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Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity

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Computational Optimal Control

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 115))

Abstract

Optimal control problems which describe realistic technical applications exhibit various features of complexity. First, the consideration of inequality constraints leads to optimal solutions with highly complex switching structures including bang-bang, singular, and control-and state-constrained sub-arcs. In addition, also isolated boundary points may occur. Techniques are surveyed for the computation of optimal trajectories with multiple subarcs. If the precise computation of the switching structure holds the spotlight, the indirect multiple shooting method is top quality. Second, the differential equations describing the dynamics may be so complicated that they have to be generated by a computer program. In this case, direct methods such as direct collocation are generally superior. Third, the task is often given in applications to solve many optimal control problems, either for parameter homotopies in the course of the solution process itself or for sensitivity investigations of the solutions with respectto various design parameters. Closely related to optimal control problems, pursuit-evasion game problems require, in a natural way, the solution of often thousands of boundary-value problems, in order to synthesize the open-loop controls by feedback strategies. In these cases, efficient homotopy methods must be used in connection with vectorized or parallelized versions of the aforementioned methods.

These three degrees of complexity in the solution of optimal control or pursuit-evasion game problems, respectively, are discussed in this survey paper by means of three examples: the abort landing of a passenger aircraft in the presence of a varying down burst, the time-and energy-optimal control of an industrial robot, and a pursuit-evasion game problem between a missile and a fighter aircraft.

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References

  1. Ba§ar, T. and Olsder, G. J.: Dynamic Noncooperative Game Theory, Academic Press, London, Great Britain, 1982.

    Google Scholar 

  2. [2]Berkmann, P. and Pesch, H. J.: Abort Landing under Different Windshear Conditions, in preparation.

    Google Scholar 

  3. Betts, J. T. and Huffman, W. P.: Trajectory Optimization Using Sparse Sequential Quadratic Programming, in: Optimal Control, Calculus of Variations, Optimal Control Theory and Numerical Methods, ed. by R. Bulirsch et. al., Birkhäuser (Inter. Series of Numer. Math. 111), Basel, Switzerland, 1993, 115–128.

    Google Scholar 

  4. Betts, J. T. and Huffman, W. P.: Path Constrained Trajectory Optimization Using Sparse Sequential Quadratic Programming, AIAA J. of Guidance, Control, and Dynamics 16 (1993) 59–68.

    Article  MATH  Google Scholar 

  5. Bock, H. G. and Plitt, K. J.: A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, in: Proc. of the 9th IFAC World Congress, Budapest, Hungary, 1984, Vol. IX, Colloquia 14.2, 09.2.

    Google Scholar 

  6. Breitner, M. H.: Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances, in: Control Applications of Optimization, München, Germany, 1992, ed. by R. Bulirsch and D. Kraft, Birkhäuser (Inter. Series of Numer. Math., this volume), Basel, Switzerland.

    Google Scholar 

  7. Breitner, M. H.: Real-Time Applicable Feedback Controller for Differential Games, to appear in Proceedings of the Sixth International Symposium on Dynamic Games and Applications, St.-Jovite, Qubéc, Canada, 1994.

    Google Scholar 

  8. Breitner, M. H. and Pesch, H. J.: Re-entry Trajectory Optimization Under Atmospheric Uncertainty as a Differential Game, in: Advances in Dynamic Games and Applications, ed. by T. Başar et al., Birkhäuser (Annals of the Inter. Society of Dynamic Games 1), Basel, Switzerland, 1993.

    Google Scholar 

  9. Breitner, M. H., Pesch, H. J., and Grimm, W.: Complex Differential Games of Pursuit-Evasion Type with State Constraints. Part 1: Necessary Conditions for Optimal Open-Loop Strategies, J. of Optim. Theory & Appl. 78 (1993), 419–441.

    Article  MathSciNet  MATH  Google Scholar 

  10. Breitner, M. H., Pesch, H. J., and Grimm, W.: Complex Differential Games of Pursuit-Evasion Type with State Constraints. Part 2: Numerical Computation of Optimal Open-Loop Strategies, J. of Optim. Theory and Appl. 78 (1993), 443–463.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bryson, A. E. and Ho, Y.-C.: Applied Optimal Control, Rev. Printing, Hemisphere Publishing Corporation, New York, New York, 1975.

    Google Scholar 

  12. Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung, Oberpfaffenhofen, Germany, Report of the Carl-Cranz Gesellschaft, 1971; Reprint: Department of Mathematics, Munich University of Technology, München, Germany, 1993.

