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SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control. (English) Zbl 0963.65070

Control systems described by nonlinear ordinary differential equations depending on parameters are considered in this paper. A performance index is minimized with respect to mixed state-control constraints.
The problem is transcribed into a nonlinear programming problem by discretization of the continuous optimization problem. Sequential quadratic programming (SQP) methods are proposed as solver. The estimation of the adjoint variables is discussed. Second-order sufficient conditions for the optimal solution are derived. A first-order Taylor expansion is used for a fast online approximation for the perturbed solution. Sensitivity analysis for the perturbed optimal control problem is investigated.
The results are illustrated by two numerical examples.

MSC:

65K10 Numerical optimization and variational techniques
49J15 Existence theories for optimal control problems involving ordinary differential equations
49M37 Numerical methods based on nonlinear programming

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[1] Barclay, A.; Gill, Ph. E.; Rosen, J. B., SQP methods and their application to numerical optimal control, Report NA 97-3, Department of Mathematics, University of California (1997), San Diego: San Diego USA
[2] Betts, J. T.; Huffmann, W. P., Path constrained trajectory optimization using sparse sequential quadratic programming, J. Guidance Control Dynamics, 16, 59-68 (1993) · Zbl 0781.49019
[3] Betts, J. T., Issues in the direct transcription of optimal control problems to sparse nonlinear programs, (Bulirsch, R.; Kraft, D., International Series of Numerical Mathematics, ISNM, Vol. 115 (1994), Birkhäuser: Birkhäuser Basel), 3-18 · Zbl 0797.49028
[4] Bock, H. G.; Plitt, K. J., A multiple shooting algorithm for direct solution of optimal control problems, IFAC nineth World Congress (1984), Budapest: Budapest Hungary
[5] Büskens, Ch., Direkte Optimierungsmethoden zur numerischen Berechnung optimaler Steuerungen, Diploma Thesis, Institut für Numerische Mathematik, Universität Münster (1993), Münster: Münster Germany
[6] Büskens, Ch., Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen, Dissertation, Institut für Numerische Mathematik, Universität Münster (1998), Münster: Münster Germany · Zbl 0907.49010
[7] Ch. Büskens, H. Maurer, Real-time control of robots with initial value perturbations via nonlinear programming methods, Special Issue of OPTIMIZATION, in press.; Ch. Büskens, H. Maurer, Real-time control of robots with initial value perturbations via nonlinear programming methods, Special Issue of OPTIMIZATION, in press.
[8] Ch. Büskens, H. Maurer, Sensitivity analysis and real-time control of nonlinear optimal control systems via nonlinear programming methods, in: International Series of Numerical Mathematics, Vol. 124, Birkhäuser, Basel, 1998.; Ch. Büskens, H. Maurer, Sensitivity analysis and real-time control of nonlinear optimal control systems via nonlinear programming methods, in: International Series of Numerical Mathematics, Vol. 124, Birkhäuser, Basel, 1998.
[9] Ch. Büskens, H. Maurer, Open-loop real-time control of an industrial robot using nonlinear programming methods, Automatica (1999), to appear.; Ch. Büskens, H. Maurer, Open-loop real-time control of an industrial robot using nonlinear programming methods, Automatica (1999), to appear.
[10] Evtushenko, Yu. G., Numerical Optimization Techniques, Translation Series in Mathematics and Engineering (1985), Optimisation Software Inc. (Publications Division): Optimisation Software Inc. (Publications Division) New York
[11] P.J. Enright, B.A. Conway, Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, AIAA Paper 90-2963-CP, 1990.; P.J. Enright, B.A. Conway, Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, AIAA Paper 90-2963-CP, 1990.
[12] Felgenhauer, U., Diskretisierung von Steuerungsproblemen unter stabilen Optimalitätsbedingungen, Habilitationsschrift, Technische Universität Cottbus, Institut für Mathematik (1998), Cottbus: Cottbus Germany
[13] Felgenhauer, U., On higher order methods for control problems with mixed inequality constraints, Preprint M-01/1998, Technische Universität Cottbus, Institut für Mathematik (1998), Cottbus: Cottbus Germany
