Optimal planar interception with fixed end conditions: Closed-form solution

  • V. Y. Glizer
Contributed Papers

Abstract

A planar constant-speed interception with prescribed end conditions is analyzed. The performance index is the time of capture penalized by the control energy. For this problem, the optimal control of the pursuer is obtained in closed form, based on solving a set of nonlinear algebraic equations involving elliptic integrals. The construction of the solution is inspired by the singularly perturbed structure of the nondimensional equations of motion.

Key Words

Planar interception fixed end conditions optimal control singular perturbations 

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References

  1. 1.
    Isaacs, R.,Differential Games, John Wiley and Sons, New York, New York, 1967.Google Scholar
  2. 2.
    Simakova, E. N.,Differential Pursuit Game, Automation and Remote Control, No. 2, pp. 173–181, 1967.Google Scholar
  3. 3.
    Green, A., Shinar, J., andGuelman, M.,Game-Optimal Guidance Law Synthesis for Short-Range Missiles, Journal of Guidance, Control, and Dynamics, Vol. 15, No. 1, pp. 191–197, 1992.Google Scholar
  4. 4.
    Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMischenko, E. F.,The Mathematical Theory of Optimal Processes, John Wiley, New York, New York, 1962.Google Scholar
  5. 5.
    Bryson, A. E., Jr. andHo, Y.,Applied Optimal Control, Hemisphere Publishing Corporation, Washington, DC, 1975.Google Scholar
  6. 6.
    Cockayne, E.,Plane Pursuit with Curvature Constraints, SIAM Journal on Applied Mathematics, Vol. 15, pp. 1511–1516, 1967.Google Scholar
  7. 7.
    Clements, J. C.,Minimum-Time Turn Trajectories to Fly-to Points, Optimal Control Applications and Methods, Vol. 11, No. 1, pp. 39–50, 1990.Google Scholar
  8. 8.
    Glizer, V. Y., andShinar, J.,On the Structure of a Class of Time-Optimal Trajectories, Optimal Control Applications and Methods, Vol. 14, No. 4, pp. 271–279, 1993.Google Scholar
  9. 9.
    Guelman, M., andShinar, J.,Optimal Guidance Law in the Plane, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 4, pp. 471–476, 1984.Google Scholar
  10. 10.
    Hull, D. G., Radke, J. J., andMack, R. E.,Time-to-Go Prediction for Homing Missiles Based on Minimum-Time Intercepts, Journal of Guidance, Control, and Dynamics, Vol. 14, No. 4, pp. 865–871, 1991.Google Scholar
  11. 11.
    Shinar, J., andSteinberg, D.,Analysis of Optimal Evasive Maneuvers Based on a Linearized Two-Dimensional Kinematic Model, Journal of Aircraft, Vol. 14, No. 8, pp. 795–802, 1977.Google Scholar
  12. 12.
    Calise, A. J.,Singular Perturbation Methods for Variational Problems in Aircraft Flight, IEEE Transactions on Automatic Control, Vol. 21, No. 3, pp. 345–353, 1976.Google Scholar
  13. 13.
    Shinar, J., andNegrin, M.,An Explicit Feedback Approximation for Medium-Range Interceptions in a Vertical Plane, Optimal Control Applications and Methods, Vol. 4, No. 4, pp. 303–323, 1983.Google Scholar
  14. 14.
    Price, D. B., Calise, A. J., andMoerder, D. D.,Piloted Simulation of an Onboard Trajectory Optimization Algorithm, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 3, pp. 355–360, 1984.Google Scholar
  15. 15.
    Visser, H. G., andShinar, J.,A Highly Accurate Feedback Approximation for Horizontal Variable-Speed Interception, Journal of Guidance, Control, and Dynamics, Vol. 9, No. 6, pp. 691–698, 1986.Google Scholar
  16. 16.
    Menon, P. K. A., andBriggs, M. M.,Near-Optimal Midcourse Guidance for Air-to-Air Missiles, Journal of Guidance, Control, and Dynamics, Vol. 13, No. 4, pp. 596–602, 1990.Google Scholar
  17. 17.
    Sheu, D., Vinh, N. X., andHowe, R. M.,Application of Singular Perturbation Methods for Three-Dimensional Minimum-Time Interception, Journal of Guidance, Control, and Dynamics, Vol. 14, No. 2, pp. 360–367, 1991.Google Scholar
  18. 18.
    Shinar, J., Well, K. M., andJarmark, B.,Near-Optimal Feedback Control for Three-Dimensional Interception, Proceedings of the 15th ICAS Congress, pp. 161–171, 1986.Google Scholar
  19. 19.
    Ioffe, A. D., andTichomirov, V. M.,Theory of Extremal Problems, Nauka, Moscow, Russia, 1974 (in Russian).Google Scholar
  20. 20.
    Ardema, M. D., Editor,Singular Perturbations in Systems and Control, Springer Verlag, New York, New York, 1982.Google Scholar
  21. 21.
    Gradshtein, I. S., andRyzhik, I. M. Tables of Integrals Series, and Products, Academic Press, New York, New York, 1965.Google Scholar
  22. 22.
    Whittaker, E. T., andWatson, G. N.,A Course of Modern Analysis, Cambridge University Press, Cambridge, England, 1927.Google Scholar
  23. 23.
    Dwight, H. B.,Tables of Integrals and Other Mathematical Data, The MacMillan Company, New York, New York, 1961.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • V. Y. Glizer
    • 1
  1. 1.Faculty of Aerospace EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael

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