Advertisement

Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Boundary Value Methods

  • Helmut Maurer
  • Dirk Augustin
Chapter

Abstract

Parametric nonlinear control problems subject to mixed control-state constraints and pure state constraints are investigated. Parameters are introduced to model perturbations of the control system and may appear in all system data. We review conditions under which the optimal solutions are differentiable functions of the parameter. In the theoretical part, these conditions are related to regularity conditions and to second order sufficient conditions. On the numerical side, the conditions are connected to shooting methods for solving the boundary value problems that characterize the optimal solution. We discuss methods for computing the sensitivity differentials of the optimal solutions with respect to parameters. The calculated sensitivity differentials can be used to construct real-time approximations of the perturbed solutions via first order Taylor expansions. Two numerical case studies are discussed in detail to illustrate the numerical methods for mixed control-state constraints and for pure state constraints.

Keywords

Boundary Value Problem Optimal Control Problem State Constraint Riccati Equation Junction Point 
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.
    A. V. Arutyunov, S. M. Aseev: Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J. Control and Optimization 35 (1997) 930–952MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    D. Augustin: Hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse bei optimalen Steuerproblemen mit Steuer- und Zustandsbeschränkungen. Diploma thesis, Institut für Numerische Mathematik, Universität Münster, Germany, 1996.Google Scholar
  3. 3.
    D. Augustin, H. Maurer: An example for computational sensitivity analysis for state constrained control problems, in Proceedings of Parametric Optimization and Related Topics V, Tokyo, 1998, Guddat, J., et al., eds, Peter Lang Verlag, Frankfurt am Main, 2000, 25–35Google Scholar
  4. 4.
    D. Augustin, H. Maurer: Computational sensitivity analysis for state constrained optimal control problems. Annals of Operations Research (2000).Google Scholar
  5. 5.
    D. Augustin, H. Maurer: Second order sufficient conditions and sensitivity analysis for optimal multiprocess control problems. Control and Cybernetics 29 (2000) 11–31MathSciNetzbMATHGoogle Scholar
  6. 6.
    D. Augustin, H. Maurer: Sensitivity analysis and real-time control of a container crane under state constraints. This volume.Google Scholar
  7. 7.
    P. Berkmann, H. J. Pesch: Abort landing in windshear: Optimal control problem with third-order state constraint and varied switching structure. J. of Optimization Theory and Applications 85 (1995) 21–57MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    H. G. Bock, P. Krämer-Eis: An Efficient Algorithm for Approximate Computation of Feedback Control Laws in Nonlinear Processes. ZAMM, 61 (1981) T 330-T 332Google Scholar
  9. 9.
    A. E. Bryson, Y. C. Ho: Applied Optimal Control. Revised Printing, Hemisphere Publishing Corporation New York, New York, 1975.Google Scholar
  10. 10.
    R. Bulirsch: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Report of the Carl-Cranz Gesellschaft, Oberpfaffenhofen, Germany, 1971. Reprinted as report R1.06 of the Son-derforschungsbereich “Transatmosphärische Flugsysteme”, TU München, Germany, 1993Google Scholar
  11. 11.
    R. Bulirsch, F. Montrone, and H. J. Pesch: 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 (1991) 1–23MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    R. Bulirsch, F. Montrone, and H. J. Pesch: 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 (1991) 223–254MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ch. Büskens: Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen. Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.Google Scholar
  14. 14.
    Ch. Büskens, H. Maurer: Sensitivity analysis and real-time control of parametric optimal control problems using nonlinear programming methods. This volumeGoogle Scholar
  15. 15.
    K. Chudej: Realistic modelled optimal control problems in aerospace engineering — a challenge to the necessary optimality conditions. Mathematical Modelling of Systems 2 (1996) 252–261MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    K. Chudej: Effiziente Lösungen zustandsbeschränkter Optimalsteuerungsaufgaben. Habilitationsschrift, Universität Bayreuth, Germany, 2000Google Scholar
  17. 17.
