Advertisement

Neural Networks Solving Free Final Time Optimal Control Problem

  • Tibor Kmet
  • Maria Kmetova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7505)

Abstract

A neural network based optimal control synthesis is presented for solving free final time optimal control problems with control and state constraints. The optimal control problem is transcribed into nonlinear programming problem which is implemented with adaptive critic neural network. The proposed simulation methods is illustrated by the optimal control problem of photosynthetic production. Results show that adaptive critic based systematic approach holds promise for obtaining the free final time optimal control with control and state constraints.

Keywords

feed forward neural network free final time optimal control problem state and control constraints numerical examples photosynthetic production 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bryson Jr., A.E.: Dynamic Optimization. Addison-Wesley Longman Inc., New York (1999)Google Scholar
  2. 2.
    Buskens, C., Maurer, H.: Sqp-methods for solving optimal control problems with control and state constraints: adjoint variable, sensitivity analysis and real-time control. Jour. Comp. Appl. Math. 120, 85–108 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Eilers, P.H.C., Peeters, J.C.H.: A model for relationship between light intensity and the rate of photosynthesis in phytoplankton. Ecol. Modelling 42, 199–215 (1988)CrossRefGoogle Scholar
  4. 4.
    Eilers, P.H.C., Peeters, J.C.H.: Dynamic behaviour of a model for photosynthesis and photoinhibition. Ecol. Modelling 69, 113–133 (1993)CrossRefGoogle Scholar
  5. 5.
    Garcia-Camacho, F., Sanchez-Miron, A., Molina-Grima, E., Camacho-Rubio, F., Merchuck, J.C.: A mechanistic model of photosynthesis in microalgal including photoacclimation dynamics. Jour. Theor. Biol. 304, 1–15 (2012)CrossRefGoogle Scholar
  6. 6.
    Hornik, M., Stichcombe, M., White, H.: Multilayer feed forward networks are universal approximators. Neural Networks 3, 256–366 (1989)Google Scholar
  7. 7.
    Kirk, D.E.: Optimal Control Theory: An Introduction. Dover Publications, Inc., Mineola (1989)Google Scholar
  8. 8.
    Kmet, T.: Neural network simulation of nitrogen transformation cycle. In: Otamendi, J., Bargiela, A., Montes, J.L., Pedrera, L. (eds.) ECMS 2009 - European Conference on Modelling and Simulation, pp. 352–358. ECMS, Madrid (2009)Google Scholar
  9. 9.
    Kmet, T.: Neural network simulation of optimal control problem with control and state constraints. In: Homelka, T., Duch, W., Girolami, M., Kaski, S. (eds.) ICANN 2011. LNCS, vol. 6791, pp. 261–268. Springer, Heidelberg (2011)Google Scholar
  10. 10.
    Kmet, T., Straskraba, M., Mauersberger, P.: A mechanistic model of the adaptation of phytoplankton photosynthesis. Bull. Math. Biol. 55, 259–275 (1993)zbMATHGoogle Scholar
  11. 11.
    Padhi, R., Balakrishnan, S.N.: A single network adaptive critic (snac) architecture for optimal control synthesis for a class of nonlinear systems. Neural Networks 19, 1648–1660 (2006)zbMATHCrossRefGoogle Scholar
  12. 12.
    Padhi, R., Unnikrishnan, N., Wang, X., Balakrishnan, S.N.: Adaptive-critic based optimal control synthesis for distributed parameter systems. Automatica 37, 1223–1234 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Polak, E.: Optimization Algorithms and Consistent Approximation. Springer, Berlin (1997)Google Scholar
  14. 14.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Process. Nauka, Moscow (1983) (in Russian)Google Scholar
  15. 15.
    Rumelhart, D.F., Hinton, G.E., Wiliams, R.J.: Learning internal representation by error propagation. In: Rumelhart, D.E., McClelland, D.E., Group, P.R. (eds.) Parallel Distributed Processing. Foundation, vol. 1, pp. 318–362. Cambridge University Press, Cambridge (1987)Google Scholar
  16. 16.
    Papáček, Š., Čelikovský, S., Rehák, B., Štys, D.: Experimental design for parameter estimation of two time-scale model of photosynthesis and photoinhibition in microalgae. Math. Comp. Sim. 80, 1302–1309 (2010)zbMATHCrossRefGoogle Scholar
  17. 17.
    Wu, X., Merchuk, J.C.: A model integrating fluid dynamics in photosynthesis and photoinhibition. Chem. Ing. Scien. 56, 3527–3538 (2001)CrossRefGoogle Scholar
  18. 18.
    Xia, Y., Feng, G.: A new neural network for solving nonlinear projection equations. Neural Network 20, 577–589 (2007)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tibor Kmet
    • 1
  • Maria Kmetova
    • 1
  1. 1.Constantine the Philosopher UniversityNitraSlovakia

Personalised recommendations