    Google Scholar 

  13. Bulirsch, R. and Callies, R.: Optimal Trajectories for an Ion Driven Spacecraft from Earth to the Planetoid Vesta, in: Proc. of the AIAA Guidance, Navigation and Control Conference, New Orleans, Louisiana, 1991, AIAA Paper 91–2683 (1991).

    Google Scholar 

  14. Bulirsch, R. and Callies, R.: Optimal Trajectories for a Multiple Rendezvous Mission to Asteroids, in: 42nd Inter. Astronautical Congress, Montreal, Canada, 1991, IAF-Paper IAF-91-342 (1991).

    Google Scholar 

  15. Bulirsch, R. and Chudej, K.: Ascent Optimization of an Airbreathing Space Vehicle, in: Proc. of the AIAA Guidance, Navigation and Control Conference, New Orleans, Louisiana, 1991, AIAA Paper 91–2656 (1991).

    Google Scholar 

  16. Bulirsch, R. and Chudej, K.: Staging and Ascent Optimization of a Dual-Stage Space Transporter, Zeitschrift für Flugwissenschaften und Weltraum-forschung 16 (1992) 143–151.

    Google Scholar 

  17. Bulirsch, R. and Chudej, K.: Guidance and Trajectory Optimization under State Constraints, in: Preprint of the 12th IFAC Symposium on Automatic Control in Aerospace -Aerospace Control 1992, München, Germany, 1992, ed. by D. B. DeBra and E. Gottzein, VDI/VDE-GMA, Düsseldorf, Germany, 1992, 533–538.

    Google Scholar 

  18. Bulirsch, R., Montrone, F., and Pesch, H. J.: Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem. Part 1: Necessary conditions, J. Optim. Theory & Appl. 70 (1991) 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  19. Bulirsch, R., Montrone, F., and Pesch, H. J.: Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem. Part 2: Multiple shooting and Homotopy, J. Optim. Theory & Appl. 70 (1991) 223–254.

    Article  MathSciNet  MATH  Google Scholar 

  20. Bulirsch, R., Nerz, E., Pesch, H. J., and von Stryk, O.: Combining Direct and Indirect Methods in Optimal Control: Range Maximization of a Hang Glider, in: Optimal Control, Calculus of Variations, Optimal Control Theory and Numerical Methods, ed. by R. Bulirsch et. al., Birkhäuser (Inter. Series of Numer. Math. 111), Basel, Switzerland, 1993, 273–288.

    Google Scholar 

  21. Callies, R.: Optimal Design of a Mission to Neptune, in: Optimal Control, Calculus of Variations, Optimal Control Theory and Numerical Methods, ed. by R. Bulirsch et. al., Birkhäuser (Inter. Series of Numer. Math. 111), Basel, Switzerland, 1993, 341–349.

    Google Scholar 

  22. Char, B. W., Geddes, K. O, Gonnet, G. H., Leong, B. L., Monagan, M. B., and Watt, S. M.: Maple V, Language Reference Manual, Springer, New York, New York, 1991.

    MATH  Google Scholar 

  23. Deuflhard, P.: A Modified Newton Method for the Solution of Ill-conditioned Systems of Nonlinear Equations with Application to Multiple Shooting, Numerische Mathematik 22 (1974) 289–315.

    Article  MathSciNet  MATH  Google Scholar 

  24. Deuflhard, P.: A Stepsize Control for Continuation Methods and its Special Application to Multiple Shooting Techniques, Numerische Mathematik 33 (1979) 115–146.

    Article  MathSciNet  MATH  Google Scholar 

  25. Deuflhard, P., Pesch, H. J., and Rentrop, P.: A Modified Continuation Method for the Numerical Solution of Nonlinear Two-Point Boundary Value Problems by Shooting Techniques, Numerische Mathematik 26 (1976) 327–343.

    Article  MathSciNet  MATH  Google Scholar 

  26. Gabler I., Miesbach S., Breitner M. H., and Pesch, H. J.: Synthesis of Optimal Strategies for Differential Games by Neural Networks, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung u. Steuerung, Munich University of Technology, München, Germany, Report No. 468, 1993.

    Google Scholar 

  27. Gill, P. E.: Large-Scale SQP Methods and Their Application in Trajectory Optimization, in: Control Applications of Optimization, München, Germany, 1992, ed. by R. Bulirsch and D. Kraft, Birkhäuser (Inter. Series of Numer. Math., this volume), Basel, Switzerland.