    A. L. Dontchev, W. W. Hager, P. A. Poore, and B. Yang: Optimality, stability and con-vergence in nonlinear control. Applied Math, and Optim. 31 (1995) 297–326MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    A. L. Dontchev, W. W. Hager: Lipschitzian stability for state constrained nonlinear optimal control. SIAM J. on Control and Optimization 36 (1998) 698–718MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    A. L. Dontchev, I. Kolmanovsky: State constraints in the linear regulator problem: a case study, J. of Optimization Theory and Applications 87 (1995) 327–347MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. L. Dontchev, I. Kolmanovsky: Best interpolation in a strip II: Reduction to unconstrained convex optimization, Computational Optimization and Applications 5 (1996) 233–251MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    J. C. Dunn: Second order optimality conditions in sets of L∞ functions with range in a polyhedron. SIAM J. Control Optimization 33 (1995) 1603–1635MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    J. C. Dunn: On L2 sufficient conditions and the gradient projection method for optimal control problems. SIAM J. Control Optimization 34 (1996) 1270–1290MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    U. Felgenhauer: Diskretisierung von Steuerungsproblemen unter stabilen Optimalitätsbedingungen. Habilitationsschrift, Dept. of Mathematics, Technische Universität Cottbus, Germany, 1999Google Scholar
  24. 24.
    U. Felgenhauer: On smoothness properties and approximability of optimal control func-tions. To appear in Annals of Operations ResearchGoogle Scholar
  25. 25.
    A. V. Fiacco: Introduction to Sensitivity and Stability Analysis in Nonlinear Program-ming, Mathematics in Science and Engineering 165, Academic Press, New York, 1983Google Scholar
  26. 26.
    R. F. Haiti, S. P. Sethi, and R. G. Vickson: A survey of the maximum principle for optimal control problems with state constraints. SIAM Review 37 (1995) 181–218MathSciNetCrossRefGoogle Scholar
  27. 27.
    M. Hestenes: Calculus of Variations and Optimal Control Theory. John Wiley, New York, 1966zbMATHGoogle Scholar
  28. 28.
    K. Ito, K. Kunisch: Sensitivity analysis of solutions to optimization problems in Hubert spaces with applications to optimal control and estimation. Journal of Differential Equations, 99 (1992) 1–40MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    D. H. Jacobson, M. M. Lele, and J. L. Speyer: New necessary conditions of optimality for control problems with state-variable inequality constraints, J. of Mathematical Analysis and Applications 35 (1971) 255–284MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    D. H. Jacobson, D. Q. Mayne: Differential Dynamic Programming. American Elsevier Publishing Company Inc., New York, 1970zbMATHGoogle Scholar
  31. 31.
    P. Krämer-Eis: Ein Mehrziel verfahren zur numerischen Berechnung optimaler Feedback-Steuerungen bei beschränkten nichtlinearen Steuerungsproblemen. Bonner Mathematische Schriften 164, 1985Google Scholar
  32. 32.
    B. Kugelmann, H. J. Pesch: A new general guidance method in constrained optimal control, Part 1: Numerical method. J. Optim. Theory and Appl. 67 (1990) 421–435MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    B. Kugelmann, H. J. Pesch: A new general guidance method in constrained optimal control, Part 2: Application to space shuttle guidance. J. Optim. Theory and Appl. 67 (1990) 437–446MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    K. Malanowski: Second order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimization problems in Hubert spaces. Applied Math. Optimization 25 (1992) 51–79MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    K. Malanowski: Two-norm approach in stability and sensitivity analysis of optimization and optimal control problems. Advances in Math. Sciences and Applications 2 (1993) 397–443MathSciNetzbMATHGoogle Scholar
  36. 36.
    K. Malanowski: Stability and sensitivity of solutions to nonlinear optimal control problems. Applied Math. Optim. 32 (1995) 111–141MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    K. Malanowski: Sufficient optimality conditions for optimal control problems subject to state constraints. SIAM J. on Control and Optimization 35 (1997), 205–227MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    K. Malanowski: Stability and sensitivity analysis for optimal control problems with control-state constraints. To appearGoogle Scholar
  39. 39.
    K. Malanowski, H. Maurer: Sensitivity analysis for parametric control problems with control-state constraints. Comput. Optim. and Applications 5 (1996) 253–283MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    K. Malanowski, H. Maurer: Sensitivity analysis for state constrained optimal control problems. Discrete and Continuous Dynamical Systems 4 (1998) 241–272MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    K. Malanowski, H. Maurer: Sensitivity analysis for optimal control problems subject to higher order state constraints. To appear in Annals of Operations Research, 2000.Google Scholar
  42. 42.
    H. Maurer: On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control and Optimization 15 (1977) 345–362MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    H. Maurer: On the minimum principle for optimal control problems with state constraints. Rechenzentrum Universität Münster, Schriftenreihe Nr. 41, Münster, Germany, 1979Google Scholar
  44. 44.
    H. Maurer: First- and second-order sufficient optimality conditions in mathematical pro-gramming and optimal control. Math. Programming Study 14 (1981) 43–62Google Scholar
  45. 45.
    H. Maurer, D. Augustin: Second order sufficient conditions and sensitivity analysis for the controlled Rayleigh problem. In Parametric Optimization and Related Topics IV, J. Guddat, H. Th. Jongen, F. Nozicka, G. Still, F. Twilt, eds., Peter Lang Verlag, 1996, 245–259Google Scholar
