远程制导火箭弹弹道优化方法研究
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摘要
在现代战争中,射程、精度、威力是火箭弹重要的性能指标,其中射程作为最重要的性能指标受到各国的重视。随着现代化武器科学技术飞速发展,世界各国都在试图采用各种技术来有效地增加火箭弹的射程,实现远程精确打击。采用滑翔弹道飞行是一种增加火箭弹射程的方法,但由于受到飞行马赫数、发动机工作时间和发射角度等条件的约束,使得滑翔弹道的弹道特性变得很复杂。弹道优化技术可以在满足约束条件前提下,以射程作为优化指标,给出最优飞行弹道。
本文对远程制导火箭弹弹道优化设计进行研究,根据远程制导火箭弹的飞行特点,将整个弹道分为推力不变段、推力可变段和滑翔段。给出了最优控制问题的一般描述,以及如何将一个最优控制问题进行参数化,并推导了梯度计算公式。针对序列二次规划方法,给出了Kuhn-Tucker(K-T)条件。对于有约束问题采用拟牛顿方法证明了一个非线性规划问题与一个二次规划问题等价,对于这个非线性规划问题的求解,等价于每一次迭代计算中对与之等价的二次规划问题的求解。对于Bk矩阵的更新,利用截断Broyden-Flecher-Goldfarb-Shanno(BFGS)修正方法,进行修正。由于序列二次规划法的局部收敛的缺点,本文采用罚函数和一维搜索相结合的方法使其满足全局收敛,并针对子问题无解以及Maratos效应给出了相应的解决方法。
在不考虑舵的约束条件下,本文以射程为性能优化指标,飞行马赫数、发射角、发动机工作时间、发动机推力、发动机燃料、末端距离地面高度和弹道倾角为约束条件,采用序列二次规划法对远程制导火箭弹飞行弹道进行优化设计并进行数学仿真。仿真结果表明,所设计的弹道在推力不变段、推力可变段和滑翔段均满足性能指标的要求。证明了本文研究方法的正确性和可行性。
In the modern warfare, range, accuracy and power are important performance indicators of the rocket, and the range, as the most important performance indicator, is paid more attention by all nations. With the rapid development of science and technology in the modern weapon, all nations are trying to use a variety of techniques to increase the range of the rocket effectively, and achieve mathematical strike in the long range. The ballistic flight used gliding flight is a way to increase the range of rocket, but because of the constraint conditions included the flight Mach number, running time of engines and the angle of emission, make the ballistic characteristics of ballistic flight become very complex. In the premise of satisfying the constraints, taken the range as the optimized index, the optimization technology of trajectory can give the optimal flight trajectory.
The optimized design of trajectory in long-range guided rocket is studied in this paper, according to flight characteristics of the long-range guided rocket, and the whole trajectory is divided into the section with the same thrust, the section with the convertible thrust and glide section. It is given the general description of optimal control problem, and how to parameterize the optimal control problem and derivation of the gradient formula. To aim at Sequential Quadratic Programming method, the conditions Kuhn-Tucker (KT) are given. For the constrained problems, using quasi-Newton method proves the equivalence of a non-linear programming problem and a quadratic programming problem, and the solution for the non-linear programming problem is equivalent to the solution of the equivalent quadratic programming problem in the calculation of each iterative computation. To the updated matrix named Bk, using amended method named Broyden-Flecher-Goldfarb-Shanno(BFGS) for correction. Because of the drawbacks of the local convergence in Sequential Quadratic Programming method, the associated method combined penalty functions and one-dimensional search method is used to satisfy the global convergence, the corresponding solutions of no solution for Subproblems and Maratos effect are also given in this paper.
Without considering the constraint condition of the rudder, taking the range as the optimized performance indicators, and taking the flight Mach number, launch angle, running hours of engines, thrust of engines, fuel of engines, the end of the height from the ground and trajectory angle as the constraints, the sequential quadratic programming is used to do optimized design and the mathematical simulation of the long-range guided rocket flight trajectory. Simulation results show that the designed trajectory with the same section of thrust, the section with the convertible thrust and glide section satisfies the performance requirements. It proves the correctness and feasibility of research methods of this paper.
