Optimal Control pp 273-288

# Combining Direct and Indirect Methods in Optimal Control: Range Maximization of a Hang Glider

• Roland Bulirsch
• Edda Nerz
• Hans Josef Pesch
• Oskar von Stryk
Chapter
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 111)

## Abstract

When solving optimal control problems, indirect methods such as multiple shooting suffer from difficulties in finding an appropriate initial guess for the adjoint variables. For, this initial estimate must be provided for the iterative solution of the multipoint boundary-value problems arising from the necessary conditions of optimal control theory. Direct methods such as direct collocation do not suffer from this problem, but they generally yield results of lower accuracy and their iteration may even terminate with a non-optimal solution. Therefore, both methods are combined in such a way that the direct collocation method is at first applied to a simplified optimal control problem where all inequality constraints are neglected as long as the resulting problem is still well-defined. Because of the larger domain of convergence of the direct method, an approximation of the optimal solution of this problem can be obtained easier. The fusion between direct and indirect methods is then based on a relationship between the Lagrange multipliers of the underlying nonlinear programming problem to be solved by the direct method and the adjoint variables appearing in the necessary conditions which form the boundary-value problem to be solved by the indirect method. Hence, the adjoint variables, too, can be estimated from the approximation obtained by the direct method. This first step then facilitates the subsequent extension and competition of the model by homotopy techniques and the solution of the arising boundary-value problems by the indirect multiple shooting method. Proceeding in this way, the high accuracy and reliability of the multiple shooting method, especially the precise computation of the switching structure and the possibility to verify many necessary conditions, is preserved while disadvantages caused by the sensitive dependence on an appropriate estimate of the solution are considerably cut down. This procedure is described in detail for the numerical solution of the maximum-range trajectory optimization problem of a hang glider in an upwind which provides an example for a control problem where appropriate initial estimates for the adjoint variables are hard to find.

## Keywords

Optimal Control Problem Collocation Method Multiple Shooting Lift Coefficient Nonlinear Programming Problem
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.