    Google Scholar 

  28. Gill, P. E., Murray, W., Saunders, M. A., and Wright, M. H.: User’s Guide for NPSOL (Version 4–0), Department of Operations Research, Stanford University, California, Report SOL 86-2, 1986.

    Google Scholar 

  29. Griewank, A.: Automatic Evaluation of Discrete Adjoints with Logarithmic Increase in Storage, in: Control Applications of Optimization, München, Germany, 1992, ed. by R. Bulirsch and D. Kraft, Birkhäuser (Inter. Series of Numer. Math., this volume), Basel, Switzerland.

    Google Scholar 

  30. Hargraves, C. R. and Paris, S. W.: Direct Trajectory Optimization Using Nonlinear Programming and Collocation, AIAA J. of Guidance, Control, and Dynamics 10 (1987) 338–342.

    Article  MATH  Google Scholar 

  31. Hartl, R. F., Sethi, S. P., and Vickson, R. G.: A Survey of the Minimum Principles for Optimal Control Problems with State Constraints, Institut für Ökonometrie, Operations Research und Systemtheorie, Vienna University of Technology, Vienna, Austria, Report Nr. 153, 1992.

    Google Scholar 

  32. Hiltmann P.: Numerische Lösung von Mehrpunkt-Randwertproblemen und Aufgaben der optimalen Steuerung mit Steuerfunktionen über endlichdimen-sionalen Räumen, Thesis, Munich University of Technology, München, Germany, 1990; see also Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung u. Steuerung, Munich University of Technology, München, Germany, Report No. 448, 1993.

    Google Scholar 

  33. Hiltmann, P., Chudej, K., and Breitner, M. H.: Eine modifizierte Mehrzielmethode zur Lösung von Mehrpunkt-Randwertproblemen -Benutzeranleitung, Sonderforschungsbereich 255 der Deutschen Forschungsgemeinschaft, Transatmosphärische Flugsysteme, Munich University of Technology, München, Germany, Report No. 14, 1993.

    Google Scholar 

  34. Jacobson, D. H., Lele, M. M., and Speyer, J. L.: New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints, J. of Math. Anal. and Appl. 35 (1971) 255-284.

    Article  MathSciNet  MATH  Google Scholar 

  35. Kiehl, M.: Vectorizing the Multiple-Shooting Method for the Solution of Boundary-Value Problems and Optimal Control Problems, in: Proc. of the 2nd Inter. Conference on Vector and Parallel Computing Issues in Applied Research and Development, Tromsø, Norway, 1988, ed. by J. Dongarra et. al., Ellis Horwood, London, Great Britain, 1989, 179-188.

    Google Scholar 

  36. Kraft, D.: FORTRAN Computer Programs for Solving Optimal Control Problems, Report 80-03, Institute for Flight Systems Dynamics, German Aerospace Research Establishment DLR, Oberpfaffenhofen, Germany, 1980.

    Google Scholar 

  37. Kraft, D.: On Converting Optimal Control Problems into Nonlinear Programming Codes, in: Computational Mathematical Programming, ed. by K. Schitt-kowski, Springer (NATO ASI Series 15), Berlin, Germany, 1985, 261–280.

    Google Scholar 

  38. Kugelmann, B., Mihatsch, O., Mikulski, L., and Schmidt, W.: Optimal Design of Elastic Arches in Combination with Bifurcation Theory, submitted for publication; see also Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung u. Steuerung, Munich University of Technology, München, Germany, Report No. 477, 1993.

    Google Scholar 

  39. Kugelmann, B. and Pesch, H. J.: New General Guidance Method in Constrained Optimal Control. Part 1: Numerical Method, J. of Optim. Theory & Appl. 67 (1990) 421–435.

    Article  MathSciNet  MATH  Google Scholar 

  40. Kugelmann, B. and Pesch, H. J.: New General Guidance Method in Constrained Optimal Control. Part 2: Application to Space Shuttle Guidance, J. of Optim. Theory & Appl. 67 (1990) 437–446.

    Article  MathSciNet  MATH  Google Scholar 

  41. Kugelmann, B. and Pesch, H. J.: Serielle und parallele Algorithmen zur Korrektur optimaler Flugbahnen in Echtzeit-Rechnung, in: Jahrestagung der Deutschen Gesellschaft für Luft-und Raumfahrt, Friedrichshafen, Germany, 1990, DGLR-Jahrbuch 1990 1 (1990) 233–241.

    Google Scholar 

  42. Lachner, R., Breitner, M. H., Pesch, H. J.: Efficient Numerical Solution of Differential Games with Application to Air-Combat, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung u. Steuerung, Munich University of Technology, München, Germany, Report No. 466, 1993.

