Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T19:18:10.673Z Has data issue: false hasContentIssue false
  • English
  • Français

The control parameterization enhancing transform for constrained optimal control problems

Published online by Cambridge University Press:  17 February 2009

K. L. Teo ,
L. S. Jennings ,
H. W. J. Lee  and
V. Rehbock
K. L. Teo
Affiliation:
School of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987 Perth, WA 6845, Australia
L. S. Jennings
Affiliation:
Centre for Applied Dynamics and Optimization, University of Western Australia, WA 6907, Australia
H. W. J. Lee
Affiliation:
Department of Systems Engineering and Engineering Management, Chinese University of Hong Kong, Shatin, Hong Kong
V. Rehbock
Affiliation:
School of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987 Perth, WA 6845, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider a general class of constrained optimal control problems in canonical form. Using the classical control parameterization technique, the time (planning) horizon is partitioned into several subintervals. The control functions are approximated by piecewise constant or piecewise linear functions with pre-fixed switching times. However, if the optimal control functions to be obtained are piecewise continuous, the accuracy of this approximation process greatly depends on how fine the partition is. On the other hand, the performance of any optimization algorithm used is limited by the number of decision variables of the problem. Thus, the time horizon cannot be partitioned into arbitrarily many subintervals to reach the desired accuracy. To overcome this difficulty, the switching points should also be taken as decision variables. This is the main motivation of the paper. A novel transform, to be referred to as the control parameterization enhancing transform, is introduced to convert approximate optimal control problems with variable switching times into equivalent standard optimal control problems involving piecewise constant or piecewise linear control functions with pre-fixed switching times. The transformed problems are essentially optimal parameter selection problems and hence are solvable by various existing algorithms. For illustration, two non-trivial numerical examples are solved using the proposed method.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 3 , January 1999 , pp. 314 - 335
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Ahmed, N. U., Elements of Finite-Dimensional Systems and Control Theory, Volume 37 of Pitman monographs and Surveys in Pure and Applied Mathematics (Longman Scientific and Technical, England, 1988).Google Scholar
[2]Bellman, R. E. and Dreyfus, R. E., Dynamic Programming and Modern Control Theory (Academic Press, Orlando, Florida, 1977).Google Scholar
[3]Craven, B. D., Control and Optimization (Chapman & Hall, London, 1995).CrossRefGoogle Scholar
[4]Cullum, J., “Finite-dimensional approximations of state constrained continuous optimal control problems”, SIAM Journal on Control 10 (1972) 649–670.CrossRefGoogle Scholar
[5]Hasdorff, L., Gradient Optimization and Nonlinear Control (John Wiley, New York, 1976).Google Scholar
[6]Hicks, G. A. and Ray, W. H., “Approximation methods for optimal control systems”, Canad. J. Chem. Eng. 49 (1971) 522–528.CrossRefGoogle Scholar
[7]Jennings, L. S., Fisher, M. E., Teo, K. L. and Goh, C. J., “MISER3: Solving optimal control problems—an update”, Advances in Engineering Software 13 (1991) 190–196.Google Scholar
[8]Jennings, L. S., Wong, K. H. and Teo, K. L., “An optimal control problem in biomechanics”, in IFAC World Congress, Volume 4, Sydney, Australia, Institute of Engineers Australia, 1993, 311–314.Google Scholar
[9]Kaji, K. and Wong, K. H., “Nonlinearly constrained time-delayed optimal control problems J.”, of Optimization Theory and Applications 82 (1994) 295–313.CrossRefGoogle Scholar
[10]Lee, H. W. J., Teo, K. L., Jennings, L. S. and Rehbock, V., “Control parametrization enhancing technique for time optimal control problem”, Dynamical Systems and Applications, 6 (1997), 243–261.Google Scholar
[11]Miele, A., “Gradient algorithms for the optimization of dynamic systems”, in Control and Dynamic Systems: Advances in Theory and Applications (ed. Leondes, C. T.), Volume 16, (Academic Press, New York, 1980) 1–52.CrossRefGoogle Scholar
[12]Miele, A. and Wang, T., “Primal-dual properties of sequential gradient-restoration algorithms for optimal control problems, part 1, basic problems”, in Integral Methods in Science and Engineering (ed. Payne, F. R. et al. ), (Hemisphere Publishing Corporation, Washington, DC, 1986) 577–607.Google Scholar
[13]Miele, A. and Wang, T., “Primal-dual properties of sequential gradient-restoration algorithms for optimal control problems, Part 2, General problem”, J. Math. Anal. Appl. 119 (1986) 21–54.CrossRefGoogle Scholar
[14]Polak, E., Computational Methods in Optimization (Academic Press, New York, 1971).Google Scholar
[15]Polak, E. and Mayne, D. Q., “A feasible directions algorithm for optimal control problems with control and terminal inequality constraints”, IEEE Trans. on Automatic Control AC-22 (1977) 741–751.CrossRefGoogle Scholar
[16]Sakawa, Y. and Shindo, Y., “On the global convergence of an algorithm for optimal control”, IEEE Trans. on Automatic Control AC-25 (1980) 1149–1153.CrossRefGoogle Scholar
[17]Sakawa, Y. and Shindo, Y., “Optimal control of container cranes”, Automatica 18 (1982) 257–266.CrossRefGoogle Scholar
[18]Schittkowski, K., “NLPQL: A FORTRAN subroutine solving constrained nonlinear programming problems”, Operations Research Annals 5 (1985) 485–500.CrossRefGoogle Scholar
[19]Schittkowski, K., “On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function”, Math. Operationsforschung und Statistik, Ser. Optimization 14 (1993) 197–216.CrossRefGoogle Scholar
[20]Sirisena, H. R. and Tan, K. S., “Computation of constrained optimal controls using parametrization techniques”, IEEE Trans. on Automatic Control AC-19 (1974) 431–433.CrossRefGoogle Scholar
[21]Teo, K. L., Goh, C. J. and Wong, K. H., A Unified Computational Approach to Optimal Control Problems, Volume 55 of Pitman monographs and Surveys in Pure and Applied Mathematics (Longman Scientific and Technical, England, 1991).Google Scholar
[22]Teo, K. L., Rehbock, V. and Jennings, L. S., “A new computational algorithm for functional inequality constrained optimization problems”, Automatica 29 (1993) 789–792.CrossRefGoogle Scholar
[23]Zhou, J. L., Tits, A. L. and Lawrence, C. T., “User's guide for ffsqp version 3.6: A fortran code for solving constrained nonlinear (minimax) optimization problems, generating iterates satisfying all inequality and linear constraints”, Report no. Systems Research Center TR-92–107r2, Electrical Engineering Dept and Institute for Systems Research, University of Maryland, College Park, MD 20742, 1996.Google Scholar