本文对远程制导火箭弹弹道优化设计进行研究,根据远程制导火箭弹的飞行特点,将整个弹道分为推力不变段、推力可变段和滑翔段。给出了最优控制问题的一般描述,以及如何将一个最优控制问题进行参数化,并推导了梯度计算公式。针对序列二次规划方法,给出了Kuhn-Tucker(K-T)条件。对于有约束问题采用拟牛顿方法证明了一个非线性规划问题与一个二次规划问题等价,对于这个非线性规划问题的求解,等价于每一次迭代计算中对与之等价的二次规划问题的求解。对于Bk矩阵的更新,利用截断Broyden-Flecher-Goldfarb-Shanno(BFGS)修正方法,进行修正。由于序列二次规划法的局部收敛的缺点,本文采用罚函数和一维搜索相结合的方法使其满足全局收敛,并针对子问题无解以及Maratos效应给出了相应的解决方法。
在不考虑舵的约束条件下,本文以射程为性能优化指标,飞行马赫数、发射角、发动机工作时间、发动机推力、发动机燃料、末端距离地面高度和弹道倾角为约束条件,采用序列二次规划法对远程制导火箭弹飞行弹道进行优化设计并进行数学仿真。仿真结果表明,所设计的弹道在推力不变段、推力可变段和滑翔段均满足性能指标的要求。证明了本文研究方法的正确性和可行性。
In the modern warfare, range, accuracy and power are important performance indicators of the rocket, and the range, as the most important performance indicator, is paid more attention by all nations. With the rapid development of science and technology in the modern weapon, all nations are trying to use a variety of techniques to increase the range of the rocket effectively, and achieve mathematical strike in the long range. The ballistic flight used gliding flight is a way to increase the range of rocket, but because of the constraint conditions included the flight Mach number, running time of engines and the angle of emission, make the ballistic characteristics of ballistic flight become very complex. In the premise of satisfying the constraints, taken the range as the optimized index, the optimization technology of trajectory can give the optimal flight trajectory.
The optimized design of trajectory in long-range guided rocket is studied in this paper, according to flight characteristics of the long-range guided rocket, and the whole trajectory is divided into the section with the same thrust, the section with the convertible thrust and glide section. It is given the general description of optimal control problem, and how to parameterize the optimal control problem and derivation of the gradient formula. To aim at Sequential Quadratic Programming method, the conditions Kuhn-Tucker (KT) are given. For the constrained problems, using quasi-Newton method proves the equivalence of a non-linear programming problem and a quadratic programming problem, and the solution for the non-linear programming problem is equivalent to the solution of the equivalent quadratic programming problem in the calculation of each iterative computation. To the updated matrix named Bk, using amended method named Broyden-Flecher-Goldfarb-Shanno(BFGS) for correction. Because of the drawbacks of the local convergence in Sequential Quadratic Programming method, the associated method combined penalty functions and one-dimensional search method is used to satisfy the global convergence, the corresponding solutions of no solution for Subproblems and Maratos effect are also given in this paper.
Without considering the constraint condition of the rudder, taking the range as the optimized performance indicators, and taking the flight Mach number, launch angle, running hours of engines, thrust of engines, fuel of engines, the end of the height from the ground and trajectory angle as the constraints, the sequential quadratic programming is used to do optimized design and the mathematical simulation of the long-range guided rocket flight trajectory. Simulation results show that the designed trajectory with the same section of thrust, the section with the convertible thrust and glide section satisfies the performance requirements. It proves the correctness and feasibility of research methods of this paper.