  46. 46.
    H. Maurer, W. Gillessen: Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables. Computing 15 (1975) 105–126MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    H. Maurer, H. D. Mittelmann: The nonlinear beam via optimal control with bounded state variables, Optimal Control Applications & Methods 12 (1991) 19–31MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    H. Maurer, H. J. Oberle: Second order sufficient conditions for optimal control problems with free final time: the Riccati approach. Submitted to SIAM J. Control and Optimiza-tionGoogle Scholar
  49. 49.
    H. Maurer, H. J. Pesch: Solution differentiability for parametric nonlinear control problems with controlstate constraints. J. Optimization Theory and Applications 86 (1995) 285–309MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    H. Maurer, S. Pickenhain: Second order sufficient conditions for optimal control problems with mixed control-state constraints. J. Optim. Theory and Applications 86 (1995) 649–667MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    A. A. Milyutin, N. P. Osmolovskii: Calculus of Variations and Optimal Control. Translations of Mathematical Monographs, Vol. 180, American Mathematical Society, Providence, 1998Google Scholar
  52. 52.
    L. W. Neustadt: Optimization: A Theory of Necessary Conditions. Princeton University Press, Princeton, 1976zbMATHGoogle Scholar
  53. 53.
    H. J. Oberle: Numerical solution of minimax optimal control problems by multiple shooting technique. J. of Optimization Theory and Applications 50 (1986) 331–357MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    H. J. Oberle, W. Grimm: BNDSCO — A program for the numerical solution of optimal control problems. Institute for Right Systems Dynamics, DLR, Oberpfaffenhofen, Germany, Internal Report 515–89/22, 1989Google Scholar
  55. 55.
    G. Opfer, H. J. Oberle: The derivation of cubic splines with obstacles by methods of optimization and optimal control. Numerische Mathematik 52 (1988) 17–31.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    N. P. Osmolovskii: Quadratic conditions for nonsingular extremals in optimal control (A theoretical treatment). Russian J. of Mathematical Physics 2 (1995) 487–516MathSciNetGoogle Scholar
  57. 57.
    H. J. Pesch: Numerical Computation of Neighboring Optimum Feedback Control Schemes in Real-Time. Applied Mathematics and Optimization 5 (1979), 231–252MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    H. J. Pesch: Neighboring Optimum Guidance of a Space-Shuttle-Orbiter-Type Vehicle. J. of Guidance and Control 3 (1980), 386–391zbMATHCrossRefGoogle Scholar
  59. 59.
    H. J. Pesch: Real-time computation of feedback controls for constrained optimal control problems, Part 1: Neighbouring extremals. Optimal Control Applications & Methods 10 (1989) 129–145MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    H. J. Pesch: Real-time computation of feedback controls for constrained optimal control problems, Part 2: A correction method based on multiple shooting. Optimal Control Applications & Methods 10 (1989) 147–171MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    S. Pickenhain: Sufficiency conditions for weak local minima in multidimensional optimal control problems with mixed control-state restrictions. Zeitschrift für Analysis und ihre Anwendungen 11 (1992) 559–568MathSciNetzbMATHGoogle Scholar
  62. 62.
    S. Pickenhain, K. Tammer: Sufficient conditions for local optimality in multidimensional control problems with state restrictions. Zeitschrift für Analysis and ihre Anwendungen 10 (1991) 397–405MathSciNetzbMATHGoogle Scholar
  63. 63.
    L. S. Pontrjagin, V. G. Boltjanskij, R. V. Gamkrelidze, E. F. Miscenko: Mathematische Theorie optimaler Prozesse. R.Oldenbourg, München Wien, 1967Google Scholar
  64. 64.
    F. Rampazzo, R. B. Vinter: Degenerate optimal control problems with state constraints. SIAM J. on Control and Optimization 39 (2000) 989–1007MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    J. Stoer, R. Bulirsch: Introduction to Numerical Analysis. Springer-Verlag, New York, 1980Google Scholar
  66. 66.
    T. Tun, T. S. Dillon: Extensions of the differential dynamic programming method to include systems with state dependent control constraints and state variable inequality constraints. J. of Applied Science and Engineering A, 3 (1978) 171–192Google Scholar
  67. 67.
    V. S. Vassiliadis, R. W. H. Sargent, and C. C. Pantelides: Solution of a Class of Mul-tistage Dynamic Optimization Problems. Part 2: Problems with Path Constraints, Ind. Eng. Chem. Res. 33, No.9 (1994) 2123–2133CrossRefGoogle Scholar
  68. 68.
    V. Zeidan: The Riccati equation for optimal control problems with mixed state-control constraints: necessity and sufficiency. SIAM J. Control and Optimization 32 (1994) 1297–1321MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Helmut Maurer
    • 1
  • Dirk Augustin
    • 1
  1. 1.Institut für Numerische MathematikWestfälische Wilhelms-Universität MünsterGermany

Personalised recommendations