Learning Techniques and Neural Networks for the Solution of N-Stage Nonlinear Nonquadratic Optimal Control Problems

  • R. Zoppoli
  • T. Parisini
Chapter
Part of the Progress in Systems and Control Theory book series (PSCT, volume 12)

Abstract

This paper deals with the problem of designing closed-loop feed-forward control strategies to drive the state of a dynamic system (in general, nonlinear) so as to track any desired trajectory joining the points of given compact sets, while minimizing a certain cost function (in general, nonquadratic). Due to the generality of the problem, conventional methods (e.g., dynamic programming, maximum principle, etc.) are difficult to apply. Then, an approximate solution is sought by constraining control strategies to take on the structure of multi-layer feed-forward neural networks. After discussing the approximation properties of neural control strategies, a particular neural architecture is presented, which is based on what has been called the “Linear-Structure Preserving Principle” (the LISP principle). The original functional problem is then reduced to a nonlinear programming one, and backpropagation is applied to derive the optimal values of the synaptic weights. Recursive equations to compute the gradient components are presented, which generalize the classical adjoint system equations of N-stage optimal control theory. Simulation results related to non-LQ problems show the effectiveness of the proposed method.

Keywords

Cost Function Optimal Control Problem Synaptic Weight State Trajectory Optimal Control Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A. P. Sage. Optimum Systems Control. Prentice-Hall, Englewood Cliffs, 1968.Google Scholar
  2. [2]
    A. E. Bryson and Y. C. Ho. Applied Optimal Control. Blaisdell Publishing Company, 1969.Google Scholar
  3. [3]
    R. E. Larson. State Increment Dynamic Programming. American Elsevier Publishing Company, 1968.Google Scholar
  4. [4]
    B. R. Eisenberg and A. P. Sage. “Closed-Loop Optimization of Fixed Configuration Systems”. International Journal of Control, Vol. 3, pp. 183–194, 1966.CrossRefGoogle Scholar
  5. [5]
    D.L. Kleinman and M. Athans. “The Design of Suboptimal Linear Time-Varying Systems”. IEEE Trans. Automatic Control, Vol. AC-13, pp. 150–159, 1968.Google Scholar
  6. [6]
    P. M. Mäkilä and H. T. Toivonen. “Computational Methods for Parametric LQ Problems- A survey”. IEEE Trans. Automatic Control, Vol. AC-32, pp. 658–671, 1987.Google Scholar
  7. [7]
    T. Parisini and R. Zoppoli. “Neural Networks for the Solution of N-Stage Optimal Control Problems”, Artificial Neural Networks, T. Kohonen, K. Mäkisara, O. Simula, and J. Kangas, Eds., North-Holland, 1991.Google Scholar
  8. [8]
    T. Parisini and R. Zoppoli. “Multi-Layer Neural Networks for the Optimal Control of Nonlinear Dynamic Systems”, Proc. First IFAC Symposium on Design Methods of Control Systems, Zurich, Switzerland, 1991.Google Scholar
  9. [9]
    D. H. Nguyen and B. Widrow. “Neural Networks for Self-Learning Control Systems”. IEEE Control System Magazine, Vol. 10, pp. 18–23, 1990.CrossRefGoogle Scholar
  10. [10]
    K. Hornik, M. Stinchombe, and H. White. “Multilayer Feedforward Networks are Universal Approximators”. Neural Networks, Vol. 2, pp. 359–366, 1989.CrossRefGoogle Scholar
  11. [11]
    R. Hecht-Nielsen. “Theory of the. Backpropagation Neural Network”. Proc. Int. Joint Conf. on Neural Networks, San Diego, CA, 1989, pp. 593–605.CrossRefGoogle Scholar
  12. [12]
    A. R. Barron. “Universal Approximation Bounds for Superpositions of a Sigmoidal Function”. Technical Report No. 58, Department of Statistics, University of Illinois, 1991.Google Scholar
  13. [13]
    T. Parisini and R. Zoppoli. “Backpropagation and Dynamic Programming in N-Stage Optimal Control Problems”. DIST Int. Rep. No. 90/4, University of Genoa, Italy, 1991.Google Scholar
  14. [14]
    Ya. Z. Tsypkin. Adaptation and Learning in Automatic Systems. Academic Press, 1971.Google Scholar
  15. [15]
    D. E. Rumelhart and J. L. McClelland. Parallel Distributed Processing, Vol. 1, Chapter 8, Cambridge, MA, MIT Press, 1986.Google Scholar
  16. [16]
    P. J. Werbos. “Beyond Regression: New Tools for Prediction and Analysis in the Behavioural Sciences”. Ph.D. Dissertation, Harvard University, Cambridge, MA, 1974.Google Scholar
  17. [17]
    G. Frisiani, T. Parisini, L. Siccardi, and R. Zoppoli. “Team Theory and Back-Propagation for Dynamic Routing in Communication Networks”. Proc. Int. Joint Conf. on Neural Networks, Seattle, WA, 1991, Vol. 1, pp. 325–334.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • R. Zoppoli
    • 1
  • T. Parisini
    • 1
  1. 1.Department of Communications, Computers and Systems SciencesUniversity of GenoaItaly

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