景和數值結果將會在文章中呈現。

摘要(英)

The trajectory optimization is a commonly-used technique of applications in aerospace engineering, such as find the optimal trajectory to maximize the weight of the payload or minimize the time for satellite insertion the target orbit, etc. This type of problem can be modeled mathematically as some form of continuous time optimal control problems. This thesis focuses on the study of the full-space quasi Lagrange-Newton-Krylov method as our numerical solver. First, we introduce the Lagrangian multiplier to convert the constrained optimization problem into unconstrained optimization problem, and solve the first-order necessary condition(KKT condition). By the Newton method combined with the backtrack technique, in each Newton iteration, all KKT system variables (state variables, control variables, design variables, Lagrangian multipliers) will be solved at the same time by Krylov-subspace method with the precondition technology. In this thesis, we improve the computational efficiency of Hessian matrix constructor in KKT system, and solve the problem of inequality constraint by introducing slack variable. The three-dimensional case of the three stage rocket problems and two-dimensional case of Taurus rocket as a numerical example, the relevant physical background and numerical results will be presented in the thesis.

關鍵字(中)

★ 最佳化控制
★ 軌跡最佳化問題
★ 全空間類牛頓法
★ 衛星載具動力學

關鍵字(英)

論文目次

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Full-space quasi-Lagrange-Newton-Krylov algorithm . . . . . . . . . . . . 5
2.1 A description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 KKT matrix construction . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Newton step computation . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Globalization strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Multistage satellite launch vehicle problem . . . . . . . . . . . . . . . . . . 12
3.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Launch point inertial (LPI) frame (xL; yL; zL) . . . . . . . . . 15
3.3 Mathematical model for launch vehicle system . . . . . . . . . . . . . . 15
3.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.2 Angle definition and control variables . . . . . . . . . . . . . . . 17
3.3.3 Coriolis force on LPI frame . . . . . . . . . . . . . . . . . . . . 18
3.3.4 Orbit insertion conditions . . . . . . . . . . . . . . . . . . . . . 18
3.3.5 Dynamic equation and optimal control problem . . . . . . . . . . 20
3.4 A parameter optimization problem . . . . . . . . . . . . . . . . . . . . . 24
4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 Non-dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 three-stage rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.1 Initial guess for control . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.2 Grid test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.3 Typical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.5 Consider inequality constraints . . . . . . . . . . . . . . . . . . . 36
4.2.6 Compare with two-dimensional case . . . . . . . . . . . . . . . . 38
4.3 Other examples: Taurus rocket . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.2 Dynamic equation . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.3 Initial guess for control . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.4 Grid test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.5 Typical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.6 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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指導教授

黃楓南

審核日期

2017-8-24

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