登月飞行器软着陆的制导与控制
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摘要
导航、制导与控制是登月计划的关键性技术,而软着陆问题又是月球探测面临的第一个关键性问题。本文以登月飞行的制导与软着陆控制问题为背景,在系统学习前人的研究成果的基础之上,应用最优控制、最优化方法、非线性控制以及切换系统方法,依照最优性、鲁棒性的本质要求,系统的进行分析和研究。
本文研究的重点是设计登月飞行器软着陆的制导律和控制律。主要内容可以分为两大部分,一部分是非定点软着陆的制导与控制问题,另一部分是定点软着陆的制导与控制问题。对于非定点软着陆的制导和控制,主要采用最优显式制导和反馈跟踪控制的思路。文中分别给出两种参数化的最优显式制导律设计方法,为保证飞行器在实际飞行中受干扰的作用下仍能很好的跟踪已设计的标称软着陆轨道,文中给出一种基于切换系统模型参考的跟踪控制方法。对于定点软着陆控制问题,给出一种基于交会对接思想的定点软着陆反馈控制方法。最后,利用STK仿真平台对前面所提出的制导律方法进行仿真验证,并对仿真结果进行细致的分析。具体内容如下:
首先,对于参数化显式制导律设计方法,文中利用经典最优控制理论中的Eular方程,构造出一种迭代的分段恒值的参数化最优控制律。该控制律可以无限逼近全局最优解,且其参数个数与系统的阶数相同。将这种参数化控制律代入原系统,可将原来复杂的最优控制问题转化为一系列参数优化问题。考虑到很多经典优化算法都要利用指标泛函和约束泛函关于参数的梯度,文中严密推导出该参数化控制律的参数梯度显式公式。最后,分别利用遗传算法和BFGS优化算法优化给出仿真结果,并对不同工况下的仿真结果进行轨道特性分析,对于软着陆过程中控制律参数变化对软着陆轨道特性的影响进行详细的描述。
其次,考虑到实际中的发动机多为常推力的情况,文中利用参数化控制及强化技术构造出一种分段的参数化最优控制律,使软着陆问题得到很好的解决,具有较强的工程意义。参数化控制及强化技术的主要思想是利用若干个分段的常数去逼近最优解。在此情况下,最优控制问题将转化为一系列参数优化问题。利用经典的参数优化方法即可求得最优控制函数的一个近似解。通过不断的增加参数的个数,缩小每段参数的持续时间,重复优化就可以得到一组无线逼近连续最优解的参数化解序列。本文将登月飞行器软着陆的最优显式制导律设计问题进行若干变换后,构造成具有Canonical标准型的最优控制问题。然后,利用上述参数化控制及强化技术给出最优显式制导律的设计方法。
考虑到实际飞行过程中扰动的作用,本文给出一种鲁棒切换模型参考跟踪控制器设计方法,该方法基于鲁棒模型参考理论并结合切换系统的H∞控制方法。文中利用线性矩阵不等式给出该控制器存在的条件。将其应用到软着陆的跟踪控制问题,取得满意的效果。
针对定点软着陆问题,本文给出一种基于交会对接思想的软着陆控制方法,该方法将软着陆控制问题转化为线性受限系统的二次调解问题,通过飞行器相对于落点的位置和速度状态反馈实现软着陆控制。文中首先以预定落点为坐标原点建立坐标系,求得飞行器的非线性运动学模型。经过合理的近似后得到相应的线性系统,考虑到燃料的最优,控制推力受限以及状态受限的要求文中利用线性受限系统二次调节理论给出最优状态反馈控制器设计方法。该方法的优点是可以通过直接的速度和位置状态反馈实现控制,控制器形式简单,且易于导航计算,可以大大减少导航计算量。另外,该方法在初始建立坐标系时,利用飞行器相对落点的速度和位置偏差作为飞行器的状态,这样将软着陆控制问题转化成控制系统稳定性问题,使问题的复杂度大大降低,理论上更容易求解。
最后,本文利用STK仿真平台将前面章节中所得出的软着陆制导律进行仿真验证。本文详细的给出了仿真的全过程,并对结果进行了科学的分析。从仿真结果可以看出,前面所提出的定点软着以及非定点软着陆制导律设计方法能够有效的实现软着陆控制。
Navigation, guidance and control are the key techniques for lunar landing plan,moreover the soft landing is the first problem of lunar exploration. This dissertationsystematically studies navigation and guidance and control of the lunar landing byoptimal control, optimization, nonlinear control, linear matrix inequality and switchedsystem methods based on precursors’results.