[14] A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering, Vol. 165, Academic Press, New York, 1983.; A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering, Vol. 165, Academic Press, New York, 1983. · Zbl 0543.90075
[15] Gill, P. E.; Murray, W.; Saunders, M. A.; Wright, M. H., Model building and practical aspects of nonlinear programming, (Schittkowski, K., Computational Mathematical Programming (1985), Springer: Springer Berlin), 209-247
[16] Grimm, W., Convergence relations between optimal control and optimal para-metric control, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft: Anwendungsbezogene Optimierung und Steuerung, Report No. 420, Institut für Flugmechanik und Flugregelung, Universität Stuttgart (1993), Stuttgart: Stuttgart Germany
[17] Hartl, R. F.; Sethi, S. P.; Vickson, R. G., A survey of the maximum principles for optimal control problems with state constraints, SIAM Rev., 37, 181-218 (1995) · Zbl 0832.49013
[18] Kluever, C. A., Optimal feedback guidance for low-thrust orbit insertion, Opt. Control Appl. Methods, 16, 155-173 (1995) · Zbl 0839.70018
[19] Kugelmann, B.; Pesch, H. J., A new general guidance method in constrained optimal control, Part 1: Numerical method, J. Optim. Theory Appl., 67, 421-435 (1990) · Zbl 0697.49026
[20] Kugelmann, B.; Pesch, H. J., A new general guidance method in constrained optimal control, Part 2: application to space shuttle guidance, J. Optim. Theory Appl., 67, 437-446 (1990) · Zbl 0697.49027
[21] Malanowski, K.; Maurer, H., Sensitivity analysis for parametric control problems with control-state constraints, Comput. Optim. Appl., 5, 253-283 (1996) · Zbl 0864.49020
[22] Malanowski, K.; Maurer, H., Sensitivity analysis for state constrained optimal control problems, Discrete Continuous Dynamical Systems, 4, 241-272 (1998) · Zbl 0952.49022
[23] Malanowski, K.; Büskens, Ch.; Maurer, H., Convergence of approximations to nonlinear optimal control problems, (Fiacco, A. V., Mathematical Programming with Data Perturbations. Mathematical Programming with Data Perturbations, Lecture Notes in Pure and Applied Mathematics, Vol. 195 (1997), Marcel Dekker: Marcel Dekker New York), 253-284 · Zbl 0883.49025
[24] Maurer, H.; Pesch, H. J., Solution differentiability for parametric nonlinear control problems, SIAM J. Control Optim., 32, 1542-1554 (1994) · Zbl 0820.49012
[25] Maurer, H.; Pesch, H. J., Solution differentiability for parametric nonlinear control problems with control-state constraints, J. Optim. Theory Appl., 23, 285-309 (1995) · Zbl 0835.49017
[26] Maurer, H.; Büskens, Ch.; Feichtinger, G., Solution techniques for periodic control problems: a case study in production planning, Opt. Control Appl. Methods, 19, 185-203 (1998)
[27] Michalska, H.; Rehman, F., Guiding functions and discontinuous control: the underwater vehicle example, Internat. J. Control, 69, 1-30 (1998) · Zbl 1041.93547
[28] Y. Nakamuira, S. Savant, Nonlinear tracking control of autonomous underwater vehicles, Proceedings of the 1992 IEEE International Conference on Robotics and Automation, Nice, France, 1992, pp. A4-A9.; Y. Nakamuira, S. Savant, Nonlinear tracking control of autonomous underwater vehicles, Proceedings of the 1992 IEEE International Conference on Robotics and Automation, Nice, France, 1992, pp. A4-A9.
[29] Neustadt, L. W., Optimization: A Theory of Necessary Conditions (1976), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0166.09401
[30] Pesch, H. J., Real-time computation of feedback controls for constrained optimal control problems, Part 1: neighbouring extremals, Opt. Control Appl. Methods, 10, 129-145 (1989) · Zbl 0675.49023
[31] Pesch, H. J., Real-time computation of feedback controls for constrained optimal control problems, Part 2: a correction method based on multiple shooting, Opt. Control Appl. Methods, 10, 147-171 (1989) · Zbl 0675.49024
[32] Pontryagin, L. S., Mathematische Theorie optimaler Prozesse (1967), Oldenbourg Verlag: Oldenbourg Verlag München-Wien · Zbl 0115.09001
[33] von Stryk, O., Numerische Lösung optimaler Steuerungsprobleme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variablen, Fortschritt-Berichte VDI, Reihe 8, Nr. 441 (1995), VDI Verlag: VDI Verlag Germany
[34] Teo, K. L.; Goh, C. J.; Wong, K. H., A Unified Computational Approach to Optimal Control Problems (1991), Longman Scientific and Technical: Longman Scientific and Technical New York · Zbl 0747.49005
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