Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-29T16:31:55.222Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonzero-sum differential game guidance laws in target-missile-defender engagement

Published online by Cambridge University Press:  12 August 2025

J.L. Xie  and
K.M. Ma Open the ORCID record for K.M. Ma [Opens in a new window]
J.L. Xie
Affiliation:
Control and Simulation Center (National Key Laboratory of Complex System Control and Intelligent Agent Cooperation), Harbin Institute of Technology, Harbin, China
K.M. Ma*
Affiliation:
Control and Simulation Center (National Key Laboratory of Complex System Control and Intelligent Agent Cooperation), Harbin Institute of Technology, Harbin, China
*
Corresponding author: K.M. Ma; Email: makemao@hit.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The guidance law design is considered for an engagement scenario of a missile attacking an actively defended target. A three-player nonzero-sum differential game is formulated to describe players’ respective behaviours including the attacking missile’s evading the target’s defender and thereafter attacking the target, the target’s evading the attacking missile under the assistance of the defender, and the defender’s intercepting the attacking missile to assist the target evading the attacking missile. It is considered that the missile hits the target at a specific terminal angle to enhance the destructive effect by attacking the vulnerable regions of the target. The advantages of norm-bounded and linear quadratic differential game guidance laws are combined and the cost functions segmented by relative distances between the players are proposed. The equilibrium solutions of the nonzero-sum differential game are derived and used as the respective guidance laws of the players. Simulations in various engagement scenarios are conducted to verify the reliability and robustness of the design.

Keywords

terminal guidancenonzero-sum differential gameterminal angle constrainttarget-missile-defender engagement