## References

1. [1]
Bock, H. G. and Plitt, K. J.: A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Proceedings of the 9th IFAC Worldcongress, Budapest, 1984, Vol. IX, Colloquia 14.2, 09.2, 1984.Google Scholar
2. [2]
Bryson, A. E. and Ho, Y. C.: Applied Optimal Control, New York: Hemisphere (Rev. Printing), 1975.Google Scholar
3. [3]
Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung, Carl-Cranz Gesellschaft, Oberpfaffenhofen, Report der Carl-Cranz Gesellschaft, 1971; Munich University of Technology, Department of Mathematics, Munich, Reprint, 1985.Google Scholar
4. [4]
Bulirsch, R. and Callies, R.: Optimal Trajectories for an Ion Driven Spacecraft from Earth to the Planetoid Vesta, Proc. of the AIAA Guidance, Navigation and Control Conference, New Orleans, 1991, AIAA Paper No. 91–2683, 1991.Google Scholar
5. [5]
Bulirsch, R. and Callies, R.: Optimal Trajectories for a Multiple Rendezvous Mission to Asteroids, 42nd International Astronautical Congress, Montreal, 1991, IAF-Paper No. IAF-91–342, 1991.Google Scholar
6. [6]
Bulirsch, R., Chudej, K., and Reinsch, K. D.: Optimal Ascent and Staging of a Two-Stage Space Vehicle System, Jahrestagung der Deutschen Gesellschaft für Luft-und Raumfahrt, Friedrichshafen, 1990, DGLR-Jahrbuch 1990, Vol. 1, 243–249, 1990.Google Scholar
7. [7]
Bulirsch, R. and Chudej, K.: Ascent Optimization of an Airbreathing Space Vehicle, Proc. of the AIAA Guidance, Navigation and Control Conference, New Orleans, 1991, AIAA Paper No. 91–2656, 1991.Google Scholar
8. [8]
Bulirsch, R., Montrone, F., and Pesch, H. J.: Abort Landing in the Presence of a Windshear as a Minimax Optimal Control Problem, Part 1: Necessary Conditions, J. of Optimization Theory and Applications 70, 1–23, 1991.
9. [9]
Bulirsch, R., Montrone, F., and Pesch, H. J.: Abort Landing in the Presence of a Windshear as a Minimax Optimal Control Problem, Part 2: Multiple Shooting and Homotopy, J. of Optimization Theory and Applications 70, 221–252, 1991.Google Scholar
10. [10]
Callies, R.: Optimal Design of a Mission to Neptune, in: Bulirsch, R., Miele, A., Stoer, J., and Well, K. H. (eds): Optimal Control, Proc. of the Conf. in Optimal Control and Variational Calculus, Oberwolfach, 1991, Lecture Notes in Control and Information Sciences, Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer, this issue.Google Scholar
11. [11]
Chernousko, F. L. and Lyubushin, A. A.: Method of Successive Approximation for Solution of Optimal Control Problems, Optimal Control Applications and Methods 3, 101–114, 1982.
12. [12]
Deuflhard, P.: A Relaxation Strategy for the Modified Newton Method, in: Bulirsch, R., Oettli, W., and Stoer, J. (eds.), Optimization and Optimal Control, Proceedings of a Conference Held at Oberwolfach, 1974, Lecture Notes in Mathematics 477, Berlin, Heidelberg, New York: Springer, 59–73, 1975.
13. [13]
Deuflhard, P.: A Modified Newton Method for the Solution of Ill-conditioned Systems of Nonlinear Equations with Application to Multiple Shooting, Numerische Mathematik 22, 289–315, 1974.
14. [14]
Deuflhard, P. and Bader, G.: Multiple Shooting Techniques Revisited, in: Deuflhard, P. and Hairer, E. (eds.), Numerical Treatment of Inverse Problems in Differential and Integral Equations, Proceedings of an International Workshop, Heidelberg, 1982, Progress in Scientific Computing 2, Boston: Birkhäuser, 74–94, 1983.
15. [15]
Gottlieb, R. G.: Rapid Convergence to Optimum Solutions Using a Min-H Strategy, AIAA J. 5, 322–329, 1967.
16. [16]
Hargraves, C. R. and Paris, S. W.: Direct Trajectory Optimization Using Nonlinear Programming and Collocation, AIAA Journal of Guidance and Control 10, 338–342, 1987.
17. [17]
Hiltmann, P.: Numerische Lösung von Mehrpunkt-Randwertproblemen und Aufgaben der optimalen Steuerung mit Steuerfunktionen über endlichdimensionalen Räumen, Munich University of Technology, Department of Mathematics, Doctoral Thesis, 1990.Google Scholar
18. [18]
Horn, K.: Solution of the Optimal Control Problem Using the Software Package STOMP, to appear in: Bernhard, P. and Bourdache-Siguerdidjane, H. (eds.), Proc. of the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, 1989, Oxford: IFAC Publications, 1991.Google Scholar