Main works of the paper include the guidance and control law of soft landing,which can be divided into two parts that are the guidance and control of fix-pointsoft landing and that of unfix-point soft landing. The paper use the optimal analyticalguidance law and model reference control method to deal with the unfix-point softlanding problem. For design of the optimal analytical guidance law, the paper devel-ops two kinds of parameterizing guidance law. Considering the exist of disturbance inthe space, a model reference tracking controller is developed to guarantee the landerrobustly to track the normal orbit. For dealing with the fix-point soft landing problem,the paper develops a feedback control method based on the intersecting method. Atlast, STK is used to validate the feasibility and advantage of the proposed method.At first, a recursive piece-wise constant parametric controller is developed byEular equation of the classic optimal control theory in this dissertation, which canbe infinitely approaching the continuous optimal solution. The number of parametersis equal to the system rank. By substituting the controller into the given system, theoriginal optimal control problem is transferred into a parameter selection problem.The analytic gradient functions of the parameters to the cost function and constraintsare developed for some classic optimization methods. Then, the simulating results aregiven based on the genetic algorithm and the BFGS algorithm, respectively.
Essence of the parametric control and the enhancing technique is to use thepiece-wise constant value to approach the optimal solution of optimal control prob-lem. Then, the optimal control problem can be transformed into the parameter selec-tion problem by these methods, moreover the optimal parameters can be solved by thecommon mathematic programming. The approximating precision can be enhanced byincreasing the number of parameters so that an acceptable solution is derived. Based on this method, design of the optimal guidance law for soft landing is transferred intothe normal optimal control problem having Canonical form according to the specificfeature of the soft landing through some transformations and deductions. By usingsome mathematic optimization method, the optimal trajectories of soft landing aregiven.
Other, a kind of robust switched model reference tracking controller is developedbased on the combination of the model reference control theory, switched system,linear inequality matrix method and H∞control theory. The existing condition ofthe controller is described by the linear matrix inequality. We linearize the kinematcismodel of the lander and the normal trajectory to construct the normal system model theswitched system with uncertainties for lander. Then, we utilize the proposed resultsof model reference control to solve the tracking controller of the soft landing.
For the fixed-point soft landing problem, the paper give a method of design-ing the feedback controller based on the intersecting method. The kinematcis modelof the lander is given based on the coordinate of the fixed-point, and transformedinto the linear system by using the rational approximation. The feedback controllersignals include the position and velocity signals of the lander in the fixed-point co-ordinate. Then, we utilize the quadratic linear regulating method of systems withinput constraints to solve the optimal feedback controller. The great advantage of thismethod is that the feedback controller can be directly realized by the RV of landerso that it can greatly reduce the numeration of the navigation system. Other, the pro-posed method transforms the soft landing problem into the stabilization problem ofthe control system, then the complexity of the problem is reduced greatly. At last, thesimulating results illustrate the feasibility and advantage of the proposed method.
At last, the feasibility and advantages of the guidance laws including fixed-pointlanding and unfixed-point landing are confirmed by the STK simulating results. At thesame time, an analyzation of the results given by STK are described, which illustratesthe performance of the proposed methods.
