Abstract
To handle final state equality constraints, two types of penalty functions are evaluated and compared in solving three optimal control problems. Both the absolute value penalty function and the quadratic penalty function with shifting terms yielded the optimal control policy in each case. The quadratic penalty function with shifting terms gave the optimum more accurately, and the approach to the optimum is less oscillatory than with the use of absolute value penalty function. In solving for the optimal control in the third example, the use of the absolute value penalty function required a series of runs to get the performance index to within 0.035% of the optimum. A further advantage in using the quadratic penalty function is obtaining sensitivity information with respect to the final state constraints from the shifting terms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bojkov, B. and Luus, R. (1994), Application of iterative dynamic programming to time optimal control, Chem. Eng. Res. Des., Vol. 72, pp. 72–80.
Chen, G. and Mills, W. H. (1981), Finite elements and terminal penalization for quadratic cost optimal control problems governed by ordinary differential equations, SIAM J. Control and Optimization, Vol. 19, pp. 744–764.
Dadebo, S. A. and McAuley, K. B. (1995), Dynamic optimization of constrained chemical engineering problems using dynamic programming, Computers and Chemical Engineering, Vol. 19, pp. 513–525.
Edgar, T. F. and Lapidus, L. (1972), The computation of optimal singular bang-bang control II. Nonlinear systems, AIChE Journal, Vol. 18, pp. 780–785.
Goh, C. J. and Teo, K.L. (1988), Control parametrization: a unified approach to optimal control problems with generalized constraints, Automatica, Vol. 24, pp. 3–18.
Hull, T. E., Enright, W. D. and Jackson, K. R. (1976), User Guide to DVERK-a Subroutine for Solving Nonstiff ODE’s. Report 100 (1976); Department of Computer Science, University of Toronto, Canada.
Lapidus, L. and Luus, R. (1967), Optimal Control of Engineering Processes, Blaisdell, Waltham, Mass.
Luus, R. (1974a), Optimal control by direct search on feedback gain matrix, Chemical Engineering Science, Vol. 29, pp. 1013–1017.
Luus, R. (1974b), A practical approach to time-optimal control of nonlinear systems, Industrial and Engineering Chemistry Process Design and Development, Vol. 13, pp. 405–408.
Luus, R. (1989), Optimal control by dynamic programming using accessible grid points and region reduction, Hungarian Journal of Industrial Chemistry, Vol. 17, pp. 523–543.
Luus, R. (1991), Application of iterative dynamic programming to state constrained optimal control problems, Hungarian Journal of Industrial Chemistry, Vol. 19, pp. 245–254.
Luus, R. (1996a), Handling difficult equality constraints in direct search optimization, Hungarian Journal of Industrial Chemistry, Vol. 24, pp. 285–290.
Luus, R. (1996b), Use of iterative dynamic programming with variable stage lengths and fixed final time, Hungarian Journal of Industrial Chemistry, Vol. 24, pp. 279–284.
Luus, R. (2000), Iterative Dynamic Programming, Chapman &amo; Hall, CRC, London, UK.
Luus, R. and Hennessy, D. (1999), Optimization of fed-batch reactors by the Luus-Jaakola optimization procedure, Ind. Eng. Chem. Res., Vol. 38, pp. 1948–1955.
Luus, R. and Jaakola, T. H. I. (1973), Optimization by direct search and systematic reduction of the size of search region, AIChE Journal, Vol. 19, pp. 760–766.
Luus, R., Mekarapiruk, W. and Storey, C. (1998), Evaluation of penalty functions for optimal control, in: Proc. of International Conference on Optimization Techniques and Applications (ICOTA ’98), Perth, Western Australia, July 1–3, 1998, Curtin Printing Services, pp. 724–731.
Luus, R. and Rosen, O. (1991), Application of dynamic programming to final state constrained optimal control problems, Ind. Eng. Chem. Res., Vol. 30, pp. 1525–1530.
Luus, R. and Storey, C. (1997), Optimal control of final state constrained systems, in: Proc. IASTED International Conference on Modelling, Simulation and Control, Singapore, Aug.11–14, 1997, pp. 245–249.
Luus, R. and Wyrwicz, R. (1996), Use of penalty functions in direct search optimization, Hungarian Journal of Industrial Chemistry, Vol. 24, pp. 273–278.
Mekarapiruk, W. and Luus, R. (1997a), Optimal control of inequality state constrained systems, Ind. Eng. Chem. Res., Vol. 36, pp. 1686–1694.
Mekarapiruk, W. and Luus, R. (1997b), Optimal control of final state constrained systems, Canadian Journal of Chemical Engineering, Vol. 75, pp. 806–811.
Nishida, N., Liu, Y. A., Lapidus, L. and Hiratsuka, S. (1976), An effective computational algorithm for suboptimal singular and/or bang-bang control, AIChE Journal, Vol. 22, pp. 505–523.
Rosen, O. and Luus, R. (1991), Evaluation of gradients for piecewise constant optimal control, Computers and Chemical Engineering, Vol. 15, pp. 273–281.
Sakawa, Y. and Shindo, Y. (1982), Optimal control of container cranes, Automatic, Vol. 18, pp. 257–266.
Teo, K. L., Goh, C. J. and Wong, K. H. (1991), A Unified Computational Approach to Optimal Control Problems, Longman Scientific &amo; Technical, Wiley, New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Luus, R., Mekarapiruk, W., Storey, C. (2001). Evaluation of Penalty Functions for Optimal Control. In: Yang, X., Teo, K.L., Caccetta, L. (eds) Optimization Methods and Applications. Applied Optimization, vol 52. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3333-4_5
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3333-4_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4850-2
Online ISBN: 978-1-4757-3333-4
eBook Packages: Springer Book Archive