• Previous Article
    Modelling and optimal control of blood glucose levels in the human body
  • JIMO Home
  • This Issue
  • Next Article
    Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem
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

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.
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, (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. 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. 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. 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. Google Scholar

[6]

S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach,, Third Edition, (1981). Google Scholar

[7]

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

[8]

A. Friedman, Differential Games,, American Mathematical Society, (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, (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, 19 (1996), 592. doi: 10.2514/3.21662. Google Scholar

[11]

K. Horie, Collocation with Nonlinear Programming for Two-Sided Flight Path Optimization,, Ph.D. Thesis, (2002). Google Scholar

[12]

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

[13]

R. Isaacs, Differential Games,, John Wiley and Sons, (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, (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. 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. 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. 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. 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. 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, 2 (2012), 571. 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. 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, 31 (2008), 1111. Google Scholar

[24]

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

[25]

K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems,, Annals of Operations Research, 5 (1986), 485. 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. 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, (1999), 651. doi: 10.2514/6.1999-4065. Google Scholar

[28]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Third Edition, (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, (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. 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, (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. 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. 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. 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. Google Scholar

[6]

S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach,, Third Edition, (1981). Google Scholar

[7]

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

[8]

A. Friedman, Differential Games,, American Mathematical Society, (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, (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, 19 (1996), 592. doi: 10.2514/3.21662. Google Scholar

[11]

K. Horie, Collocation with Nonlinear Programming for Two-Sided Flight Path Optimization,, Ph.D. Thesis, (2002). Google Scholar

[12]

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

[13]

R. Isaacs, Differential Games,, John Wiley and Sons, (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, (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. 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. 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. 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. 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. 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, 2 (2012), 571. 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. 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, 31 (2008), 1111. Google Scholar

[24]

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

[25]

K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems,, Annals of Operations Research, 5 (1986), 485. 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. 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, (1999), 651. doi: 10.2514/6.1999-4065. Google Scholar

[28]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Third Edition, (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, (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. doi: 10.3934/jimo.2009.5.705. Google Scholar

[1]

Martino Bardi, Shigeaki Koike, Pierpaolo Soravia. Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 361-380. doi: 10.3934/dcds.2000.6.361

[2]

John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018

[3]

Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275

[4]

Li Jin, Hongying Huang. Differential equation method based on approximate augmented Lagrangian for nonlinear programming. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2019053

[5]

Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487

[6]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235

[7]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019045

[8]

Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021

[9]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1

[10]

Yanqin Bai, Chuanhao Guo. Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems. Journal of Industrial & Management Optimization, 2014, 10 (2) : 543-556. doi: 10.3934/jimo.2014.10.543

[11]

Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193

[12]

Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353

[13]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[14]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455

[15]

Hamid Reza Marzban, Hamid Reza Tabrizidooz. Solution of nonlinear delay optimal control problems using a composite pseudospectral collocation method. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1379-1389. doi: 10.3934/cpaa.2010.9.1379

[16]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183

[17]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[18]

Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071

[19]

Wai-Ki Ching, Sin-Man Choi, Min Huang. Optimal service capacity in a multiple-server queueing system: A game theory approach. Journal of Industrial & Management Optimization, 2010, 6 (1) : 73-102. doi: 10.3934/jimo.2010.6.73

[20]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]