本文研究的重点是设计登月飞行器软着陆的制导律和控制律。主要内容可以分为两大部分,一部分是非定点软着陆的制导与控制问题,另一部分是定点软着陆的制导与控制问题。对于非定点软着陆的制导和控制,主要采用最优显式制导和反馈跟踪控制的思路。文中分别给出两种参数化的最优显式制导律设计方法,为保证飞行器在实际飞行中受干扰的作用下仍能很好的跟踪已设计的标称软着陆轨道,文中给出一种基于切换系统模型参考的跟踪控制方法。对于定点软着陆控制问题,给出一种基于交会对接思想的定点软着陆反馈控制方法。最后,利用STK仿真平台对前面所提出的制导律方法进行仿真验证,并对仿真结果进行细致的分析。具体内容如下:
首先,对于参数化显式制导律设计方法,文中利用经典最优控制理论中的Eular方程,构造出一种迭代的分段恒值的参数化最优控制律。该控制律可以无限逼近全局最优解,且其参数个数与系统的阶数相同。将这种参数化控制律代入原系统,可将原来复杂的最优控制问题转化为一系列参数优化问题。考虑到很多经典优化算法都要利用指标泛函和约束泛函关于参数的梯度,文中严密推导出该参数化控制律的参数梯度显式公式。最后,分别利用遗传算法和BFGS优化算法优化给出仿真结果,并对不同工况下的仿真结果进行轨道特性分析,对于软着陆过程中控制律参数变化对软着陆轨道特性的影响进行详细的描述。
其次,考虑到实际中的发动机多为常推力的情况,文中利用参数化控制及强化技术构造出一种分段的参数化最优控制律,使软着陆问题得到很好的解决,具有较强的工程意义。参数化控制及强化技术的主要思想是利用若干个分段的常数去逼近最优解。在此情况下,最优控制问题将转化为一系列参数优化问题。利用经典的参数优化方法即可求得最优控制函数的一个近似解。通过不断的增加参数的个数,缩小每段参数的持续时间,重复优化就可以得到一组无线逼近连续最优解的参数化解序列。本文将登月飞行器软着陆的最优显式制导律设计问题进行若干变换后,构造成具有Canonical标准型的最优控制问题。然后,利用上述参数化控制及强化技术给出最优显式制导律的设计方法。
考虑到实际飞行过程中扰动的作用,本文给出一种鲁棒切换模型参考跟踪控制器设计方法,该方法基于鲁棒模型参考理论并结合切换系统的H∞控制方法。文中利用线性矩阵不等式给出该控制器存在的条件。将其应用到软着陆的跟踪控制问题,取得满意的效果。
针对定点软着陆问题,本文给出一种基于交会对接思想的软着陆控制方法,该方法将软着陆控制问题转化为线性受限系统的二次调解问题,通过飞行器相对于落点的位置和速度状态反馈实现软着陆控制。文中首先以预定落点为坐标原点建立坐标系,求得飞行器的非线性运动学模型。经过合理的近似后得到相应的线性系统,考虑到燃料的最优,控制推力受限以及状态受限的要求文中利用线性受限系统二次调节理论给出最优状态反馈控制器设计方法。该方法的优点是可以通过直接的速度和位置状态反馈实现控制,控制器形式简单,且易于导航计算,可以大大减少导航计算量。另外,该方法在初始建立坐标系时,利用飞行器相对落点的速度和位置偏差作为飞行器的状态,这样将软着陆控制问题转化成控制系统稳定性问题,使问题的复杂度大大降低,理论上更容易求解。
最后,本文利用STK仿真平台将前面章节中所得出的软着陆制导律进行仿真验证。本文详细的给出了仿真的全过程,并对结果进行了科学的分析。从仿真结果可以看出,前面所提出的定点软着以及非定点软着陆制导律设计方法能够有效的实现软着陆控制。
Navigation, guidance and control are the key techniques for lunar landing plan,moreover the soft landing is the first problem of lunar exploration. This dissertationsystematically studies navigation and guidance and control of the lunar landing byoptimal control, optimization, nonlinear control, linear matrix inequality and switchedsystem methods based on precursors’results.
Main works of the paper include the guidance and control law of soft landing,which can be divided into two parts that are the guidance and control of fix-pointsoft landing and that of unfix-point soft landing. The paper use the optimal analyticalguidance law and model reference control method to deal with the unfix-point softlanding problem. For design of the optimal analytical guidance law, the paper devel-ops two kinds of parameterizing guidance law. Considering the exist of disturbance inthe space, a model reference tracking controller is developed to guarantee the landerrobustly to track the normal orbit. For dealing with the fix-point soft landing problem,the paper develops a feedback control method based on the intersecting method. Atlast, STK is used to validate the feasibility and advantage of the proposed method.At first, a recursive piece-wise constant parametric controller is developed byEular equation of the classic optimal control theory in this dissertation, which canbe infinitely approaching the continuous optimal solution. The number of parametersis equal to the system rank. By substituting the controller into the given system, theoriginal optimal control problem is transferred into a parameter selection problem.The analytic gradient functions of the parameters to the cost function and constraintsare developed for some classic optimization methods. Then, the simulating results aregiven based on the genetic algorithm and the BFGS algorithm, respectively.
Essence of the parametric control and the enhancing technique is to use thepiece-wise constant value to approach the optimal solution of optimal control prob-lem. Then, the optimal control problem can be transformed into the parameter selec-tion problem by these methods, moreover the optimal parameters can be solved by thecommon mathematic programming. The approximating precision can be enhanced byincreasing the number of parameters so that an acceptable solution is derived. Based on this method, design of the optimal guidance law for soft landing is transferred intothe normal optimal control problem having Canonical form according to the specificfeature of the soft landing through some transformations and deductions. By usingsome mathematic optimization method, the optimal trajectories of soft landing aregiven.
