EVOLUTIONARY COMPUTATION FOR DISCRETE AND CONTINUOUS TIME OPTIMAL CONTROL PROBLEMS

Yechiel Crispin

2005

Abstract

Nonlinear discrete time and continuous time optimal control problems with terminal constraints are solved using a new evolutionary approach which seeks the control history directly by evolutionary computation. Unlike methods that use the first order necessary conditions to determine the optimum, the main advantage of the present method is that it does not require the development of a Hamiltonian formulation and consequently, it eliminates the requirement to solve the adjoint problem which usually leads to a difficult two-point boundary value problem. The method is verified on two benchmark problems. The first problem is the discrete time velocity direction programming problem with the effects of gravity, thrust and drag and a terminal constraint on the final vertical position. The second problem is a continuous time optimal control problem in rocket dynamics, the Goddard’s problem. The solutions of both problems compared favorably with published results based on gradient methods.

References

  1. A.S. Bondarenko, D. B. and More, J. (1999). Cops: Large scale nonlinearly constrained optimization problems. In Argonne National Laboratory Technical Report ANL/MCS-TM-237.
  2. Betts, J. (2001). Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA.
  3. Bryson, A. (1999). Dynamic Optimization. AddisonWesley Longman, Menlo Park, CA.
  4. Coleman, T. and Liao, A. (1995). An efficient trust region method for unconstrained discrete-time optimal control problems. In Computational Optimization and Applications, 4:47-66.
  5. Dolan, E. and More, J. (2000). Benchmarking optimization software with cops. In Argonne National Labo-ratory Technical Report ANL/MCS-TM-246.
  6. Dunn, J. and Bertsekas, D. (1989). Efficient dynamic programming implementations of newton's method for unconstrained optimal control problems. In J. of Optimization Theory and Applications,63, pp. 23-38.
  7. Fogel, D. (1998). Evolutionary Computation, The Fossil Record. IEEE Press, New York.
  8. Fox, C. (1950). An Introduction to the Calculus of Variations. Oxford University Press, London.
  9. J. Betts, S. E. and Huffman, W. (1993). Sparse nonlinear programming test problems. In Technical Report BCSTECH-93-047. Boeing Computer Services, Seattle, Washington.
  10. Jacobson, D. and Mayne, D. (1970). Differential Dynamic Programming. Elsevier Science Publishers, Amsterdam, Netherland.
  11. Kirkpatrick, G. and Vecchi (1983). Optimization by simulated annealing. In Science, 220: 671-680. AAAS.
  12. Laarhoven, P. and Aarts, E. (1989). Simulated Annealing: Theory and Applications. Kluwer, Amsterdam.
  13. Liao, L. and Shoemaker, C. (1991). Convergence in unconstrained discrete-time differential dynamic programming. In IEEE Transactions on Automatic Control, 36, pp. 692-706. IEEE.
  14. L.S. Pontryagin, V.G. Boltyanskii, R. G. and Mishchenko, E. (1962). The Mathematical Theory of Optimal Processes. translated by K.N. Trirogoff, L.W. Neustadt (Ed.), Moscow, interscience, new york edition.
  15. Mayne, D. (1966). A second-order gradient method for determining optimal trajectories of nonlinear discrete time systems. In International Journal of Control, 3, pp. 85-95.
  16. Michalewicz, Z. (1992). Genetic Algorithms + Data Structures= Evolution Programs. Springer-Verlag, Berlin.
  17. Murray, D. and Yakowitz, S. (1984). Differential dynamic programming and newton's method for discrete optimal control problems. In J. of Optimization Theory and Applications, 43:395-414.
  18. Ohno, K. (1978). A new approach of differential dynamic programming for discrete time systems. In IEEE Transactions on Automatic Control, 23, pp. 37-47. IEEE.
  19. Pantoja, J. (1988). Differential dynamic programming and newton's method. In International Journal of Control, 53: 1539-1553.
  20. Schwefel, H. (1995). Evolution and Optimum Seeking. Wiley, New York.
  21. Yakowitz, S. and Rutherford, B. (1984). Computational aspects of discrete-time optimal control. In Appl. Math. Comput., 15, pp. 29-45.
  22. Z. Michalewicz, C. J. and Krawczyk, J. (1992). A modified genetic algorithm for optimal control problems. In Computers Math. Appl., 23(12),8394.
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Paper Citation


in Harvard Style

Crispin Y. (2005). EVOLUTIONARY COMPUTATION FOR DISCRETE AND CONTINUOUS TIME OPTIMAL CONTROL PROBLEMS . In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO, ISBN 972-8865-29-5, pages 45-54. DOI: 10.5220/0001171200450054


in Bibtex Style

@conference{icinco05,
author={Yechiel Crispin},
title={EVOLUTIONARY COMPUTATION FOR DISCRETE AND CONTINUOUS TIME OPTIMAL CONTROL PROBLEMS},
booktitle={Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},
year={2005},
pages={45-54},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0001171200450054},
isbn={972-8865-29-5},
}


in EndNote Style

TY - CONF
JO - Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,
TI - EVOLUTIONARY COMPUTATION FOR DISCRETE AND CONTINUOUS TIME OPTIMAL CONTROL PROBLEMS
SN - 972-8865-29-5
AU - Crispin Y.
PY - 2005
SP - 45
EP - 54
DO - 10.5220/0001171200450054