半直接配点法在航天器追逃问题求解中的应用
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摘要
采用半直接配点法求解时间固定两航天器追逃问题,提出一种新的数值求解追逃双方最优控制策略的方式,避免了求解非线性两点边值问题。在两航天器均为连续小推力假设条件下,以终端距离为支付函数,给出了半直接配点法求解此追逃问题的过程。在此数值方法中,根据半直接转换将微分对策问题转化为一个最优控制问题,由Gauss-Lobbato配点法最终将此最优问题转化为非线性规划问题,继而通过序列二次规划方法求解。这种半直接配点法避免微分对策问题最优策略的必要条件(两点边值问题)求解,并且数值稳定性好。数值仿真给出了追逃双发的最优控制策略和相应的追逃轨迹。
The semi-direct collocation method is adopted for solving the pursuit-evasion problem with fixed timehorizon. A new numerical way to solve the optimal control strategies of the pursuit and evasion spacecraft is proposed such that a two-point boundary value problem is not necessary to be solved. Under the assumption of the continuous low-thrust,the procedure solving such a pursuit-evasion problem is given with the payoff of the terminal distance of two spacecraft. In such a numerical method, the differential game is reduced to an optimal control problem according to the semitransformation. Then,by the Gauss-Lobbato collocation method the optimal control problem is reduced to a nonlinear programming problem which is solved by the sequential quadratic programming method. Such a semi-direct collocation method does not need to solve the necessary condition( a two-point boundary value problem) for the optimal strategies of the differential games,and it is numerically stable. The numerical simulation result shows the optimal control strategies and the associated pursuit-evasion trajectory for a pursuit-evasion problem of spacecraft.
The semi-direct collocation method is adopted for solving the pursuit-evasion problem with fixed timehorizon. A new numerical way to solve the optimal control strategies of the pursuit and evasion spacecraft is proposed such that a two-point boundary value problem is not necessary to be solved. Under the assumption of the continuous low-thrust,the procedure solving such a pursuit-evasion problem is given with the payoff of the terminal distance of two spacecraft. In such a numerical method, the differential game is reduced to an optimal control problem according to the semitransformation. Then,by the Gauss-Lobbato collocation method the optimal control problem is reduced to a nonlinear programming problem which is solved by the sequential quadratic programming method. Such a semi-direct collocation method does not need to solve the necessary condition( a two-point boundary value problem) for the optimal strategies of the differential games,and it is numerically stable. The numerical simulation result shows the optimal control strategies and the associated pursuit-evasion trajectory for a pursuit-evasion problem of spacecraft.
引文
[1] Isaacs R. Differential games[M]. New York:John Wiley and Sons,1965.
[2] Berkovitz L D. Necessary conditions for optimal strategies in a class of differential games and control problems[J]. SIAM Journal on Control and Optimization,1967,5(1):1-24.
[3] Friedman A. Differential games[M]. Rhode Island:American Mathematical Society,1974.
[4] Dickmanns E,Well K. Approximate solution of optimal control problems using third order hermite polynomial functions[C].Optimization Techniques IFIP Technical Conference, Novosibirsk,July 1-7,1974.
[5] Stoer J,Bulirsch R. Introduction to numerical analysis[M].New York:Springer,1983.
[6]张秋华,孙松涛,谌颖,等.时间固定的两航天器追逃策略及数值求解[J].宇航学报,2014,35(5):537-544.[Zhang Qiu-hua, Sun Song-tao, Chen Ying, et al. Strategy and numerical solution of pursuit evasion with fixed duration for two spacecraft[J]. Journal of Astronautics,2014,35(5):537-544.]
[7] Horie K,Conway B. Optimal fighter pursuit-evasion maneuvers found via two-sided optimization[J]. Journal of Guidance,Control,and Dynamics,2006,29(1):105-112.
[8] Pontani M,Conway B A. Optimal interception of evasive missile warheads:numerical solution of the differential game[J].Journal of Guidance,Control,and Dynamics,2008(31):1111-1122.
[9]常燕,陈韵,鲜勇,等.机动目标的空间交会微分对策制导方法[J].宇航学报,2016,37(7):795-801.[Chang Yan,Chen Yun,Xian Yong,et al. Differential game guidance for space rendezvous of maneuvering target[J]. Journal of Astronautics,2016,37(7):795-801.]
[10] Carr W R,Cobb G R,Pachter M,et al. Solution of a pursuitevasion game using a near-optimal strategy[J]. Journal of Guidance,Control,and Dynamics,2018,41(4):841-850.
[11] Sun S T,Zhang Q H,Loxton R,et al. Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit[J]. Journal of Industrial and Management Optimization,2015,11(4):1127-1147.
[12] Herman A,Conway B. Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules[J]. Journal of Guidance,Control,and Dynamics,1996,19(3):592-599.
[13] Gill P,Murray W,Saunders M. User’s guide for SNOPT version7:software for large-scale nonlinear programming[R]. USA:University of California,San Diego,June 2008.
[2] Berkovitz L D. Necessary conditions for optimal strategies in a class of differential games and control problems[J]. SIAM Journal on Control and Optimization,1967,5(1):1-24.
[3] Friedman A. Differential games[M]. Rhode Island:American Mathematical Society,1974.
[4] Dickmanns E,Well K. Approximate solution of optimal control problems using third order hermite polynomial functions[C].Optimization Techniques IFIP Technical Conference, Novosibirsk,July 1-7,1974.
[5] Stoer J,Bulirsch R. Introduction to numerical analysis[M].New York:Springer,1983.
[6]张秋华,孙松涛,谌颖,等.时间固定的两航天器追逃策略及数值求解[J].宇航学报,2014,35(5):537-544.[Zhang Qiu-hua, Sun Song-tao, Chen Ying, et al. Strategy and numerical solution of pursuit evasion with fixed duration for two spacecraft[J]. Journal of Astronautics,2014,35(5):537-544.]
[7] Horie K,Conway B. Optimal fighter pursuit-evasion maneuvers found via two-sided optimization[J]. Journal of Guidance,Control,and Dynamics,2006,29(1):105-112.
[8] Pontani M,Conway B A. Optimal interception of evasive missile warheads:numerical solution of the differential game[J].Journal of Guidance,Control,and Dynamics,2008(31):1111-1122.
[9]常燕,陈韵,鲜勇,等.机动目标的空间交会微分对策制导方法[J].宇航学报,2016,37(7):795-801.[Chang Yan,Chen Yun,Xian Yong,et al. Differential game guidance for space rendezvous of maneuvering target[J]. Journal of Astronautics,2016,37(7):795-801.]
[10] Carr W R,Cobb G R,Pachter M,et al. Solution of a pursuitevasion game using a near-optimal strategy[J]. Journal of Guidance,Control,and Dynamics,2018,41(4):841-850.
[11] Sun S T,Zhang Q H,Loxton R,et al. Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit[J]. Journal of Industrial and Management Optimization,2015,11(4):1127-1147.
[12] Herman A,Conway B. Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules[J]. Journal of Guidance,Control,and Dynamics,1996,19(3):592-599.
[13] Gill P,Murray W,Saunders M. User’s guide for SNOPT version7:software for large-scale nonlinear programming[R]. USA:University of California,San Diego,June 2008.