Applied Mathematics and Optimization

, Volume 5, Issue 1, pp 231–252 | Cite as

Numerical computation of neighboring optimum feedback control schemes in real-time

  • Hans Josef Pesch


A modification of the theory of neighboring extremals is presented which leads to a new formulation of a linear boundary value problem for the perturbation of the state and adjoint variables around a reference trajectory. On the basis of the multiple shooting algorithm, a numerical method for stable and efficient computation of perturbation feedback schemes is developed. This method is then applied to guidance problems in astronautics. Using as much stored a priori information about the precalculated flight path as possible, the only computational work to be done on the board computer for the computation of a regenerated optimal control program is a single integration of the state differential equations and the solution of a few small systems of linear equations. The amount of computation is small enough to be carried through on modern board computers for real-time. Nevertheless, the controllability region is large enough to compensate realistic flight disturbances, so that optimality is preserved.


Multiple Shooting Flight Path Reference Trajectory Feedback Scheme Optimum Feedback 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    J. V. Breakwell, and Y. C. Ho, On the Conjugate Point Condition for the Control Problem,Int. J. Eng. Sci., 2, 565–579, 1965.Google Scholar
  2. 2.
    J. V. Breakwell, J. L. Speyer, and A. E. Bryson, Optimization and Control of Nonlinear Systems Using the Second Variation,SIAM Journal on Control, Ser. A, 1, 193–223, 1963.Google Scholar
  3. 3.
    A. E. Bryson, and Y. C. Ho,Applied Optimal Control, Ginn and Company, Waltham, Massachusetts, 1969.Google Scholar
  4. 4.
    R. Bulirsch, Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung, Report der Carl-Cranz Gesellschaft, 1971.Google Scholar
  5. 5.
    R. Bulirsch, and J. Stoer, Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods,Numer. Math., 8, 1–13, 1966.Google Scholar
  6. 6.
    R. Bulirsch, J. Stoer, and P. Deuflhard, Numerical Solution of Nonlinear Two-Point Boundary Value Problems I,Handbook Series Approximation, Numer. Math. (to appear). Google Scholar
  7. 7.
    H. J. Diekhoff, P. Lory, H. J. Oberle, H. J. Pesch, P. Rentrop, and R. Seydel, Comparing Routines for the Numerical Solution of Initial Value Problems of Ordinary Differential Equations in Multiple Shooting,Numer. Math., 27, 449–469, 1977.Google Scholar
  8. 8.
    P. Dyer, and S. R. McReynolds, Optimization of Control Systems with Discontinuities and Terminal Constraints,IEEE Transactions on Automatic Control, AC-14, 223–229, 1969.Google Scholar
  9. 9.
    R. England, Error Estimates for Runge-Kutta Type Solutions to Systems of Ordinary Differential Equations,Comp. J., 12, 166–170, 1969.Google Scholar
  10. 10.
    W. H. Enright, R. Bedet, I. Farkas, and T. E. Hull, Test Results on Initial Value Methods for Non-Stiff Ordinary Differential Equations, Tech. Rep. No. 68, Department of Computer Science, University of Toronto, 1974.Google Scholar
  11. 11.
    E. Fehlberg, Classical Fifth-, Sixth-, Seventh-, and Eighth-Order Runge-Kutta Formulas with Stepsize Control,Computing, 4, 93–106, 1969.Google Scholar
  12. 12.
    E. Fehlberg, Low-Order Classical, Runge-Kutta Formulas with Stepsize Control and Their Application to Some Heat Transfer Problems,Computing, 6, 61–71, 1970.Google Scholar
  13. 13.
    E. Fehlberg, Private Communication, 1978.Google Scholar
  14. 14.
    F. R. Gantmacher, Matrizenrechnung II, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959.Google Scholar
  15. 15.
    W. B. Gragg, On Extrapolation Algorithms for Ordinary Initial Value Problems,SIAM J. Num. Anal. Ser. B, 2, 384–403, 1965.Google Scholar
  16. 16.
    T. E. Hull, Numerical Solutions of Initial Value Problems for Ordinary Differential Equations in: Aziz, A. K. (ed.):Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations, Academic Press, London, 1975.Google Scholar
  17. 17.
    H. G. Hussels, Schrittweitensteuerung bei der Integration gewöhnlicher Differentialgleichungen mit Extrapolation, Diplomarbeit, Universität zu Köln, 1973.Google Scholar
  18. 18.
    H. B. Keller, Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, London, 1968.Google Scholar
  19. 19.
    H. J. Kelley, Guidance Theory and Extremal Fields,IRE Transactions on Automatic Control, AC-7, 75–82, 1962.Google Scholar
  20. 20.
    H. J. Kelley, An Optimal Guidance Approximation Theory,IEEE Transactions on Automatic Control, AC-9, 375–380, 1964.Google Scholar
  21. 21.
    I. Lee, Optimal Trajectory, Guidance, and Conjugate Points,Information and Control, 8, 589–606, 1965.CrossRefGoogle Scholar
  22. 22.
    S. R. McReynolds, and A. E. Bryson, A Successive Sweep Method for Solving Optimal Programming Problems, 6th Joint Automatic Control Conference, Troy, N. Y., 551–555, 1965.Google Scholar
  23. 23.
    S. K. Mitter, Successive Approximation Methods for the Solution of Optimal Control Problems,Automatica, 3, 135–149, 1966.Google Scholar
  24. 24.
    H. J. Pesch, Numerische Berechnung optimaler Flugbahnkorrekturen in Echtzeit-Rechnung, Dissertation, Technische Universität Müchen, 1978; see also TUM-Report 7820.Google Scholar
  25. 25.
    W. F. Powers, A Method for Comparing Trajectories in Optimum Linear Perturbation Guidance Schemes,AIAA Journal, 6, 2451–2452, 1968.Google Scholar
  26. 26.
    W. F. Powers, Techniques for Improved Convergence in Neighboring Optimum Guidance,AIAA Journal, 8, 2235–2241, 1970.Google Scholar
  27. 27.
    W. E. Schmitendorf and S. J. Citron, On the Applicability of the Sweep Method to Optimal Control Problems,IEEE Transactions on Automatic Control, AC-14, 69–72, 1969.Google Scholar
  28. 28.
    A. E. Sedgwick, An Effective Variable Order Variable Step Adams Method, Tech. Rep. No. 53, Department of Computer Science, University of Toronto, 1973.Google Scholar
  29. 29.
    J. L. Speyer and A. E. Bryson, A Neighboring Optimum Feedback Control Scheme Based on Estimated Time-to-Go with Application to Re-Entry Flight Paths,AIAA Journal, 6, 769–776, 1968.Google Scholar
  30. 30.
    J. Stoer and R. Bulirsch, Einführung in die Numerische Mathematik II, Heidelberger Taschenbuch, Bd. 114, Springer Verlag, Berlin, Heidelberg, New York, 1973.Google Scholar
  31. 31.
    L. J. Wood, Perturbation Guidance for Minimum Time Flight Paths of Spacecraft, AIAA Paper No. 72-915, 1972.Google Scholar
  32. 32.
    L. J. Wood and A. E Bryson, Second Order Optimality Conditions for Variable End Time Terminal Control Problems,AIAA Journal, 11, 1241–1246, 1973.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Hans Josef Pesch
    • 1
  1. 1.Institut für MathematikTechnischen Universität MünchenMünchen 2Federal Republic of Germany

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