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

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：32

、訪客IP：3.135.216.174

姓名

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)

相關論文

★ 非線性塊狀高斯消去牛頓演算法在噴嘴流體的應用	★ 以平行 Newton-Krylov-Schwarz 演算法解 Poisson-Boltzmann 方程式的有限元素解在膠體科學上的應用
★ 最小平方有限元素法求解對流擴散方程以及使用Bubble函數的改良	★ Bifurcation Analysis of Incompressible Sudden Expansion Flows Using Parallel Computing
★ Parallel Jacobi-Davidson Algorithms and Software Developments for Polynomial Eigenvalue Problems in Quantum Dot Simulation	★ An Inexact Newton Method for Drift-DiffusionModel in Semiconductor Device Simulations
★ Numerical Simulation of Three-dimensional Blood Flows in Arteries Using Domain Decomposition Based Scientific Software Packages in Parallel Computers	★ A Parallel Fully Coupled Implicit Domain Decomposition Method for the Stabilized Finite Element Solution of Three-dimensional Unsteady Incompressible Navier-Stokes Equations
★ A Study for Linear Stability Analysis of Incompressible Flows on Parallel Computers	★ Parallel Computation of Acoustic Eigenvalue Problems Using a Polynomial Jacobi-Davidson Method
★ Numerical Study of Algebraic Multigrid Methods for Solving Linear/Nonlinear Elliptic Problems on Sequential and Parallel Computers	★ A Parallel Multilevel Semi-implicit Scheme of Fluid Modeling for Numerical Low-Temperature Plasma Simulation
★ Performance Comparison of Two PETSc-based Eigensolvers for Quadratic PDE Problems	★ A Parallel Two-level Polynomial Jacobi-Davidson Algorithm for Large Sparse Dissipative Acoustic Eigenvalue Problems
★ A Full Space Lagrange-Newton-Krylov Algorithm for Minimum Time Trajectory Optimization	★ Parallel Two-level Patient-specific Numerical Simulation of Three-dimensional Rheological Blood Flows in Branching Arteries