19. [19]
Jänsch, C. and Paus, M.: Aircraft Trajectory Optimization with Direct Collocation Using Movable Gridpoints, in: Proceedings of the American Control Conference, San Diego, 262–267, 1990.Google Scholar
20. [20]
Jänsch, C., Schnepper, K., and Well, K. H.: Ascent and Descent Trajectory Optimization of Ariane V/Hermes, in: AGARD Conf. Proc. No. 489 on Space Vehicle Flight Mechanics, 75th Symp. of the AGARD Flight Mechanics Panel, Luxembourg, 1989.Google Scholar
21. [21]
Kiehl, M.: Vectorizing the Multiple-Shooting Method for the Solution of Boundary-Value Problems and Optimal-Control Problems, in: Dongarra, J., Duff, I., Gaffney, P., and McKee, S. (eds.), Proceedings of the 2nd International Conference on Vector and Parallel Computing Issues in Applied Research and Development, Troms0, 1988, London: Ellis Horwood, 179–188, 1989.Google Scholar
22. [22]
Kelley, H. J., Kopp, R. E., and Moyer, H. G.: Successive Approximation Techniques for Trajectory Optimization, Proc. Symp. on Vehicle System Optimization, New York, 1961.Google Scholar
23. [23]
Kraft, D.: FORTRAN Computer Programs for Solving Optimal Control Problems, Deutsche Forschungs- und Versuchsanstalt für Luft- und Raumfahrt, Oberpfaffenhofen, Report 80–03, 1980.Google Scholar
24. [24]
Kraft, D.: On Converting Optimal Control Problems into Nonlinear Programming Codes, in: Schittkowski, K. (ed.) Computational Mathematical Programming, Berlin: Springer (NATO ASI Series 15), 261–280, 1985.
25. [25]
Lorenz, J.: Numerical Solution of the Minimum-Time Flight of a Glider Through a Thermal by Use of Multiple Shooting Methods, Optimal Control Applications and Methods 6, 125–140, 1985.
26. [26]
Miele, A.: Gradient Algorithms for the Optimization of Dynamic Systems, in: Leondes, C. T., Control and Dynamic Systems 16, New York: Academic Press, 1–52, 1980.Google Scholar
27. [27]
Nerz, E.: Optimale Steuerung eines Hängegleiters, Munich University of Technology, Department of Mathematics, Diploma Thesis, 1990.Google Scholar
28. [28]
Oberle, H. J.: Numerische Berechnung optimaler Steuerungen von Heizung und Kühlung für ein realistisches Sonnenhausmodell, Habilitationsschrift, Munich University of Technology, Munich, Germany, 1982.Google Scholar
29. [29]
Oberle, H. J.: Numerical Computation of Singular Control Functions for a Two-Link Robot Arm, in: Bulirsch, R., Miele, A., Stoer, J., and Well, K. H. (eds): Optimal Control, Proc. of the Conf. in Optimal Control and Variational Calculus, Oberwolfach, 1986, Lecture Notes in Control and Information Sciences 95, Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer, 244–253, 1987.
30. [30]
Oberle, H. J.: Numerical Computation of Singular Functions in Trajectory Optimization Problems, J. Guidance and Control 13, 153–159, 1990.
31. [31]
Renes, J. J.: On the Use of Splines and Collocation in a Trajectory Optimization Algorithm Based on Mathematical Programming, National Aerospace Laboratory, Amsterdam, Report No. NLR-TR-78016 U, 1978.Google Scholar
32. [32]
Stoer, J. and Bulirsch, R.: Introduction to Numerical Analysis, New York: Springer, 1980.Google Scholar
33. [33]
von Stryk, O.: Numerical Solution of Optimal Control Problems by Direct Collocation, in: Bulirsch, R., Miele, A., Stoer, J., and Well, K. H. (eds): Optimal Control, Proc. of the Conf. in Optimal Control and Variational Calculus, Oberwolfach, 1991, Lecture Notes in Control and Information Sciences, Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer, this issue.Google Scholar
34. [34]
von Stryk, O. and Bulirsch, R.: Direct and Indirect Methods for Trajectory Optimization, to appear in Annals of Operations Research, 1991.Google Scholar
35. [35]
Tolle, H.: Optimierungsverfahren, Berlin: Springer, 1971.Google Scholar
36. [36]
Drachenflieger m agazin, München: Ringier Verlag, issue 7, 1988.Google Scholar

## Authors and Affiliations

• Roland Bulirsch
• 1
• Edda Nerz
• 1
• Hans Josef Pesch
• 1
• Oskar von Stryk
• 1
1. 1.Mathematisches InstitutTechnische Universität MünchenMünchen 2Germany