引文
1 Betts, J. T. Survey of numerical methods for trajectory optimization. Journal of Guidance Control and Danymics. 1998, 21(2):193-206
2 David, G. Hull. Conversion of Optimal Control Problems into Parameter Optimization Problems. Journal of Guidance Control and Dynamics. 1997, 20(1):57-60
3雍恩米,唐国金,陈磊.飞行器轨迹优化数值方法综述.宇航学报, 2008, 29(2):398-406
4陈聪,关成启,史宏亮.飞行器轨迹优化的直接数值解法综述.战术火箭弹控制技术, 2009, 31(2):34-40
5 David, G. Hull, J. L. Speyer. Optimal Reentry and Plane-Change Trajectories. Journal of the Astronautical Sciences. 1982, 30(2):117-130
6王华,唐国经,雷勇军.有限推力轨迹优化问题的直接打靶法研究.中国空间科学技术, 2003, 10(5):51-56
7 Brain, C. Frabien. Some Tools For the Direct Solution of Optimal Control Problems. Advances in Engineering Software. 1998, 29(1):45-61
8 H.G. Bock, K.J. Plitt. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, IFAC 9th World Congress, Budapest, Hungary. 1984:22-30
9 David, Marcelo, Garza. Application of Automatic Differentiation to Trajectory Optimization via Direct Multiple Shooting. PH. D. Dissertation, The University of Texas at Austin, 2003:11-18
10 Hans, Seywald. Trajectory Optimization Based on Differential Inclusion. Journal of Guidance Control and Dynamics. 1994, 17(3):480-487
11 B. A. Conway, K. M. Larson. Collocation Versus Differential Inclusion in Direct Optimization. Journal of Guidance Control and Dynamics. 1998, 21(5):780-785
12 C. R. Hargraves, S. W. Paris. Direct Trajectory Optimization using Nonlinear Programming and Collocation. Journal of Guidance Control and Dynamics. 1987, 10(4):338-342
13 D. Tieu, W. R. Cluett, A. Penlidis. A Comparison of Collocation Methods for Solving Dynamic Optimization Problems. Compurers Chem Engng. 1995, 19(4):375-381
14 Anhtuan, D. Ngo. A Fuel-Optimal Trajectory for a Constrained Hypersonic Vehicle using a Direct Transcription Method. Proceedings on Aerospace Conference. IEEE 2004, 4:2704-2709
15 Lianghui Tu, Jian-ping Yuan. Reentry Trajectory Optimization Using DirectCollocation Method and Nonlinear Programming. 57th International Astronautical Congress, Valencia, Spain. Oct.2-6,2006
16 Cheng P ,Shen Z , Lavalle S M RRT—Based Trajectory Design for Autonomous Automobiles and Spacecraft. Archives of Control Sciences. 2001, 11(3-4):51-78
17席裕庚.预测控制.北京:国防工业出版社, 1993
18 Mayne, D. Rawling J ,Rao C ,et a1.Constrained model predictive control: Stability and optimality. Automatica. 1987, 36(6):789-814
19 N. X. Vinh, J. S. Chern, C. F. Lin. Phugoid Oscillations in Optimal Reentry Trajectories. Acta Astronautica. 1981, 8:311-324
20 Vasile, Istratie. Optimal Skip Entry with Terminal Maximum Velocity and Heat Constraint. AIAA/ASME Joint Thermophysics and Heat Transfer Conference, 7th, Albuquerque, NM, June 15-18, 1998.AIAA-1998-2457
21 Wilson, R.B. A Simplicial Algorithm for Concave Programming. PH. D. Dissertation Harvard university. 1963:30-50