Other, a kind of robust switched model reference tracking controller is developedbased on the combination of the model reference control theory, switched system,linear inequality matrix method and H∞control theory. The existing condition ofthe controller is described by the linear matrix inequality. We linearize the kinematcismodel of the lander and the normal trajectory to construct the normal system model theswitched system with uncertainties for lander. Then, we utilize the proposed resultsof model reference control to solve the tracking controller of the soft landing.
For the fixed-point soft landing problem, the paper give a method of design-ing the feedback controller based on the intersecting method. The kinematcis modelof the lander is given based on the coordinate of the fixed-point, and transformedinto the linear system by using the rational approximation. The feedback controllersignals include the position and velocity signals of the lander in the fixed-point co-ordinate. Then, we utilize the quadratic linear regulating method of systems withinput constraints to solve the optimal feedback controller. The great advantage of thismethod is that the feedback controller can be directly realized by the RV of landerso that it can greatly reduce the numeration of the navigation system. Other, the pro-posed method transforms the soft landing problem into the stabilization problem ofthe control system, then the complexity of the problem is reduced greatly. At last, thesimulating results illustrate the feasibility and advantage of the proposed method.
At last, the feasibility and advantages of the guidance laws including fixed-pointlanding and unfixed-point landing are confirmed by the STK simulating results. At thesame time, an analyzation of the results given by STK are described, which illustratesthe performance of the proposed methods.
引文
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56 J. Wang, N. G. Cui and D. Liu, Preliminary Study on Minimum-Fuel LunarProbe Trajectories, Flight Dynamics, 2000, 18(2):46-54
57 Y. C. Tian, Automatic control algorithm of spacecraft fixed-point landing, Jour-nal of Astronautics, 1998, 19(2):86-90
58 C. N. Souza, An optimal guidance law for planetary landing, AIAA Guidance,Navigation and Control Conference, USA, 1997, 1376-1381
59 X. Y. Huang, H. T. Cui and P. Y. Cui, An autonomous optical navigation andguidance for soft landing on asteroids, Acta Astronautica, 2004, 54:763-771
60 L. L. Show, J. C. Juang and Y. W. Jan, An LMI-Based Nonlinear AttitudeControl Approach, IEEE Transactions On Control Systems Technology, 2003,11(1):73-83
61 M. Basso and R. Max, Robust attitude orbit control for large ?imsy appendages,IEEE Conference on Decision and Control, 2004, 278-283
62 W. C. Luo, Y. C. Chu and K. V. Ling, H∞tracking control of a rigid spacecraft,Proceedings of American Control Conference, USA, 2004, 2861-2686
63 Colin R. Mcinnes, Path Shaping Guidance For Terminal Lunar Descent, ActaAstronautica, 1995, 36(7):367-377
64 D. Y. Wang, T. S. Li, H. Yan and M. R. Ma, Guidance Control for LunarGravity-Turn Descent, Chinese Space Science And Technology, 2000, 5:17-29
65 G. Q. Zeng, H. Y. Zhao and X. N. Xi, A Study on Soft Landing Orbit of LunarDetector, Chinese Space Science And Technology, 1996, 18(4):40-43
66 M. Xu and J. F. Li, Optimal Control of Lunar Soft Landing, Journal of TsinghuaUniversity(Science and Technology), 2001, 41(8):87-89
67 M. S. James, On the Problem of Optimal Thrust Programming For A Lunar SoftLanding, IEEE Transactions On Automatic Control, 1964, 477-484
68 D. Y. Wang and X. R. Ma, Neuro-optimal Guidance Law for Lunar Soft Land-ing, Systems Engineering and Electronics, 1999, 21(12):31-36
69 H. T. Cui, X. Y. Shi, P. Y. Cui and A. S. Li, Guidance and Control Law for SoftLanding Asteroid, Flight Dynamics, 2002, 20(2):35-38