Information

Type
Research Article
Information
The Aeronautical Journal , First View , pp. 1 - 17
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Royal Aeronautical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Faruqi, F. Differential Game Theory with Applications to Missiles and Autonomous Systems Guidance, John Wiley and Sons, Hoboken, NJ, USA, 2017.10.1002/9781119168485CrossRefGoogle Scholar
Luo, B.H., Ji, H.B. and Wang, X.H. Cooperative robust line-of-sight guidance law based on high-gain observers for active defense, Int J Robust Nonlinear Control, 2023, 33, (16), pp 9602–9617.10.1002/rnc.6877CrossRefGoogle Scholar
Isaacs, R. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Wiley, New York and London, 1965.Google Scholar
Friedman, A. Differential Games: A Survey. Courier Corporation. SIAM Review, 1972, 14, (4).10.1137/1014117CrossRefGoogle Scholar
Garcia, E., Casbeer, D.W. and Pachter, M. Cooperative strategies for optimal aircraft defense from an attacking missile, J Guid Control Dyn, 2017, 38, (8), pp 1510–1520.10.2514/1.G001083CrossRefGoogle Scholar
Shalumov, V. and Shima, T. Weapon-target-allocation strategies in multiagent target–missile–defender engagement, J Guid Control Dyn 2017, 40, (10), pp 2452–2464.10.2514/1.G002598CrossRefGoogle Scholar
Rubinsky, S. and Gutman, S. Three-player pursuit and evasion conflict, J Guid Control Dyn, 2014, 37, (1), pp 98–110.10.2514/1.61832CrossRefGoogle Scholar
Ratnoo, A. and Shima, T. Guidance strategies against defended aerial targets, J Guid Control Dyn, 2012, 35, (4), pp 1059–1068.10.2514/1.56924CrossRefGoogle Scholar
Turetsky, V. and Shinar, J. Missile guidance laws based on pursuit-evasion game formulations, Automatica, 2003, 39, (4), pp 607–618.10.1016/S0005-1098(02)00273-XCrossRefGoogle Scholar
Weiss, M., Shima, T., Castaneda, D., et al. Combined and cooperative minimum-effort guidance algorithms in an active aircraft defense scenario, J Guid Control Dyn, 2017, 40, (5), pp 1241–1254.10.2514/1.G002315CrossRefGoogle Scholar
Gutman, S. On optimal guidance for homing missiles, J Guidance Control, 1979, 2, (4), pp 296–300.10.2514/3.55878CrossRefGoogle Scholar
Shinar, J. and Gutman, S. Three-dimensional optimal pursuit and evasion with bounded controls, IEE Trans Autom Control, 1980, 25, (3), pp 492–496.10.1109/TAC.1980.1102372CrossRefGoogle Scholar
Qi, N.M., Sun, Q.L. and Zhao, J. Evasion and pursuit guidance law against defended target, Chin J Aeronaut, 2017, 30, (6), pp 1958–1973.10.1016/j.cja.2017.06.015CrossRefGoogle Scholar
Sun, Q.L., Zhang, C.F., Liu, N., et al. Guidance laws for attacking defended target. Chin J Aeronaut, 2019, 32, (10), pp 2337–2353.10.1016/j.cja.2019.05.011CrossRefGoogle Scholar
Sun, Q.L., Shen, M.H., Gu, X.L., et al. Evasion-pursuit strategy against defended aircraft based on differential game theory, Int J Aerosp Eng, 2019, 2019, pp 1–12.Google Scholar
Sun, Q.L., Qi, N.M. and Huo, M.Y. Optimal strategy for target protection with a defender in the pursuit-evasion scenario, J Def Model Simul, 2018, 15, (3), pp 289–301.10.1177/1548512918756220CrossRefGoogle Scholar
Chao, T., Wang, X.T., Wang, S.Y. and Yang, M. Linear-quadratic and norm-bounded differential game combined guidance strategy against active defense aircraft in three-player engagement, Chin J Aeronaut, 2023, 36, (8), pp 331–350.10.1016/j.cja.2023.04.012CrossRefGoogle Scholar
Liang, H.Z., Wang, J.Y., Wang, Y.H., et al. Optimal guidance against active defense ballistic missiles via differential game strategies, Chin J Aeronaut, 2019, 33, (3), pp 978–989.10.1016/j.cja.2019.12.009CrossRefGoogle Scholar
Ben-Asher, J.Z. and Yaesh, I. Advances in missile guidance theory, In Progress in Astronautics and Aeronautics, 1998, pp 25–90.Google Scholar
Weiss, M. and Shima, T. Linear quadratic optimal control-based missile guidance law with obstacle avoidance. IEE Trans Aerosp Electron Syst, 2019, 55, (1), pp 205–214.10.1109/TAES.2018.2849901CrossRefGoogle Scholar
Kumar, S.R., Weiss, M. and Shima, T. Minimum-effort intercept angle guidance with multiple-obstacle avoidance. J Guid Control Dyn, 2018, 41, (6), pp 1355–1369.10.2514/1.G003223CrossRefGoogle Scholar
Perelman, A. and Shima, T. Cooperative Differential Games Strategies for Active Aircraft Protection from a Homing Missile, J Guid Control Dyn, 2011, 34, (3), pp 761–773.10.2514/1.51611CrossRefGoogle Scholar