22 John, T. Betts and William P. Huffman. Path-Constrained Trajectory Optimization Using Sparse Sequential Quadratic Programming. Journal of Guidance Control and Dynamics. 1993, 16(1):59-68
23 V. Brinda, S. Dasgupta, Madan Lal. Trajectory Optimization and Guidance of an Air Breathing Hypersonic Vehicle. 14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference, Canberra, Australia, Nov. 6-9, 2006. AIAA-2006-7997
24 Bellman, R.E. Dynamic Programming.Princeton, USA: Princeton University Press, 1957
25 S.N. Sivannadam, S.N. Deepa.Introduction to Genetic Agorithms. India: PSG College of Technology,2008
26陈刚,万自明,徐敏,陈士橹.飞行器轨迹优化应用遗传算法的参数化与约束处理方法研究.系统仿真学报. 2005, 17(11):2737-2740
27徐成贤,林卫东.无约束连续最优控制问题的离散序列二次规划方法.高等学校计算数学学报, 1995, 6(2):100-107
28 C. R. Hargraves, S.W.Paris,Direct Trajectory Optimization Using Nolinear Programming And Collocation.Journal of Guidance. 1986
29 Paul, J. Enright, Bruce A. Conway. Optimal Finite-Thrust Spacecraft Trajectories Using Collocation And Nolinear Programming.Journal of Guidance. 1991
30 Paul, J. Enright, Bruce A. Conway. Discrete Approximations To Optimal Trajectories Using Direct Transcription And Nolinear Programming.Journal of Guidance. 1992
31 Wayne, A Scheel, Bruce, A. Conway.Optimization Of Very-Low-Thrust,Many-Revolution Spacecraft Trajectories.Journal of Guidance.1994
32 Sean Tang, Bruce A.Conway.Optimization Of Low-Thrust Interplanetary Spacecraft Trajectories Using Collocation And Nolinear Programming.Journal of Guidance. 1995
33 Albert L. Herman, Bruce A. Conway. Direct Optimization Using Collocation Based On High-Order Gauss-Lobatto Quadrature Pulses. 1996
34黄奕勇,张育林.用高精度配点法进行弹道优化.推进技术. 1998, 19(4):66-69
35屈香菊.直接多重打靶法在轨迹优化方面的应用.飞行力学. 1992, 10(1):13-21
36林国华,胡朝江.序列二次规划法解最优控制问题.飞行力学. 1994, 12(4):45-49
37 Chen G,Hu Y,Wan Z M,et al.RLV Reentry Trajectory Multi-ob-jective Optimization Design Based on NSGA-II Algorithm. In. AIAA Atmospheri Flight Mechanis Conference and Exhibit. San Francisco, Calliforina, USA. 2005:1-6.
38 Hu Y,Chen G, Wan ZM, et al. Multi-objective Pareto Collaborative optimization And Its Application. In. 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Portsmouth, Virginia. 2006
39 Akhtar S, Linshu H. An Efficient Evolutionary Multi-Objective Approach for Robust Design of Multi-Stage Space Launch Vehicle. In 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Portsmouth, Virginia. 2006
40 Mettler B, Bachelder E. Combining On-and Offline Optimization Techniques for Efficient Autonomous Vehicle’s Trajectory Planning. In. AIAA Guidance, Navigation, and Control Conference and Exhibit. San Francisco, CA, USA. 2005
41 Kuwata Y, Schouwenaars T, Richards A, et al. Robust Constrained Receding Horizon Control for Trajectory Planning. In. AIAA Guidance, Navigation, and Control Conference and Exhibit. San Francisco, CA, USA.2005
42唐强,张新国,刘锡成.一种用于低空飞行的在线航迹重规划方法.西北工业大学学报. 2005, 23(4):271-275.