70 H. T. Cui, X. Y. Shi, P. Y. Cui and A. S. Li, Line-of-Sight Guidance for AdhesionAsteroid, Chinese Journal of Space Science, 2002, 22(3):256-260
71 K. M. Ma, L. J. Chen and Z. C. Wang, Practical Design of Control Law forFlight Vehicle Soft Landing, Missiles and Space Vehicles, 2001, 2:39-43
72 J. Wang, J. F. Li, N. G. Cui and D. Liu, Genetic Algorithm Optimization ofLunar Probe Soft Landing Trajectories, Journal of Tsinghua University (Scienceand Technology), 2003, 43(8):14-36
73 J. Wang, J. F. Li, N. G. Cui and D. Liun, On Initial Point Selection of theContinued-Propulsion Soft-Landing Trajectories of Lunar Probes, EngineeringMechanics, 2003, 20(6):145-148
74 X. G. Ruan, A Nonlinear Neuro-control Scheme for Lunar Soft Landing, Jour-nal of Astronautics, 1998, 19(1):35-43
75 X. G. Ruan and S. F. Guo, Studies on Neuro-Optimal Statefeedback Controland Its Applications to Lunar Soft Landing, Journal of Nanjing University ofAeronautics and Astronautics, 1994, 26(6):721-729
76刘暾,航天器轨道动力学。哈尔滨工业大学,1992,15-65
77蔡宣三,最优化与最优控制,清华大学出版社,1984
78俞立,鲁棒控制-线性矩阵不等式处理方法,清华大学出版社, 2002
79 Y. Kuroda and T. Kurosawa, Accurate Localization in Combination with PlanetObservation and Dead Reckoning for Lunar Rover, Proceedings of the 2004IEEE International Conference on Robotics and Automation, New Orieans,2004, 2092-2097
80 A. Miele, Optimal trajectories for earth-moon-earth ?ight, Acta Astronautica,2001, 49(2):59-71
81 C. Huang, X. G. Hu and X. Li, Design of lunar landing trajectory under con-straints, Chinease astronomy and astrophysics, 2001, 25:464-477
82 R. B. Wilson, A Simplicial Method For Convex Programming, Ph.D. Thesis,Harvard University, Cambridge, MA, 1963
83 S. P. Han, Super linearity convergent variable metric algorithm for general non-linear programming problems, Mathematical Programming, 1967, 11:263-282
84 S. P. Han, A globally convergent method for nonlinear programming, Journalof Optimization Theory and Applications, 1976, 22:297-309
85 M. J. D. Powell, A fast algorithm for nonlinearity constrained optimization cal-culations, Numerical Analysis: Lecture Notes in Mathematics, Springer-Verlag,1978, 6:30-41
86 M. J. D. Powell, The convergence of variable metric methods for nonlinearlyconstrained optimization calculations, Nonlinear Programming, 1978, 3:27-64
87 K. Schittkiwski, On the convergence of a sequential quadratic programmingmethod with an augmented Lagrangian line search function, Math. Operationsand Statistic, Ser. Optimization, 1983, 14:197-216
88 K. Schittkiwski, NLPQL: A FORTRAN Subroutine solving constrained nonlin-ear programming problems, Operations Research Annuals, 1985, 5:485-500
89 C. C. Chiu and P. L. Hsu, A constraint-based genetic algorithm approachfor mining classification rules, IEEE Transaction on Systems, Man andCybernetics-Part C:Application and Reviews, 2005, 35(2):205-221
90 Z. E. Christoforos, G. B. Anastasios, B. T. John and P. Vasilios, A genetic algo-rithm solution approach to hydrothermal coordination problem, IEEE Transac-tion on Power Systems, 2004, 19(2):1356-1365
91 S. S. Sancho and Y. Xin, A hybrid hopfield network-genetic algorithm approachfor the terminal assignment problem, IEEE Transaction on Systems, Man andCybernetics-Part B:Cyberneticas, 2004, 34(6):2343-2354
92 C. C. Lo and W. H. Chang, A multiobejective hybrid genetic algorithm for thecapacitated multipoiint network design problem, IEEE Transaction on Systems,Man and Cybernetics-Part B:Cyberneticas, 2000, 30(2):461-471
93 F. L. Yu, F. Tu and P. R. Krishna, A noval congruent organizational designmethodology using group technology and a nested genetic algorithm, IEEETransaction on Systems, Man and Cybernetics-Part A: Systems and Humans,2006, 36(1):5-19
94 K. M. Eric, and D. W. William, Aircraft antenna coupling minimization usinggenetic algorithm approximations, IEEE Transaction on Aerospace and Elec-tronic Systems, 2004, 40(2):742-752