    Google Scholar 

  43. Maurer, H.: Optimale Steuerprozesse mit Zustandsbeschränkungen, Habilitationsschrift, University of Würzburg, Würzburg, Germany, 1976.

    Google Scholar 

  44. Miele, A., Wang, T., Melvin, and W. W.: Optimal Abort Landing Trajectories in the Presence of Windshear, J. of Optim. Theory & Appl. 55 (1987) 165–202.

    Article  MATH  Google Scholar 

  45. Oberle, H. J.: Numerische Berechnung optimaler Steuerungen von Heizung und Kühlung für ein realistisches Sonnenhausmodell, Habilitationsschrift, Munich University of Technology, München, Germany, 1982.

    Google Scholar 

  46. Oberle, H. J. and Grimm, W.: BNDSCO-A Program for the Numerical Solution of Optimal Control Problems, Internal Report No. 515-89/22, Institute for Flight Systems Dynamics, German Aerospace Research Establishment DLR, Oberpfaffenhofen, Germany, 1989.

    Google Scholar 

  47. Otter, M. and Türk, S.: The DFVLR Models 1 and 2 of the Manutec r3 Robot, DLR-Mitteilungen 88-13, Institute for Flight Systems Dynamics, German Aerospace Research Establishment DLR, Oberpfaffenhofen, Germany, 1988.

    Google Scholar 

  48. Pesch, H. J.:Real-time Computation of Feedback Controls for Constrained Optimal Control Problems. Part 1: Neighbouring Extremals, Optimal Control Applications and Methods 10 (1989) 129–145.

    Google Scholar 

  49. 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 10 (1989) 147–171.

    Google Scholar 

  50. Pesch, H. J.: Offline and Online Computation of Optimal Trajectories in the Aerospace Field, in: Applied Mathematics in Aerospace Science and Engineering, 12th Course of the International School of Mathematics “G. Stampacchia”, Erice, Italy, 1991, ed. by A. Miele and A. Salvetti, Plenum Publishing Corporation, New York, New York, 1994; see also Sonderforschungsbereich 255 der Deutschen Forschungsgemeinschaft, Transatmosphärische Flugsysteme, Munich University of Technology, München, Germany, Report No. 9, 1992.

    Google Scholar 

  51. [51]Pesch, H. J., Schlemmer, M., and von Stryk, O.: Minimum-Energy and Minimum-Time Control of Three-Degrees-Of-Freedom Robots. Part 1: Mathematical Model and Necessary Conditions. Part 2: Numerical Methods and Results for the Manutec r3 Robot, in preparation.

    Google Scholar 

  52. Stoer, J. and Bulirsch, R.: Introduction to Numerical Analysis, 2nd Ed., Springer, New York, New York, 1993.

    MATH  Google Scholar 

  53. von Stryk, O.: Numerical Solution of Optimal Control Problems by Direct Collocation, in: Optimal Control, Calculus of Variations, Optimal Control Theory and Numerical Methods, ed. by R. Bulirsch et. al., Birkhäuser (Inter. Series of Numer. Math. 111), Basel, Switzerland, 1993, 129–143.

    Google Scholar 

  54. von Stryk, O. and Bulirsch, R.: Direct and Indirect Methods for Trajectory Optimization, Annals of Operations Research 37 (1992) 357–373.

    Google Scholar 

  55. von Stryk, O. and Schlemmer, M.: Optimal Control of the Industrial Robot Manutec r3, in: Control Applications of Optimization, München, 1992, ed. by R. Bulirsch and D. Kraft, Birkhäuser (Inter. Series of Numer. Math., this volume), Basel, Switzerland.

    Google Scholar 

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  1. Department of Mathematics, Munich University of Technology, D-80290, München, Germany

    Hans Josef Pesch

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  1. Hans Josef Pesch
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  1. Mathematisches Institut, TH München, Postfach 20 24 20, D-80290, München, Germany

    R. Bulirsch

  2. Labor für Regelungs-und Steuerungslechnik Fachbereich Maschinenbau, Fachhochschule München, Postfach 20 01 13, D-80001, München, Germany

    D. Kraft

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Pesch, H.J. (1994). Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity. In: Bulirsch, R., Kraft, D. (eds) Computational Optimal Control. ISNM International Series of Numerical Mathematics, vol 115. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-8497-6_4

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  • DOI: https://doi.org/10.1007/978-3-0348-8497-6_4

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-7643-5015-4

  • Online ISBN: 978-3-0348-8497-6

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