Abstract
An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.
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PONTRYAGIN, L. S., BOLTYANSKII, R.V., GAMKRELIDZE, R. V., and MISHENKO, E.F., The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, NY, 1962.
KAYA, C.Y., Control in the Plane and Time-Optimal Bang-Bang Control, PhD Thesis, University of Western Australia, Nedlands, Western Australia, Australia, 1995.
KAYA, C. Y., and NOAKES, J. L., Computations and Time-Optimal Controls, Optimal Control Applications and Methods, Vol. 17, pp. 171-185, 1996.
SCRIVENER, S. L., and THOMPSON, R.C., Survey of Time-Optimal Attitude Maneuvers, Journal of Guidance, Control, and Dynamics, Vol. 17, pp. 225-233, 1994.
MOHLER, R.R., Bilinear Control Processes, Chapter 3, Academic Press, New York, NY, 1973.
MOHLER, R.R., Nonlinear Systems, Vol.2: Applications to Bilinear Control, Chapter 7, Prentice Hall, Englewood Cliffs, New Jersey, 1991.
MEIER, E. B., and BRYSON, Jr., A. E., Efficient Algorithm for Time-Optimal Control of a Two-Link Manipulator, Journal of Guidance, Control, and Dynamics, Vol. 13, pp. 859-866, 19
TEO, K. L., GOH, C. J., and WONG, K.H., AUnified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, England, 1991.
WONG, K.H., CLEMENTS, D. J., and TEO, K. L., Optimal Control Computation for Nonlinear Time-Lag Systems, Journal of Optimization Theory and Applications, Vol. 47, pp. 91-107, 1985.
KAYA, C. Y., and NOAKES, J.L., A Global Control Law with Implications in Time-Optimal Control, Proceedings of the 33rd IEEE Conference on Decision and Control, Orlando, Florida, pp. 3823-3824, 1994.
JENNINGS, L. S., FISHER, M. E., TEO, K. L., and GOH, C.J., MISER3 Optimal Control Software: Theory and User Manual, EMCOSS, Carine, Western Australia, Australia, 1990.
MISER3.1 Optimal Control Software, Department of Mathematics, University of Western Australia, Australia, 1991.
MISER 3.2 Optimal Control Software, Department of Mathematics, University of Western Australia, Australia, 1997.
VINCENT, T. L., and GRANTHAM, W.J., Optimality in Parametric Systems, Wiley-Interscience, New York, NY, 1981.
LEE, H. W. J., TEO, K. L., REHBOCK, V., and JENNINGS, L.S., Control Parametrization Enhancing Technique for Time-Optimal Control Problems, Dynamic Systems and Applications, Vol. 6, pp. 243-262, 1997.
LUENBERGER, D.G., Linear and Nonlinear Programming, Addison-Wesley, Reading, Massachusetts, 1989.
GARRARD, W. L., and JORDAN, J.M., Design of Nonlinear Automatic Control Systems, Automatica, Vol. 13, pp. 497-505, 1977.
DESROCHERS, A. A., and AL-JAAR, R.Y., Nonlinear Model Simplification in Flight Control System Design, AIAA Journal of Guidance, Control, and Dynamics, Vol. 7, pp. 684-689, 1984.
BANKS, S. P., and MHANA, K.J., Optimal Control and Stabilization of Nonlinear Systems, IMA Journal of Mathematical Control and Information, Vol. 9, pp. 179-196, 1992.
REHBOCK, V., TEO, K. L., and JENNINGS, L.S., Suboptimal Feedback Control for a Class of Nonlinear Systems Using Spline Interpolation, Discrete and Continuous Dynamical Systems, Vol. 1, pp. 223-236, 1994.
LUCAS, S. K., and KAYA, C.Y., Switching Time Computation for Bang-Bang Control Laws, Proceedings of the 2001 American Control Conference, pp. 176-181, 2001.
MAURER, H., Numerical Solution of Singular Control Problems Using Multiple-Shooting Techniques, Journal of Optimization Theory and Applications, Vol. 18, pp. 235-257, 1976.
FRASER-ANDREWS, G., Finding Candidate Singular Optimal Controls: A State of the Art Survey, Journal of Optimization Theory and Applications, Vol. 60, pp. 173-190, 1989.
FRASER-ANDREWS, G., Numerical Methods for Singular Optimal Control, Journal of Optimization Theory and Applications, Vol. 61, pp. 377-401, 1989.
BÜSKENS, C., PESCH, H. J., and WINDERL, S., Real-Time Solutions of Bang-Bang and Singular Optimal Control Problems, Online Optimization of Large Scale Systems, Edited by M. Grotschel, S. O. Krumke, and J. Rambau, Springer Verlag, Berlin, Germany, pp. 129-142, 2001.
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Kaya, C., Noakes, J. Computational Method for Time-Optimal Switching Control. Journal of Optimization Theory and Applications 117, 69–92 (2003). https://doi.org/10.1023/A:1023600422807
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DOI: https://doi.org/10.1023/A:1023600422807