95 C. W. Lee, and C. S. Yung, Construction of fuzzy systems using least-squaresmethod and genetic algorithm Fuzzy sets and systems, 2003, 137:297-323
96 K. L. Teo, C. J. Goh and K. H. Wong. A Unified Computational Approach toOptimal Control Problems. Longman Scientific and Technical, England, 1991
97 H. W. J. Lee, K. L. Teo, L. S. Jennings and V. Rehbock, Control parametrizationenhancing technique for time optimal control problem, Dynamical Syst. Appl.,1977, 6:243-261
98 K. L. Teo, L. S. Jennings, H. W. Lee and V. Rehbock, The Control Parame-terization Enhancing Transform for Constrained Optimal Control Problems, J.Austral. Math. Soc., 1999, 40:314-335
99 H. W. J. Lee, L. S. Jennings, K. L. Teo and V. Rehbock, Control Parameteriza-tion Enhancing Technique for Time Optimal Control Problems, Dynamic Sys-tem and Applications, 1997, 6:243-262
100 K. L. Teo and L. S. Jennings, Nonlinear Optimal Control Problems with Con-tinuous State Inequality Constraints, Journal of Optimization Theory and Ap-plications, 1989, 63:1-22
101 L. S. Jennings and K. L. Teo, A Computational Algorithm for Functional In-equality Constrained Optimization Problems, Automatica, 1990,26:371-375
102 K. L. Teo, V. Rehbock and L. S. Jennings, A New Computational Algorithm forFunctional Inequality Constrained Optimization Problems, Automatica, 1993,29:789-792
103 W. X. Zheng, B. Vo, A. Cantoni and K. L. Teo, Recursive Procedures for Con-strained Optimization Problems and Its Application to Signal Processing, IEEProceeding: Vision, Image and Signal Processing, 1995, 142:161-168
104 C. B. Sorin and A. D. Raymond, Optimal Control of Switching System, Auto-matica, 2005, 41:11-27
105 X. Xu and P. J. Antsaklis, Optimal control of switched systems: new resultsand open problems, Proceedings of the American control conference, Chicago,2000, 2683-2687
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21 D. Y. Wang,T. S. Li,X. R. Ma and et al., Neuro optimal guidance controlfor Lunar Soft Landing, Journal of Systems Engineering and Electronics, 1999,10(3):222-231
22 D. Y. Wang, T. S. Li and H. Yan, A Suboptimal Fuel Guidance Law For LunarSoft Landing, Journal of Astronautics, 2000, 21(4):55-63
23 A. Miele, Optimal Trajectories For Earth-Moon-Earth Flight, Acta Astronautica, 2001, 49(2)-59-71
24 X. Y. Huang, P. Y. Cui, H. T. Cui and E. J. Luan, A Navigation and GuidanceAlgorithm for Small Celestial Body Landing Using the Gauss-Markov Process,Journal of Astronautics, 2004, 25(3):338-342
25 B. Bussey and S. Paul, Small Spacecraft Exploration of the Moon, Acta Astro-nautica, 2004, 55:637-641
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27 Y. Kian and M. Ernst, Analysis of Parking Orbits and Transfer Trajectories forMission Design of Cis-Lunar Space Stations, Acta Astronautica, 2004, 55:759-771
28 H. J. Wu, X. Z. Feng, Y. D. Jiang and et. al., Acrjet-Engine Performance TestUsing Hydrogen/Nitrogen Mixtures as Propellant, Chinease Space Science andTechnology, 2002, 4:65-72
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31 J. Wang, N. G. Cui and D. Liu, On Constant-amplitude Low-Thrust Lunar ProbeTrajectories, Acta Aeronautica and Astronautica Sinica, 2001, 22(2):6-9
32 G. Q. Zeng, X. N. Xi and X. Ren, A Study On the Optimal Low-thrust OrbitManeuver of Lunar Satellite, Acta Astronomica Sinica, 2000, 41(30):289-299
33 W. Y. Zhou and W. L. Yang, Mid-correction of trans-lunar trajectory of lunarexplorer, Journal of Astrounautics, 2004, 25:89-93
34 T. L. Li, D. Yang and H. T. Cui, One method of calculating cislunar transfertrajectory, Journal of Astronautics, 2003, 24(2):150-156
35 H. T. Cui and P. Y. Cui, Autonomous Navigation and Guidance for Soft LandingAsteroid, Journal of Astronautics, 2002, 23(5):1-10
36 X. Y. Huang, H. T. Cui and P. Y. Cui, An Autonomous Optical Navigation andGuidance for Soft Landing on Asteroids, Acta Astronautica, 2004, 54:763-771
37 L. Yoshihiko, Orbit Determination by Means of Kalman Filter, Proceedings ofthe 1999 IEEE International Conference on Control Applications, USA, 1999,968-972
