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October  2015, 11(4): 1127-1147. doi: 10.3934/jimo.2015.11.1127

Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit

Songtao Sun 1, , Qiuhua Zhang 1, , Ryan Loxton 2, and  Bin Li 3,

1. 

Department of Astronautical Science and Mechanics, Harbin Institute of Technology, Harbin, China, China
2. 

Department of Mathematics and Statistics, Curtin University, Perth 6845
3. 

Department of Mathematics and Statistics, Curtin University, Perth, Australia

Received  November 2013 Revised  July 2014 Published  March 2015

This paper considers a spacecraft pursuit-evasion problem taking place in low earth orbit. The problem is formulated as a zero-sum differential game in which there are two players, a pursuing spacecraft that attempts to minimize a payoff, and an evading spacecraft that attempts to maximize the same payoff. We introduce two associated optimal control problems and show that a saddle point for the differential game exists if and only if the two optimal control problems have the same optimal value. Then, on the basis of this result, we propose two computational methods for determining a saddle point solution: a semi-direct control parameterization method (SDCP method), which is based on a piecewise-constant control approximation scheme, and a hybrid method, which combines the new SDCP method with the multiple shooting method. Simulation results show that the proposed SDCP and hybrid methods are superior to the semi-direct collocation nonlinear programming method (SDCNLP method), which is widely used to solve pursuit-evasion problems in the aerospace field.
Keywords: nonlinear programming, Pursuit-evasion problem, optimal control, multiple shooting method., differential game.
Mathematics Subject Classification: Primary: 91A25; Secondary: 49M37, 49N9.
Citation: Songtao Sun, Qiuhua Zhang, Ryan Loxton, Bin Li. Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1127-1147. doi: 10.3934/jimo.2015.11.1127
References:
[1]

M. Bardi, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[2]

L. D. Berkovitz, Necessary conditions for optimal strategies in a class of differential games and control problems, SIAM Journal on Control and Optimization, 5 (1967), 1-24. doi: 10.1137/0305001.  Google Scholar

[3]

L. D. Berkovitz, The existence of value and saddle point in games of fixed duration, SIAM Journal on Control and Optimization, 23 (1985), 172-196. doi: 10.1137/0323015.  Google Scholar

[4]

M. Breitner, H. Pesch and W. Grimm, Complex differential games of pursuit-evasion type with state constraints, part 2: Necessary conditions for optimal open-loop strategies, Journal of Optimization Theory and Applications, 78 (1993), 443-463. doi: 10.1007/BF00939877.  Google Scholar

[5]

W. H. Clohessy and R. S. Wiltshire, Terminal guidance system for satellite rendezvous, Journal of the Aerospace Sciences, 11 (1960), 653-658. Google Scholar

[6]

S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third Edition, McGraw-Hill, New York, 1981.  Google Scholar

[7]

K. Deb, A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197. Google Scholar

[8]

A. Friedman, Differential Games, American Mathematical Society, Rhode Island, 1974.  Google Scholar

[9]

P. E. Gill, W. Murray, M. Saunders and M. H. Wright, User's Guide for NPSOL (Version 5.0): A Fortran Package for Nonlinear Programming, Systems and Optimization Lab, Stanford University, California, 1998. Google Scholar

[10]

A. L. Herman and B. A. Conway, Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules, Journal of Guidance, Control, and Dynamics, 19 (1996), 592-599. doi: 10.2514/3.21662.  Google Scholar

[11]

K. Horie, Collocation with Nonlinear Programming for Two-Sided Flight Path Optimization, Ph.D. Thesis, University of Illinois at Urbana-Champaign, Champaign, 2002. Google Scholar

[12]

K. Horie and B. A. Conway, Optimal fighter pursuit-evasion maneuvers found via two-sided optimization, Journal of Guidance, Control, and Dynamics, 29 (2006), 105-112. doi: 10.2514/1.3960.  Google Scholar

[13]

R. Isaacs, Differential Games, John Wiley and Sons, New York, 1965.  Google Scholar

[14]

L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3 Optimal Control Software: Theory and User Manual, Department of Mathematics, The University of Western Australia, 2002. Google Scholar

[15]

C. Jiang, Q. Lin, C. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.  Google Scholar

[16]

B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minmax optimal control problems with applications, ANZIAM Journal, 51 (2009), 162-177. doi: 10.1017/S1446181110000040.  Google Scholar

[17]

B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092.  Google Scholar

[18]

B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.  Google Scholar

[19]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275.  Google Scholar

[20]

R. C. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numerical Algebra, Control, and Optimization, 2 (2012), 571-599. doi: 10.3934/naco.2012.2.571.  Google Scholar

[21]

R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

[22]

H. J. Oberle and W. Grimm, BNDSCO: A Program for the Numerical Solution of Optimal Control Problems, Inst. für Angewandte Math. der Univ. Hamburg, 2001. Google Scholar

[23]

M. Pontani and B. A. Conway, Optimal interception of evasive missile warheads: Numerical solution of the differential game, Journal of Guidance, Control, and Dynamics, 31 (2008), 1111-1122. Google Scholar

[24]

