Abstract
A very fast numerical method is developed for the computation of neighboring optimum feedback controls. This method is applicable to a general class of optimal control problems (for example, problems including inequality constraints and discontinuities) and needs no on-line computation, except for one matrix-vector multiplication. The method is based on the so-called accessory minimum problem. The necessary conditions for this auxiliary optimal control problem form a linear multipoint boundary-value problem with linear jump conditions, which is especially well suited for numerical treatment. In the second part of this paper, the performance of the guidance scheme is shown for the heating-constrained cross-range maximization problem of a space-shuttle-orbiter-type vehicle.
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References
Pesch, H. J.,Numerische Berechnung optimaler Flugbahnkorrekturen in Echtzeitrechnung, Munich University of Technology, Doctoral Dissertation, 1978.
Pesch, H. J.,Numerical Computation of Neighboring Optimum Feedback Control Schemes in Real Time, Applied Mathematics and Optimization, Vol. 5, pp. 231–252, 1979.
Pesch, H. J.,Echtzeitberechnung fastoptimaler Rückkopplungssteuerungen bei Steuerungsproblemen mit Beschränkungen, Munich University of Technology, Habilitationsschrift, 1986.
Pesch, H. J.,Real-Time Computation of Feedback Controls for Constrained Optimal Control Problems, Part 2: A Correction Method Based on Multiple Shooting, Optimal Control Applications and Methods, Vol. 10, pp. 147–171, 1989.
Kelley, H. J.,Guidance Theory and Extremal Fields, IRE Transactions on Automatic Control, Vol. AC-7, pp. 75–82, 1962.
Breakwell, J. V., andHo, Y. C.,On the Conjugate Point Condition for the Control Problem, International Journal of Engineering Science, Vol. 2, pp. 565–579, 1965.
Pesch, H. J.,Real-Time Computation of Feedback Controls for Constrained Optimal Control Problems, Part 1: Neighboring Extremals, Optimal Control Applications and Methods, Vol. 10, pp. 129–145, 1989.
Breakwell, J. V., Speyer, J. L., andBryson, A. E.,Optimization and Control of Nonlinear Systems Using the Second Variation, SIAM Journal on Control, Vol. 1, pp. 193–223, 1963.
Kelley, H. J.,An Optimal Guidance Approximation Theory, IEEE Transactions on Automatic Control, Vol. AC-9, pp. 375–380, 1964.
Bryson, A. E., andHo, Y. C.,Applied Optimal Control, Ginn and Company, Waltham, Massachusetts, 1969.
Lee, I.,Optimal Trajectory, Guidance, and Conjugate Points, Information and Control, Vol. 8, pp. 589–606, 1965.
Kugelmann, B.,Numerische Berechnung optimaler Flugbahnkorrekturen für den Wiederaufstieg einer Mondfähre und das Bremsmanöver eines Raumfahrzeuges in der Erdatmosphäre, Munich University of Technology, Diploma Thesis, 1983.
Coddington, E. A., andLevinson, N.,Theory of Ordinary Differential Equations, McGraw-Hill, New York, New York, 1955.
Bulirsch, R.,Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung, DLR, Oberpfaffenhofen, Germany, Report of the Carl-Cranz Gesellschaft, 1971.
Stoer, J., andBulirsch, R.,Introduction to Numerical Analysis, Springer, New York, New York, 1980.
Oberle, H. J.,Numerische Berechnung optimaler Steuerungen von Heizung und Kühlung für ein realistisches Sonnenhausmodell, Munich University of Technology, Habilitationsschrift, 1982.
Kugelmann, B.,Zeitminimale Berechnung von Rückkopplungssteuerungen für optimale Lenkungsprobleme mit Anwendung in der Raumfahrt, Munich University of Technology, Doctoral Dissertation, 1986.
Speyer, J. L., andBryson, A. E.,A Neighboring Optimum Feedback Control Scheme Based on Estimated Time-to-Go with Application to Reentry Flight Paths, AIAA Journal, Vol. 6, pp. 769–776, 1968.
Powers, W. F.,A Method for Comparing Trajectories in Optimum Linear Perturbation Guidance Schemes, AIAA Journal, Vol. 6, pp. 2451–2452, 1968.
Powers, W. F.,Techniques for Improved Convergence in Neighboring Optimum Guidance, AIAA Journal, Vol. 8, pp. 2235–2241, 1970.
Krämer-Eis, P.,Ein Mehrzielverfahren zur numerischen Berechnung optimaler Feedback-Steuerungen bei beschränkten nichtlinearen Steuerungsproblemen, University of Bonn, Doctoral Dissertation, 1985.
Hymas, C. E., Cavin, R. K., andColumnga, D.,Neighboring Extremals for Optimal Control Problems, AIAA Journal, Vol. 11, pp. 1101–1109, 1973.
Wood, L. J.,Comment on: Neighboring Extremals for Optimal Control Problems, AIAA Journal, Vol. 12, pp. 1452–1454, 1974.
Lee, A. Y.,Neighboring Extremals of Dynamic Optimization Problems with Path Equality Constraints, Journal of Optimization Theory and Applications, Vol. 57, pp. 519–536, 1988.
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Communicated by D. G. Hull
This research was supported in part by the Deutsche Forschungsgemeinschaft under the Schwerpunktprogramm “Anwendungsbezogene Optimierung und Steuerung.”
The authors wish to express their sincere and grateful appreciation to Professor Roland Bulirsch who encouraged this work.
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Kugelmann, B., Pesch, H.J. New general guidance method in constrained optimal control, part 1: Numerical method. J Optim Theory Appl 67, 421–435 (1990). https://doi.org/10.1007/BF00939642
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DOI: https://doi.org/10.1007/BF00939642