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54 A. E. Finzi, Automatic Optimum Moon Landing, Proc. of 48th Int . Astronauti-cal Congress , Italy , 1997 , IAF2972
55 D. Y. Wang, T. S. Li and H. Yan, A Suboptimal Fuel Guidance Law For LunarSoft Landing, Journal of Astronautics, 2000, 21(4):55-63
56 J. Wang, N. G. Cui and D. Liu, Preliminary Study on Minimum-Fuel LunarProbe Trajectories, Flight Dynamics, 2000, 18(2):46-54
57 Y. C. Tian, Automatic control algorithm of spacecraft fixed-point landing, Jour-nal of Astronautics, 1998, 19(2):86-90
58 C. N. Souza, An optimal guidance law for planetary landing, AIAA Guidance,Navigation and Control Conference, USA, 1997, 1376-1381
59 X. Y. Huang, H. T. Cui and P. Y. Cui, An autonomous optical navigation andguidance for soft landing on asteroids, Acta Astronautica, 2004, 54:763-771
60 L. L. Show, J. C. Juang and Y. W. Jan, An LMI-Based Nonlinear AttitudeControl Approach, IEEE Transactions On Control Systems Technology, 2003,11(1):73-83
61 M. Basso and R. Max, Robust attitude orbit control for large ?imsy appendages,IEEE Conference on Decision and Control, 2004, 278-283
62 W. C. Luo, Y. C. Chu and K. V. Ling, H∞tracking control of a rigid spacecraft,Proceedings of American Control Conference, USA, 2004, 2861-2686
63 Colin R. Mcinnes, Path Shaping Guidance For Terminal Lunar Descent, ActaAstronautica, 1995, 36(7):367-377
64 D. Y. Wang, T. S. Li, H. Yan and M. R. Ma, Guidance Control for LunarGravity-Turn Descent, Chinese Space Science And Technology, 2000, 5:17-29
65 G. Q. Zeng, H. Y. Zhao and X. N. Xi, A Study on Soft Landing Orbit of LunarDetector, Chinese Space Science And Technology, 1996, 18(4):40-43
66 M. Xu and J. F. Li, Optimal Control of Lunar Soft Landing, Journal of TsinghuaUniversity(Science and Technology), 2001, 41(8):87-89
67 M. S. James, On the Problem of Optimal Thrust Programming For A Lunar SoftLanding, IEEE Transactions On Automatic Control, 1964, 477-484
68 D. Y. Wang and X. R. Ma, Neuro-optimal Guidance Law for Lunar Soft Land-ing, Systems Engineering and Electronics, 1999, 21(12):31-36
69 H. T. Cui, X. Y. Shi, P. Y. Cui and A. S. Li, Guidance and Control Law for SoftLanding Asteroid, Flight Dynamics, 2002, 20(2):35-38
70 H. T. Cui, X. Y. Shi, P. Y. Cui and A. S. Li, Line-of-Sight Guidance for AdhesionAsteroid, Chinese Journal of Space Science, 2002, 22(3):256-260
71 K. M. Ma, L. J. Chen and Z. C. Wang, Practical Design of Control Law forFlight Vehicle Soft Landing, Missiles and Space Vehicles, 2001, 2:39-43
72 J. Wang, J. F. Li, N. G. Cui and D. Liu, Genetic Algorithm Optimization ofLunar Probe Soft Landing Trajectories, Journal of Tsinghua University (Scienceand Technology), 2003, 43(8):14-36
73 J. Wang, J. F. Li, N. G. Cui and D. Liun, On Initial Point Selection of theContinued-Propulsion Soft-Landing Trajectories of Lunar Probes, EngineeringMechanics, 2003, 20(6):145-148
74 X. G. Ruan, A Nonlinear Neuro-control Scheme for Lunar Soft Landing, Jour-nal of Astronautics, 1998, 19(1):35-43
75 X. G. Ruan and S. F. Guo, Studies on Neuro-Optimal Statefeedback Controland Its Applications to Lunar Soft Landing, Journal of Nanjing University ofAeronautics and Astronautics, 1994, 26(6):721-729
76刘暾,航天器轨道动力学。哈尔滨工业大学,1992,15-65
77蔡宣三,最优化与最优控制,清华大学出版社,1984
78俞立,鲁棒控制-线性矩阵不等式处理方法,清华大学出版社, 2002
79 Y. Kuroda and T. Kurosawa, Accurate Localization in Combination with PlanetObservation and Dead Reckoning for Lunar Rover, Proceedings of the 2004IEEE International Conference on Robotics and Automation, New Orieans,2004, 2092-2097
80 A. Miele, Optimal trajectories for earth-moon-earth ?ight, Acta Astronautica,2001, 49(2):59-71
81 C. Huang, X. G. Hu and X. Li, Design of lunar landing trajectory under con-straints, Chinease astronomy and astrophysics, 2001, 25:464-477
