Advertisement

Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Nonlinear Programming Methods

  • Christof Büskens
  • Helmut Maurer
Chapter

Abstract

We discuss nonlinear programming (NLP) methods for solving optimal control problems with control and state inequality constraints. Suitable discretizations of control and state variables are used to transform the optimal control into a finite dimensional NLP problem. In [8] we have proposed numerical methods for the post-optimal calculations of parameter sensitivity derivatives of optimal solutions to NLP problems. The purpose of this paper is to extend the methods of post-optimal sensitivity analysis and real-time optimization to discretized control problems. The dimension of the discretized control problem should be kept small to obtain accurate sensitivity results. This can be achieved by taking only the discretized control variables as optimization variables whereas the state variables are computed recursively through an appropriate integration routine. We discuss the implications of this approach for the calculations of parameter sensitivity derivatives with respect to optimal control, state and adjoint functions. The efficiency of the proposed methods are illustrated by two numerical examples.

Keywords

Control Problem Optimal Control Problem Optimization Variable Adjoint Variable Adjoint Function 
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.
    D. Augustin, H. Maurer: Sensitivity Analysis and Real-Time Control of a Container Crane under State Constraints. This volumeGoogle Scholar
  2. 2.
    A. Barclay, P. E. Gill, J. B. Rosen: SQP Methods and their Application to Nu-merical Optimal Control. In Variational Calculus, Optimal Control and Applications, W. H. Schmidt, Heier, K., Bittner, L., Bulirsch, R., eds., Birkhäuser Basel, Boston, Berlin (1998) 207–222CrossRefGoogle Scholar
  3. 3.
    J. T. Betts: Survey of Numerical Methods for Trajectory Optimization. Journal of Guidance, Control, and Dynamics, 21 (1998) 193–207MATHCrossRefGoogle Scholar
  4. 4.
    H. G. Bock, K. J. Plitt: A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems. IFAC 9th World Congress, Budapest, Hungary (1984)Google Scholar
  5. 5.
    C. Büskens: Direkte Optimierungsmethoden zur numerischen Berechnung optimaler Steuerungen. Diploma thesis, Institut für Numerische Mathematik, Universität Münster, Münster, Germany (1993)Google Scholar
  6. 6.
    C. Büskens: Real-Time Solutions for Perturbed Optimal Control Problems by a Mixed Open- and Closed-Loop Strategy. This volume.Google Scholar
  7. 7.
    C. Büskens: Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen. Dissertation, Institut für Numerische Mathematik, Universität Münster, Münster, Germany (1998)Google Scholar
  8. 8.
    C. Büskens, H. Maurer: Sensitivity Analysis and Real-Time Optimization of Parametric Nonlinear Programming Methods. This volume.Google Scholar
  9. 9.
    C. Büskens, H. Maurer: Real-Time Control of Robots with Initial Value Perturbations via Nonlinear Programming Methods. Optimization 47 (2000) 383–405MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    C. Büskens, H. Maurer: Real-Time Control of an Industrial Robot. This volume.Google Scholar
  11. 11.
    A. L. Dontchev, W. W. Hager, K. Malanowski: Error Bounds for Euler Approximation of a State and Control Constrained Optimal Control Problem. Functional Analysis and Optimization 21 (2000) 653–682MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    P. J. Enright, B. A. Conway: Discrete Approximations to Optimal Trajectories Using Direct Transcription and Nonlinear Programming. AIAA Paper 90–2963-CP (1990)Google Scholar
  13. 13.
    Yu. G. Evtushenko: Numerical Optimization Techniques. Translation Series in Mathematics and Engineering, Optimisation Software Inc., Publications Division, New York (1985)CrossRefGoogle Scholar
  14. 14.
    U. Felgenhauer: Diskretisierung von Steuerungsproblemen unter stabilen Optimalitätsbedingungen. Institut für Mathematik, Habilitation, Technische Universität Cottbus, Cottbus, Germany (1998)Google Scholar
  15. 15.
    U. Felgenhauer: On Higher Order Methods for Control Problems with Mixed Inequality Constraints. Institut für Mathematik, Preprint M-01/1998, Technische Universität Cottbus, Cottbus, Germany (1998)Google Scholar
  16. 16.
    R. F. Haiti, S. P. Sethi, R. G. Vickson: A Survey of the Maximum Principles for Optimal Control Problems with State Constraints. SIAM Review 37 (1995) 181–218MathSciNetCrossRefGoogle Scholar
  17. 17.
    K. Malanowski, C. Büskens, H. Maurer: Convergence of Approximations to Nonlinear Optimal Control Problems. In: Mathematical Programming with Data Perturbations, A. V. Fiacco, ed., Lecture notes in pure and applied mathematics, Vol. 195, Marcel Dekker, Inc. (1998) 253–284Google Scholar
  18. 18.
    H. Maurer: Optimale Steuerprozesse mit Zustandsbeschränkungen. Mathematisches Institut, Habilitation, Universität Würzburg, Würzburg, Germany (1976).Google Scholar
  19. 19.
    H. Maurer, D. Augustin: Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Boundary Value Methods. This volume.Google Scholar
  20. 20.
    L. W. Neustadt: Optimization: A Theory of Necessary Conditions. Princeton University Press, Princeton, New Jersey (1976)MATHGoogle Scholar
  21. 21.
    L. S. Pontrjagin, V. G. Boltjanskij, R. V. Gamkrelidze, E. F. Miscenko: Mathematische Theorie optimaler Prozesse. R. Oldenbourg, München, Wien (1967)Google Scholar
  22. 22.
    O. von Stryk, Numerische Lösung optimaler Steuerungsprobleme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variablen. Fortschritt-Berichte VDI, Reihe 8, Nr. 441 VDI Verlag, Germany (1995)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Christof Büskens
    • 1
  • Helmut Maurer
    • 2
  1. 1.Lehrstuhl für IngenieurmathematikUniversität BayreuthGermany
  2. 2.Institut für numerische MathematikUniversität MünsterGermany

Personalised recommendations