M. Pontani and B. A. Conway, Numerical solution of the three-dimensional orbital pursuit-evasion game, Journal of Guidance, Control, and Dynamics, 32 (2009), 474-487. Google Scholar

[25]

K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems, Annals of Operations Research, 5 (1986), 485-500. doi: 10.1007/BF02739235.  Google Scholar

[26]

T. Shima and J. Shinar, Time-varying linear pursuit-evasion game models with bounded controls, Journal of Optimization Theory and Applications, 25 (2002), 607-618. doi: 10.2514/2.4927.  Google Scholar

[27]

J. Shinar and T. Shima, Guidance law evaluation in highly nonlinear scenarios - comparison to linear analysis, in Proceedings of the AIAA Guidance, Navigation, and Control Conference, (1999), 651-661. doi: 10.2514/6.1999-4065.  Google Scholar

[28]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Third Edition, Springer, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[29]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991.  Google Scholar

[30]

L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.  Google Scholar

show all references

References:
[1]

M. Bardi, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[2]

L. D. Berkovitz, Necessary conditions for optimal strategies in a class of differential games and control problems, SIAM Journal on Control and Optimization, 5 (1967), 1-24. doi: 10.1137/0305001.  Google Scholar

[3]

L. D. Berkovitz, The existence of value and saddle point in games of fixed duration, SIAM Journal on Control and Optimization, 23 (1985), 172-196. doi: 10.1137/0323015.  Google Scholar

[4]

M. Breitner, H. Pesch and W. Grimm, Complex differential games of pursuit-evasion type with state constraints, part 2: Necessary conditions for optimal open-loop strategies, Journal of Optimization Theory and Applications, 78 (1993), 443-463. doi: 10.1007/BF00939877.  Google Scholar

[5]

W. H. Clohessy and R. S. Wiltshire, Terminal guidance system for satellite rendezvous, Journal of the Aerospace Sciences, 11 (1960), 653-658. Google Scholar

[6]

S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third Edition, McGraw-Hill, New York, 1981.  Google Scholar

[7]

K. Deb, A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197. Google Scholar

[8]

A. Friedman, Differential Games, American Mathematical Society, Rhode Island, 1974.  Google Scholar

[9]

P. E. Gill, W. Murray, M. Saunders and M. H. Wright, User's Guide for NPSOL (Version 5.0): A Fortran Package for Nonlinear Programming, Systems and Optimization Lab, Stanford University, California, 1998. Google Scholar

[10]

A. L. Herman and B. A. Conway, Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules, Journal of Guidance, Control, and Dynamics, 19 (1996), 592-599. doi: 10.2514/3.21662.  Google Scholar

[11]

K. Horie, Collocation with Nonlinear Programming for Two-Sided Flight Path Optimization, Ph.D. Thesis, University of Illinois at Urbana-Champaign, Champaign, 2002. Google Scholar

[12]

K. Horie and B. A. Conway, Optimal fighter pursuit-evasion maneuvers found via two-sided optimization, Journal of Guidance, Control, and Dynamics, 29 (2006), 105-112. doi: 10.2514/1.3960.  Google Scholar

[13]

R. Isaacs, Differential Games, John Wiley and Sons, New York, 1965.  Google Scholar

[14]

L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3 Optimal Control Software: Theory and User Manual, Department of Mathematics, The University of Western Australia, 2002. Google Scholar

[15]

C. Jiang, Q. Lin, C. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.  Google Scholar

[16]

B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minmax optimal control problems with applications, ANZIAM Journal, 51 (2009), 162-177. doi: 10.1017/S1446181110000040.  Google Scholar

[17]

B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092.  Google Scholar

[18]

B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.  Google Scholar

[19]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275.  Google Scholar

[20]

R. C. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numerical Algebra, Control, and Optimization, 2 (2012), 571-599. doi: 10.3934/naco.2012.2.571.  Google Scholar

[21]

R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

[22]

H. J. Oberle and W. Grimm, BNDSCO: A Program for the Numerical Solution of Optimal Control Problems, Inst. für Angewandte Math. der Univ. Hamburg, 2001. Google Scholar

[23]

M. Pontani and B. A. Conway, Optimal interception of evasive missile warheads: Numerical solution of the differential game, Journal of Guidance, Control, and Dynamics, 31 (2008), 1111-1122. Google Scholar

[24]

M. Pontani and B. A. Conway, Numerical solution of the three-dimensional orbital pursuit-evasion game, Journal of Guidance, Control, and Dynamics, 32 (2009), 474-487. Google Scholar

[25]

K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems, Annals of Operations Research, 5 (1986), 485-500. doi: 10.1007/BF02739235.  Google Scholar

[26]

T. Shima and J. Shinar, Time-varying linear pursuit-evasion game models with bounded controls, Journal of Optimization Theory and Applications, 25 (2002), 607-618. doi: 10.2514/2.4927.  Google Scholar

[27]

J. Shinar and T. Shima, Guidance law evaluation in highly nonlinear scenarios - comparison to linear analysis, in Proceedings of the AIAA Guidance, Navigation, and Control Conference, (1999), 651-661. doi: 10.2514/6.1999-4065.  Google Scholar

[28]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Third Edition, Springer, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[29]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991.  Google Scholar

[30]

L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.  Google Scholar

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