82 R. B. Wilson, A Simplicial Method For Convex Programming, Ph.D. Thesis,Harvard University, Cambridge, MA, 1963
83 S. P. Han, Super linearity convergent variable metric algorithm for general non-linear programming problems, Mathematical Programming, 1967, 11:263-282
84 S. P. Han, A globally convergent method for nonlinear programming, Journalof Optimization Theory and Applications, 1976, 22:297-309
85 M. J. D. Powell, A fast algorithm for nonlinearity constrained optimization cal-culations, Numerical Analysis: Lecture Notes in Mathematics, Springer-Verlag,1978, 6:30-41
86 M. J. D. Powell, The convergence of variable metric methods for nonlinearlyconstrained optimization calculations, Nonlinear Programming, 1978, 3:27-64
87 K. Schittkiwski, On the convergence of a sequential quadratic programmingmethod with an augmented Lagrangian line search function, Math. Operationsand Statistic, Ser. Optimization, 1983, 14:197-216
88 K. Schittkiwski, NLPQL: A FORTRAN Subroutine solving constrained nonlin-ear programming problems, Operations Research Annuals, 1985, 5:485-500
89 C. C. Chiu and P. L. Hsu, A constraint-based genetic algorithm approachfor mining classification rules, IEEE Transaction on Systems, Man andCybernetics-Part C:Application and Reviews, 2005, 35(2):205-221
90 Z. E. Christoforos, G. B. Anastasios, B. T. John and P. Vasilios, A genetic algo-rithm solution approach to hydrothermal coordination problem, IEEE Transac-tion on Power Systems, 2004, 19(2):1356-1365
91 S. S. Sancho and Y. Xin, A hybrid hopfield network-genetic algorithm approachfor the terminal assignment problem, IEEE Transaction on Systems, Man andCybernetics-Part B:Cyberneticas, 2004, 34(6):2343-2354
92 C. C. Lo and W. H. Chang, A multiobejective hybrid genetic algorithm for thecapacitated multipoiint network design problem, IEEE Transaction on Systems,Man and Cybernetics-Part B:Cyberneticas, 2000, 30(2):461-471
93 F. L. Yu, F. Tu and P. R. Krishna, A noval congruent organizational designmethodology using group technology and a nested genetic algorithm, IEEETransaction on Systems, Man and Cybernetics-Part A: Systems and Humans,2006, 36(1):5-19
94 K. M. Eric, and D. W. William, Aircraft antenna coupling minimization usinggenetic algorithm approximations, IEEE Transaction on Aerospace and Elec-tronic Systems, 2004, 40(2):742-752
95 C. W. Lee, and C. S. Yung, Construction of fuzzy systems using least-squaresmethod and genetic algorithm Fuzzy sets and systems, 2003, 137:297-323
96 K. L. Teo, C. J. Goh and K. H. Wong. A Unified Computational Approach toOptimal Control Problems. Longman Scientific and Technical, England, 1991
97 H. W. J. Lee, K. L. Teo, L. S. Jennings and V. Rehbock, Control parametrizationenhancing technique for time optimal control problem, Dynamical Syst. Appl.,1977, 6:243-261
98 K. L. Teo, L. S. Jennings, H. W. Lee and V. Rehbock, The Control Parame-terization Enhancing Transform for Constrained Optimal Control Problems, J.Austral. Math. Soc., 1999, 40:314-335
99 H. W. J. Lee, L. S. Jennings, K. L. Teo and V. Rehbock, Control Parameteriza-tion Enhancing Technique for Time Optimal Control Problems, Dynamic Sys-tem and Applications, 1997, 6:243-262
100 K. L. Teo and L. S. Jennings, Nonlinear Optimal Control Problems with Con-tinuous State Inequality Constraints, Journal of Optimization Theory and Ap-plications, 1989, 63:1-22
101 L. S. Jennings and K. L. Teo, A Computational Algorithm for Functional In-equality Constrained Optimization Problems, Automatica, 1990,26:371-375
102 K. L. Teo, V. Rehbock and L. S. Jennings, A New Computational Algorithm forFunctional Inequality Constrained Optimization Problems, Automatica, 1993,29:789-792
103 W. X. Zheng, B. Vo, A. Cantoni and K. L. Teo, Recursive Procedures for Con-strained Optimization Problems and Its Application to Signal Processing, IEEProceeding: Vision, Image and Signal Processing, 1995, 142:161-168
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