求解最优控制问题的混合变量变分方法及其航天控制应用

彭海军; 高强; 吴志刚; 钟万勰

doi:10.3724/SP.J.1004.2011.01248

求解最优控制问题的混合变量变分方法及其航天控制应用

doi: 10.3724/SP.J.1004.2011.01248

1.
大连理工大学工程力学系工业装备结构分析国家重点实验室大连 116024;
2.
大连理工大学航空航天学院工业装备结构分析国家重点实验室大连 116024

详细信息

通讯作者:
高强大连理工大学工程力学系讲师. 主要研究方向为动力系统数值方法研究. E-mail: qgao@dlut.edu.cn

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出版历程
- 收稿日期: 2011-03-04
- 修回日期: 2011-04-04
- 刊出日期: 2011-10-20

A Mixed Variable Variational Method for Optimal Control Problems with Applications in Aerospace Control

1.
Department of Engineering Mechanics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024;
2.
School of Aeronautics and Astronautics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024

摘要

摘要: 针对非线性最优控制导出的Hamiltonian系统两点边值问题,提出一种以离散区段右端状态和左端协态为混合独立变量的数值求解方法, 将非线性Hamiltonian系统两点边值问题的求解通过混合独立变量变分原理转化为非线性方程组求解.所提出的算法综合了求解最优控制的"直接法"和"间接法"的特征,既满足最优控制理论的一阶必要条件,又不需要对协态初值的准确猜测,避免了求解大规模非线性规划问题. 通过两个航天控制算例讨论了本文算法的精度和效率等问题.与近年来在航空航天控制中备受关注的高斯伪谱方法相比较,本文算法无论是在精度还是效率上都具有明显的优势.
- 非线性最优控制 /
- 两点边值 /
- 对偶变量 /
- 变分原理 /
- Hamiltonian系统
Abstract: The nonlinear optimal control problem is transformed into the Hamiltonian two point boundary value problem (TPBVP), and a numerical method is proposed based on the dual variational principle. The nonlinear Hamiltonian TPBVP is transformed into a system of nonlinear equations by using the dual variational principle and by taking the left costate and the right state as independent variables. The proposed algorithm has the feature of both "direct method" and "indirect method" for solving optimal control problem, i.e., it satisfies the first-order necessary condition of optimal control theory, and it needs no precise initial guess for costate variables and avoids the solving of large scale nonlinear programming problem. The accuracy and efficiency of the proposed method are discussed by numerical simulations in aerospace control. Comparison between the proposed algorithm and the famous Gauss pseudo-spectral method in aeronautics and astronautics control shows that the proposed algorithm has obvious advantages on accuracy and efficiency.
- Nonlinear optimal control /
- two-point boundary value problem /
- dual variable /
- variational principle /
- Hamiltonian system

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参考文献(1)

[1]

Betts J T. Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics, 1998, 21(2): 193-207[2] Bryson A E, Ho Y C. Applied Optimal Control: Optimization, Estimation and Control. New York: John Wiley and Sons, 1975[3] Kirk D E. Optimal Control Theory: An Introduction. New York: Dover Publications, 2003[4] Betts J T. Practical Methods for Optimal Control Using Nonlinear Programming. Philadelphia: Society for Industrial Mathematics, 2001[5] Leyendecker S, Ober-Blobaum S, Marsden J E, Ortiz M. Discrete mechanics and optimal control for constrained systems. Optimal Control Applications and Methods, 1998, 31(6): 505-528[6] Benson D A. A Gauss Pseudospectral Transcription for Optimal Control [Ph.D. dissertation], Massachusetts Institute of Technology, USA, 2005[7] Rao A V, Benson D, Darby C, Patterson M A, Francolin C, Sanders I, Huntington G T. Algorithm 902: GPOPS, Matlab software for solving multiple-phase optimal control problems using the Gauss pseudospectral method. ACM Transactions on Mathematical Software, 37(2): 1-39[8] Zhong Wan-Xie, Wu Zhi-Gang, Tan Shu-Jun. Theory and Computation of State-space Control. Beijing: Science Press, 2007(钟万勰, 吴志刚, 谭述君. 状态空间控制理论与计算. 北京: 科学出版社, 2007[9] Tan Shu-Jun, Zhong Wan-Xie. Computation of nonlinear optimal control via symplectic conservative perturbation method. Acta Automatica Sinica, 2007, 33(9): 1004-1008(谭述君, 钟万勰. 非线性最优控制系统的保辛摄动近似求解. 自动化学报, 2007, 33(9): 1004-1008)[10] Wang Fei-Yue. Formulation of Kalman filter and Rauch smoother by quadratic optimal method. Journal of Zhejiang University (Natural Science), 1989, 23(2): 193-204(王飞跃. 用二次最优控制推导Kalman滤波器和最优插值器. 浙江大学学报(自然科学版), 1989, 23(2): 193-204)[11] Arnold V I. Mathematical Methods of Classical Mechanics. New York: Springer-Verlag, 1989[12] Park C, Scheeres D J. Determination of optimal feedback terminal controllers for general boundary conditions using generating functions. Automatica, 2006, 42(5): 869-875[13] Park, C. The Hamilton-Jacobi Theory for Solving Optimal Feedback Control Problems with General Boundary Conditions [Ph.D. dissertation], University of Michigan, USA, 2006[14] Peng H J, Gao Q, Wu Z G, Zhong W X. Symplectic multi-level method for solving nonlinear optimal control problem. Applied Mathematics and Mechanics, 2010, 31(10): 1251-1260[15] Gao Qiang, Peng Hai-Jun, Wu Zhi-Gang, Zhong Wan-Xie. Symplectic method for solving optimal control problem of nonlinear dynamical system. Journal of Dynamics and Control, 2010, 8(1): 1-7(高强, 彭海军, 吴志刚, 钟万勰. 非线性动力学系统最优控制问题的保辛求解方法. 动力学与控制学报, 2010, 8(1): 1-7)[16] Cui Ping-Yuan, Shang Hai-Bin, Luan En-Jie. A fast search algorithm for launch window of interplanetary low-thrust exploration mission. Journal of Astronautics, 2008, 29(1): 40-45(崔平远, 尚海滨, 栾恩杰. 星际小推力转移任务发射机会的快速搜索方法. 宇航学报, 2008, 29(1): 40-45)

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