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Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems

  • N. D. Botkin
  • V. M. Kein
  • V. S. Patsko
  • V. L. Turova
  • M. A. Zarkh
Chapter
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 115)

Abstract

The majority of numerical methods for solving differential games is based on the backward procedure for computing the solvability sets [1–3]. The solvability set comprises all initial positions of a game such that one player can bring the state vector to a terminal target set for any possible actions of the opposite player. In the case of linear differential games, each step of the backward procedure consists of two operations with convex polyhedra: finding both an algebraic sum and a geometric difference of polyhedra. For two-or three-dimensional state space, these operations were implemented by the authors in terms of computing the convex hull of some piecewise-linear “almost” convex function. For this low-dimensional case, especially effective algorithms, which use information about local violation of convexity, have been created [4]. These algorithms were applied to various problems of an aircraft guidance in the presence of wind disturbances [5–7]. The way used for solving the above mentioned problems includes such steps as linearization of the corresponding nonlinear dynamic system with respect to some nominal trajectory, solving an auxiliary linear differential game, computing optimal strategies, and simulating the nonlinear dynamic system, using the strategies obtained from the auxiliary linear differential game.

Keywords

Differential Games Numerical Methods Backward Procedure Solvability Set Optimization-Based Control System Design 

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References

  1. [1]
    Krasovskii, N.N. and Subbotin, A.I.: Game-Theoretical Control Problems. New York: Springer-Verlag, 1988.zbMATHCrossRefGoogle Scholar
  2. [2]
    Pontryagin, L.S.: Linear differential games. 2, Soviet Math. Dokl., Vol. 8, pp. 910–912, 1967.zbMATHGoogle Scholar
  3. [3]
    Pshenichnii, B.N. and Sagaidak, M.I.: Differential games of prescribed duration. Cybernetics, Vol. 6, No. 2, pp. 72–83, 1970 (in Russian).CrossRefGoogle Scholar
  4. [4]
    Subbotin, A.I. and Patsko, V.S. (Editors): Algorithms and programs for solving linear differential games. Ural Scientific Center, Academy of Sciences of the USSR, Sverdlovsk, 1984 (in Russian).Google Scholar
  5. [5]
    Botkin, N.D., Kein, V.M., Krasov, A.I. and Patsko, V.S.: Control of aircraft lateral motion during landing in the presence of wind disturbances. Institute of Mathematics and Mechanics, Sverdlovsk, Civil Aviation Academy, Leningrad, Report No.81104592/02830078880, VNTI Center, 1983 (in Russian).Google Scholar
  6. [6]
    Botkin, N.D., Kein, V.M., Patsko, V.S. and Turova, V.L.: Aircraft landing control in the presence of windshear. Problems of Control and Information Theory, Vol. 18, No. 4, pp. 223–235, 1989.MathSciNetGoogle Scholar
  7. [7]
    Botkin, N.D., Zarkh, M.A., Kein, V.M., Patsko, V.S. and Turova, V.L.: Differential games and aircraft control problems in the presence of wind disturbances. Izvestia Akademii Nauk. Tekhnicheskaya Kibernetika, No. 1, pp. 68–76, 1993 (in Russian).Google Scholar
  8. [8]
    Bushenkov, V.A. and Lotov, A.V.: Methods and algorithms for linear system analysis on the basis of constructing the generalized attainable sets. J. of Computational Mathematics and Mathematical Physics, Vol. 20, No. 5, pp. 1130–1141, 1980 (in Russian).MathSciNetzbMATHGoogle Scholar
  9. [9]
    Chernikov, S.N.: Lineare Ungleichungen. Verlag Wissenschaft, 1971 (translated from Russian).zbMATHGoogle Scholar
  10. [10]
    Botkin, N.D.: Evaluation of numerical construction error in differential game with fixed terminal time. Problems of Control and Information Theory, Vol. 11, no. 4, pp. 283–295, 1982.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Botkin, N.D. and Patsko ,V.S.: Positional control in a linear differential game. Engineering Cybernetics, Vol. 21, No.4, pp.69–76, 1983.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Chen, Y.H. and Leitmann, G.: Robustness of uncertain systems in the absence of matching assumptions. Int. J. Control, Vol. 45, No. 5, pp. 1527–1542, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Kelly, J.M., Leitmann, G. and Soldatos, A.G.: Robust control of Based-Isolated Structures under Earthquake Excitation. J. of Optimization Theory and Applications, Vol. 53, No. 2, pp. 159–180, 1987.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 1994

Authors and Affiliations

  • N. D. Botkin
    • 1
  • V. M. Kein
    • 2
  • V. S. Patsko
    • 1
  • V. L. Turova
    • 1
  • M. A. Zarkh
    • 1
  1. 1.Institute of Mathematics and MechanicsEkaterinburgRussia
  2. 2.Civil Aviation AcademyS.-PetersburgRussia

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