43 Thomas H. Bradley, Blake A. Moffitt, David E. Parekh. Energy Management for Fuel Cell Powered Hybri-Electric Aircraft. 7th International Energy Conversion Engineering Conference 2-5 August 2009, Denver, Colorado. AIAA 2009-4590
44 Amer Farhan Rafique, He LinShu, Qasim Zeeshan, Ali Kamran, Khurram Nisar, Wang Xiaowei. Hybrid Optimization Method for Multidisciplinary Design of Air Launched Satellite Launch Vehicle. 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 2-5August 2009, Denver, Colorado. AIAA 2009-5535
2 David, G. Hull. Conversion of Optimal Control Problems into Parameter Optimization Problems. Journal of Guidance Control and Dynamics. 1997, 20(1):57-60
3雍恩米,唐国金,陈磊.飞行器轨迹优化数值方法综述.宇航学报, 2008, 29(2):398-406
4陈聪,关成启,史宏亮.飞行器轨迹优化的直接数值解法综述.战术火箭弹控制技术, 2009, 31(2):34-40
5 David, G. Hull, J. L. Speyer. Optimal Reentry and Plane-Change Trajectories. Journal of the Astronautical Sciences. 1982, 30(2):117-130
6王华,唐国经,雷勇军.有限推力轨迹优化问题的直接打靶法研究.中国空间科学技术, 2003, 10(5):51-56
7 Brain, C. Frabien. Some Tools For the Direct Solution of Optimal Control Problems. Advances in Engineering Software. 1998, 29(1):45-61
8 H.G. Bock, K.J. Plitt. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, IFAC 9th World Congress, Budapest, Hungary. 1984:22-30
9 David, Marcelo, Garza. Application of Automatic Differentiation to Trajectory Optimization via Direct Multiple Shooting. PH. D. Dissertation, The University of Texas at Austin, 2003:11-18
10 Hans, Seywald. Trajectory Optimization Based on Differential Inclusion. Journal of Guidance Control and Dynamics. 1994, 17(3):480-487
11 B. A. Conway, K. M. Larson. Collocation Versus Differential Inclusion in Direct Optimization. Journal of Guidance Control and Dynamics. 1998, 21(5):780-785
12 C. R. Hargraves, S. W. Paris. Direct Trajectory Optimization using Nonlinear Programming and Collocation. Journal of Guidance Control and Dynamics. 1987, 10(4):338-342
13 D. Tieu, W. R. Cluett, A. Penlidis. A Comparison of Collocation Methods for Solving Dynamic Optimization Problems. Compurers Chem Engng. 1995, 19(4):375-381
14 Anhtuan, D. Ngo. A Fuel-Optimal Trajectory for a Constrained Hypersonic Vehicle using a Direct Transcription Method. Proceedings on Aerospace Conference. IEEE 2004, 4:2704-2709
15 Lianghui Tu, Jian-ping Yuan. Reentry Trajectory Optimization Using DirectCollocation Method and Nonlinear Programming. 57th International Astronautical Congress, Valencia, Spain. Oct.2-6,2006
16 Cheng P ,Shen Z , Lavalle S M RRT—Based Trajectory Design for Autonomous Automobiles and Spacecraft. Archives of Control Sciences. 2001, 11(3-4):51-78
17席裕庚.预测控制.北京:国防工业出版社, 1993
18 Mayne, D. Rawling J ,Rao C ,et a1.Constrained model predictive control: Stability and optimality. Automatica. 1987, 36(6):789-814
19 N. X. Vinh, J. S. Chern, C. F. Lin. Phugoid Oscillations in Optimal Reentry Trajectories. Acta Astronautica. 1981, 8:311-324
20 Vasile, Istratie. Optimal Skip Entry with Terminal Maximum Velocity and Heat Constraint. AIAA/ASME Joint Thermophysics and Heat Transfer Conference, 7th, Albuquerque, NM, June 15-18, 1998.AIAA-1998-2457
21 Wilson, R.B. A Simplicial Algorithm for Concave Programming. PH. D. Dissertation Harvard university. 1963:30-50
22 John, T. Betts and William P. Huffman. Path-Constrained Trajectory Optimization Using Sparse Sequential Quadratic Programming. Journal of Guidance Control and Dynamics. 1993, 16(1):59-68
