Advertisement

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

  • Hans Josef Pesch
Chapter
Part of the ISNM International Series of Numerical Mathematics book series (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.

Keywords

Optimal Control Problem Differential Game Touch Point Switching Structure Direct Collocation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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. [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. [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.zbMATHCrossRefGoogle Scholar
  5. [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. [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. [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. [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. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Bryson, A. E. and Ho, Y.-C.: Applied Optimal Control, Rev. Printing, Hemisphere Publishing Corporation, New York, New York, 1975.Google Scholar
  12. [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. [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. [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. [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. [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. [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. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [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. [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. [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.zbMATHGoogle Scholar
  23. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Deuflhard, P.: A Stepsize Control for Continuation Methods and its Special Application to Multiple Shooting Techniques, Numerische Mathematik 33 (1979) 115–146.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [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. [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. [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. [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. [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.zbMATHCrossRefGoogle Scholar
  31. [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. [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. [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. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [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. [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. [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. [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. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [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.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [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. [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. [43]
    Maurer, H.: Optimale Steuerprozesse mit Zustandsbeschränkungen, Habilitationsschrift, University of Würzburg, Würzburg, Germany, 1976.Google Scholar
  44. [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.zbMATHCrossRefGoogle Scholar
  45. [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. [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. [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. [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. [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. [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. [52]
    Stoer, J. and Bulirsch, R.: Introduction to Numerical Analysis, 2nd Ed., Springer, New York, New York, 1993.zbMATHGoogle Scholar
  53. [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. [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. [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

Copyright information

© Birkhäuser Verlag Basel 1994

Authors and Affiliations

  • Hans Josef Pesch
    • 1
  1. 1.Department of MathematicsMunich University of TechnologyMünchenGermany

Personalised recommendations