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

本電子論文使用權限為同意立即開放。
已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

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

參考文獻

[1] M. Benzi, G.H. Golub, and J. Liesen. Numerical solution of saddle point problems.
Acta Numer., 14:1–137, 2005.
[2] J.T. Betts. Survey of numerical methods for trajectory optimization. J. Guid. Contr.
Dynam., 21:193–207, 1998.
[3] J.T. Betts. Very low-thrust trajectory optimization using a direct SQP method. J.
Comput. Appl. Math., 120:27–40, 2000.
[4] J.T. Betts. Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming. SIAM, Philadelphia, 2nd edition, 2010.
[5] N. Biehn, S.L. Campbell, L. Jay, and T.Westbrook. Some comments on DAE theory
for IRK methods and trajectory optimization. J. Comput. Appl. Math., 120:109–131,
2000.
[6] G. Biros and O. Ghattas. Parallel Lagrange–Newton–Krylov–Schur methods for
PDE-constrained optimization. Part II: The Lagrange–Newton solver and its application
to optimal control of steady viscous flows. SIAM J. Sci. Comput., 27:714–739,
2005.
[7] G. Biros and O. Ghattas. Parallel Lagrange-Newton-Krylov-Schur methods for
PDE-constrained optimization. part I: The Krylov–Schur solver. SIAM J. Sci. Comput.,
27:687–713, 2005.
[8] A.E. Bryson and Y.-C. Ho. Applied Optimal Control: Optimization, Estimation, and
Control. Taylor & Francis Group, 1975.
[9] J.E. Dennis Jr. and R.B. Schnabel. Numerical methods for unconstrained optimization
and nonlinear equations. SIAM, Philadelphia, 1996.
[10] D.J. Estep, D.H. Hodges, and M. Warner. The solution of a launch vehicle trajectory
problem by an adaptive finite-element method. Computer methods in applied
mechanics and engineering, 190(35):4677–4690, 2001.
[11] F. Fahroo and I.M. Ross. Costate estimation by a Legendre pseudospectral method.
J. Guid., Contr. Dynam, 24:270–277, 2001.
[12] M. Fink. Automatic Differentiation for Matlab. MATLAB Central File Exchange.
Retrieved May 18, 2015, 2007.
[13] P.E. Gill, L.O. Jay, M.W. Leonard, L.R. Petzold, and V. Sharma. An SQP method
for the optimal control of large-scale dynamical systems. J. Comput. Appl. Math.,
120:197–213, 2000.
[14] D.G. Hull. Optimal Control Theory for Applications. Springer-Verlag, New York,
2003.
[15] C.T. Kelley. Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia,
1995.
[16] Wilfried Ley, KlausWittmann, andWilli Hallmann. Handbook of space technology,
volume 22. John Wiley & Sons, 2009.
[17] P. Lu and M.A. Khan. Nonsmooth trajectory optimization-an approach using continuous
simulated annealing. J. Guid. Contr, Dynam., 17:685–691, 1994.
[18] R Timothy Marler and Jasbir S Arora. Survey of multi-objective optimization methods
for engineering. Struct. Multidisc. Optim., 26(6):369–395, 2004.
[19] J. Nocedal and S.J. Wright. Numerical Optimization. Springer-Verlag, New York,
2006.
[20] M.A. Patterson and A.V. Rao. GPOPS-II: A MATLAB software for solving
multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation
methods and sparse nonlinear programming. ACM Trans. Math. Software,
41:1, 2014.
[21] M. Pontani. Particle swarm optimization of ascent trajectories of multistage launch
vehicles. Acta Astronaut., 94:852–864, 2014.
[22] E.E. Prudencio, R. Byrd, and X.C. Cai. Parallel full space SQP Lagrange-Newton-
Krylov-Schwarz algorithms for PDE-constrained optimization problems. SIAM J.
Sci. Comput., 27:1305–1328, 2006.
[23] A.V. Rao. Trajectory optimization: A survey. In Optimization and Optimal Control
in Automotive Systems, pages 3–21. Springer, 2014.
[24] W. Roh and Y. Kim. Trajectory optimization for a multi-stage launch vehicle using
time finite element and direct collocation methods. Eng. Optim., 34:15–32, 2002.
[25] Woongrae Roh and Youdan Kim. Trajectory optimization for a multi-stage launch
vehicle using time finite element and direct collocation methods. Engineering optimization,
34(1):15–32, 2002.
[26] Y. Saad. Iterative methods for sparse linear systems. SIAM, Philadelphia, 2nd
edition, 2003.
[27] M.R. Sentinella and L. Casalino. Cooperative evolutionary algorithm for space trajectory
optimization. Celestial Mech. Dyn. Astron., 105:211–227, 2009.
[28] K. Subbarao and B.M. Shippey. Hybrid genetic algorithm collocation method for
trajectory optimization. J. Guid. Contr, Dynam., 32:1396–1403, 2009.
[29] S. Subchan and R. ˙ Zbikowski. Computational Optimal Control: Tools and Practice.
John Wiley & Sons, 2009.
[30] BN Suresh and K Sivan. Integrated Design for Space Transportation System.
Springer, 2015.
[31] O. von Stryk and R. Bulirsch. Direct and indirect methods for trajectory optimization.
Annals Oper. Res., 37:357–373, 1992.
[32] P. Williams. Jacobi pseudospectral method for solving optimal control problems. J.
Guid. Contr. Dynam., 27:293–297, 2004.
[33] A. Wuerl, T. Crain, and E. Braden. Genetic algorithm and calculus of variationsbased
trajectory optimization technique. J. Spacecraft Rockets, 40:882–888, 2003.
[34] H. Yang, F.-N. Hwang, and X-.C. Cai. Nonlinear preconditioning techniques for
full-space Lagrange-Newton solution of PDE-constrained optimization problems.
SIAM J. Sci. Comput., to appear, 2016.

指導教授

黃楓南

審核日期

2017-8-24

推文