Abstract
We investigate the most critical final homing phase for one intercepting missile versus one maneuverable ballistic missile. In the worst case a future ballistic missile will know the intercepting missile’s guidance scheme and could maximize the minimum (miss) distance. An intercepting missile with three-dimensional proportional navigation guidance is studied. An optimal small guidance upgrade for this missile is calculated as a closed-loop solution of a dynamic Stackelberg game. The intercepting missile acts as leader, whereas the ballistic missile advantageously acts as follower. An optimal small upgrade is approximated numerically along very many optimal trajectories that cover the relevant final homing scenarios. The trajectories are computed efficiently by a direct collocation method. Along the trajectories a linearized minimum—maximum principle using the adjoint variables’ estimates is applied. An upgrade with nearest points on neighboring optimal trajectories is synthesized. Alternative synthesis approaches with local Taylor-series expansions with global smoothing or with artificial neural networks are discussed. Simulations for many different scenarios show the improved interception capability for the upgraded guidance of the intercepting missile.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adler, F. P. Missile guidance by three-dimensional proportional navigation, J. Appl. Phys. 27, 1954.
Basar, T. and Olsder, G. J. Dynamic Noncooperative Game Theory, Academic Press, London, 1982 (2nd edition, 1995).
Breitner, M. H. Robust optimale Rückkopplungssteuerungen gegen unvorhersehbare Einflüsse: Differentialspielansatz, numerische Berechnung und Echtzeitapproximation. Extended Ph.d. thesis, VDI Fortschritt-Bericht 596, Series 8 “Mess -, Steuerungsund Regelungstechnik,” VDI-Verlag, Düsseldorf, 1995.
Breitner, M. H. and Heim, A. Robust optimal control of a reentering space shuttle, in DGLR-Jahrbuch 1995, Deutsche Gesellschaft für Luft-und Raumfahrt, Bonn, 1995
Breitner, M. H. and Pesch, H. J. Reentry trajectory optimization under atmospheric uncertainty as a differential game, Ann. Inter. Soc. Dyn. Games, 1, 1994.
Breitner, M. H., Pesch, H. J., and Grimm, W. Complex differential games of pursuit-evasion type with state constraints, Part 1: Necessary conditions for optimal open-loop strategies, Part 2: Numerical computation of optimal open-loop strategies, J. Opt. Th. Appl., 78, 1993.
Ehtamo, H. and Raivio, T. A feasible direction method for saddlepoint problems. Proc. 8th Intl. Symp. Dyn. Games and Appl., Vaals, 1998.
Imado, F. Some aspects of a realistic three-dimensional pursuit-evasion game. J. Guidance, Cont. Dyn., 16, 1993.
Imado, F. and Ishihara, T. Pursuit-evasion geometry analysis between two missiles and an aircraft, Comp. Math. Appl.. 26, 1993.
Imado, F., Kuroda, T., and Miwa, S. Optimal midcourse guidance for medium-range air-to-air missiles, J. Guidance, Cont. Dyn., 13, 1990.
Imado, F., Kuroda, T. and Tahk, M. J. A trade-off study between conventional and augmented proportional navigation, submitted to J. Guidance, Cont. Dyn., 1998.
Gill, P. E., Murray, W., and Wright, M. H. Practical optimization, Academic Press, London 1997 (1st ed., 1981).
Krasovskii, N. N. and Subbotin, A. I. Game-Theoretical Control Problems, Springer-Verlag, New York, 1988.
Lachner, R., Breitner, M. H., and Pesch, H. J. Three-dimensional air combat analysis—An example for the numerical solution of complex differential games, Ann. Intl. Soc. Dyn. Games, 3, 1996.
Lachner, R., Breitner, M. H., and Pesch, H. J. Real-time collision avoidance against wrong drivers: Differential game approach, numerical solution and synthesis of strategies with neural networks, Ann. Intl. Soc. Dyn. Games, 5, 1998.
Lipman, Y. and Shinar, J A linear pursuit-evasion game with a state constraint for a highly maneuverable evader, Ann. Intl. Soc. Dyn. Games, 3, 1995.
Macfadzean, R. H. M. Surface-based Air Defense System Analysis, Artech House, Boston, 1992.
Pesch, H. J., Gabler, I., Breitner, M. H., and Miesbach, S. Synthesis of optimal strategies for differential games by neural networks, Ann. Intl. Soc. Dyn. Games. 3, 1996.
Rettig, U. Verbesserung von Regelalgorithmen mit direkten numerischen Optimierungsverfahren, Diploma thesis, Technische Universität Clausthal, ClausthalZellerfeld, 1998.
Shinar, J. and Davidovitz, A. A two-target game analysis in line-of-sight coordinates, Comp. Math. Appl., 13, 1987.
Shinar, J. and Gutman, S. Three-dimensional optimal pursuit and evasion with bounded controls, IEEE Trans. Auto. Cont., 25, 1980.
Shinar, J. and Shima, T. A new application of Bernhard’s certainty equivalence principle, Proc. 8th Intl. Symp. Dyn. Games Appl., Vaals, 1998.
Shinar, J. and Tabak, R. New results in optimal missile avoidance analysis, J. Guidance, Cont. Dyn., 17, 1994.
Shinar, J. and Visser, H. G. Synthesis and validation of feedback guidance laws for air-to-air interceptions, Cont. Dyn. Sys., 38, 1990.
Shinar, J., Well, K. H., and Järmark, B., Near-Optimal Feedback Control for Three-dimensional Interceptions, 15th ICAS Congress, London, 1986.
Stoer, J. and Bulirsch, R. Introduction to Numerical Analysis, Springer-Verlag, New York 1993.
von Stryk, O. Numerische Lösung optimaler Steuerungsprobleme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variablen. Ph.d. thesis, Fortschritt-Bericht 441, Series 8 “Meß-, Steuerungs-und Regelungstechnik,” VDI-Verlag, Düsseldorf, 1994.
von Stryk, O. Numerical solution of optimal control problems by direct collocation, in Optimal Control, Calculus of Variations, Optimal Control Theory and Numerical Methods, Birkhäuser, Intl. Series of Numerical Mathematics, 111, Basel, 1993.
Zakharov, V. V. Stackelberg differential games and problem of time consistency, Intl. Yearbook Game Th. App., 1, Nova Science Publishers, New York, 1993.
Zarchan, P. Tactical and Strategic Missile Guidance, American Institute of Aeronautics and Astronautics, Progress in Astronautics and Aeronautics 157, Washington, DC. 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Breitner, M.H., Rettig, U., von Stryk, O. (2001). On Optimal Missile Guidance Upgrades with Dynamic Stackelberg Game Linearizations. In: Altman, E., Pourtallier, O. (eds) Advances in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol 6. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0155-7_16
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0155-7_16
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6637-2
Online ISBN: 978-1-4612-0155-7
eBook Packages: Springer Book Archive