23 V. Brinda, S. Dasgupta, Madan Lal. Trajectory Optimization and Guidance of an Air Breathing Hypersonic Vehicle. 14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference, Canberra, Australia, Nov. 6-9, 2006. AIAA-2006-7997
24 Bellman, R.E. Dynamic Programming.Princeton, USA: Princeton University Press, 1957
25 S.N. Sivannadam, S.N. Deepa.Introduction to Genetic Agorithms. India: PSG College of Technology,2008
26陈刚,万自明,徐敏,陈士橹.飞行器轨迹优化应用遗传算法的参数化与约束处理方法研究.系统仿真学报. 2005, 17(11):2737-2740
27徐成贤,林卫东.无约束连续最优控制问题的离散序列二次规划方法.高等学校计算数学学报, 1995, 6(2):100-107
28 C. R. Hargraves, S.W.Paris,Direct Trajectory Optimization Using Nolinear Programming And Collocation.Journal of Guidance. 1986
29 Paul, J. Enright, Bruce A. Conway. Optimal Finite-Thrust Spacecraft Trajectories Using Collocation And Nolinear Programming.Journal of Guidance. 1991
30 Paul, J. Enright, Bruce A. Conway. Discrete Approximations To Optimal Trajectories Using Direct Transcription And Nolinear Programming.Journal of Guidance. 1992
31 Wayne, A Scheel, Bruce, A. Conway.Optimization Of Very-Low-Thrust,Many-Revolution Spacecraft Trajectories.Journal of Guidance.1994
32 Sean Tang, Bruce A.Conway.Optimization Of Low-Thrust Interplanetary Spacecraft Trajectories Using Collocation And Nolinear Programming.Journal of Guidance. 1995
33 Albert L. Herman, Bruce A. Conway. Direct Optimization Using Collocation Based On High-Order Gauss-Lobatto Quadrature Pulses. 1996
34黄奕勇,张育林.用高精度配点法进行弹道优化.推进技术. 1998, 19(4):66-69
35屈香菊.直接多重打靶法在轨迹优化方面的应用.飞行力学. 1992, 10(1):13-21
36林国华,胡朝江.序列二次规划法解最优控制问题.飞行力学. 1994, 12(4):45-49
37 Chen G,Hu Y,Wan Z M,et al.RLV Reentry Trajectory Multi-ob-jective Optimization Design Based on NSGA-II Algorithm. In. AIAA Atmospheri Flight Mechanis Conference and Exhibit. San Francisco, Calliforina, USA. 2005:1-6.
38 Hu Y,Chen G, Wan ZM, et al. Multi-objective Pareto Collaborative optimization And Its Application. In. 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Portsmouth, Virginia. 2006
39 Akhtar S, Linshu H. An Efficient Evolutionary Multi-Objective Approach for Robust Design of Multi-Stage Space Launch Vehicle. In 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Portsmouth, Virginia. 2006
40 Mettler B, Bachelder E. Combining On-and Offline Optimization Techniques for Efficient Autonomous Vehicle’s Trajectory Planning. In. AIAA Guidance, Navigation, and Control Conference and Exhibit. San Francisco, CA, USA. 2005
41 Kuwata Y, Schouwenaars T, Richards A, et al. Robust Constrained Receding Horizon Control for Trajectory Planning. In. AIAA Guidance, Navigation, and Control Conference and Exhibit. San Francisco, CA, USA.2005
42唐强,张新国,刘锡成.一种用于低空飞行的在线航迹重规划方法.西北工业大学学报. 2005, 23(4):271-275.
43 Thomas H. Bradley, Blake A. Moffitt, David E. Parekh. Energy Management for Fuel Cell Powered Hybri-Electric Aircraft. 7th International Energy Conversion Engineering Conference 2-5 August 2009, Denver, Colorado. AIAA 2009-4590
44 Amer Farhan Rafique, He LinShu, Qasim Zeeshan, Ali Kamran, Khurram Nisar, Wang Xiaowei. Hybrid Optimization Method for Multidisciplinary Design of Air Launched Satellite Launch Vehicle. 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 2-5August 2009, Denver, Colorado. AIAA 2009-5535