Xu, B., Xu, J.L., Li, S., et al. Impact angle constraint guidance against active defense target based on partial differential games, Int J Aerosp Eng, 2024, 2024, pp 1–13.10.1155/2024/9991666CrossRefGoogle Scholar
Hai, Z.L., Jian, Y.W., Jia, Q.L. and Peng, L. Guidance strategies for interceptor against active defense spacecraft in two-on-two engagement, Aerosp Sci Technol, 2020, 96, pp 1–13.Google Scholar
Shalumov, V. Optimal cooperative guidance laws in a multiagent target-missile-defender engagement, J Guid Control Dyn, 2019, 42, (9), pp 1993–2006.10.2514/1.G004054CrossRefGoogle Scholar
Liu, F., Dong, X.W., Li, Q.D. and Ren, Z. Robust multi-agent differential games with application to cooperative guidance. Aerosp Sci Technol, 2021, 111, pp 1–20.10.1016/j.ast.2021.106568CrossRefGoogle Scholar
Turetsky, V., Weiss, M. and Shima, T. A combined linear-quadratic/bounded control differential game guidance law, IEEE Trans Aerosp Electron Syst, 2021, 57, (5), pp 3452–3462.10.1109/TAES.2021.3083574CrossRefGoogle Scholar
Xiong, S.F., Wang, W.H., Liu, X.D., et al. Guidance law against maneuvering targets with intercept angle constraint, ISA Trans, 2014, 53, pp 1332–1342.10.1016/j.isatra.2014.03.007CrossRefGoogle ScholarPubMed
Wang, X.H. and Lu, X. Three-dimensional impact angle constrained distributed guidance law design for cooperative attacks, ISA Trans, 2018, 73, pp 79–90.10.1016/j.isatra.2017.12.009CrossRefGoogle ScholarPubMed
Yang, X.Y., Zhang, Y.C. and Song, S.M. Three-dimensional nonsingular impact angle guidance strategy with physical constraints, ISA Trans, 2022, 131, pp 476–488.10.1016/j.isatra.2022.05.023CrossRefGoogle ScholarPubMed
Shaferman, V. and Shima, T. Linear quadratic guidance laws for imposing a terminal intercept angle, J Guid Control Dyn, 2008, 31, (5), 1400–1412.10.2514/1.32836CrossRefGoogle Scholar
Saurav, A., Kumar, S.R. and Maity, A. Cooperative guidance strategies for aircraft defense with impact angle constraints, AIAA Scitech 2019 Forum, 2019, pp 1–20.10.2514/6.2019-0356CrossRefGoogle Scholar
Xie, J.L. and Ma, K.M. Suboptimal guidance law against maneuvering target with angle and time constraint. Aerosp Sci Technol, 2024, 148, pp 1–13.10.1016/j.ast.2024.109111CrossRefGoogle Scholar
Shaferman, V. and Shima, T. Cooperative optimal guidance laws for imposing a relative intercept angle, J Guid Control Dyn, 2015, 38, (8), pp 1395–1408.10.2514/1.G000568CrossRefGoogle Scholar
Shaferman, V. and Shima, T. Cooperative differential game guidance laws for imposing a relative intercept angle, J Guid Control Dyn, 2017, 40, (10), pp 2465–2480.10.2514/1.G002594CrossRefGoogle Scholar
Zhang, H.G., Cui, L.L. and Luo, Y.H. Near-optimal control for nonzero-sum differential games of continuous-time nonlinear systems using single-network ADP, IEE Trans Cybern, 2013, 43, (1), pp 206–216.10.1109/TSMCB.2012.2203336CrossRefGoogle ScholarPubMed
Luo, B., Wu, H.N., Huang, T.W. and Liu, D.R. Data-based approximate policy iteration for affine nonlinear continuous-time optimal control design, Automatica, 2014, 50, pp 3281–3290.10.1016/j.automatica.2014.10.056CrossRefGoogle Scholar
Sun, J.L., Liu, C.S. and Ye, Q. Robust differential game guidance laws design for uncertain interceptor-target engagement via adaptive dynamic programming, Int J Control, 2017, 90, (5), pp 990–1004.10.1080/00207179.2016.1192687CrossRefGoogle Scholar
Sun, J.L., Liu, C.S. and Zhao, X. Backstepping-based zero-sum differential games for missile-target interception systems with input and output constraints, IET Control Theory Appl, 2018, 12, (2), pp 243–253.10.1049/iet-cta.2017.0501CrossRefGoogle Scholar
Sun, J.L. and Liu, C.S. Distributed zero-sum differential game for multi-agent systems in strict-feedback form with input saturation and output constraint, Neural Netw, 2018, 106, 8–19.10.1016/j.neunet.2018.06.007CrossRefGoogle ScholarPubMed
Zarchan, P. Tactical and Strategic Missile. Process in Astronautics and Aeronautics, 5th edition, AIAA, Virginia, 2006.Google Scholar
Li, Q.C., Yan, T., Gao, M.J., et al. Optimal cooperative guidance strategies for aircraft defense with impact angle constraints, Aerospace, 2022, 710, (9), pp 1–26.Google Scholar
Tan, Z., Fonod, R. and Shima, T. Cooperative guidance law for target pair to lure two pursuers into collision, J Guid Control Dyn, 2018, 41, (8), pp 1687–1699.10.2514/1.G003357CrossRefGoogle Scholar