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Adaptive Critic Neural Network Solution of Optimal Control Problems with Discrete Time Delays

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Artificial Neural Networks and Machine Learning – ICANN 2013 (ICANN 2013)

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Abstract

A neural network based optimal control synthesis is presented for solving optimal control problems with discrete time delays in state and control variables subject to a control and state constraints. The optimal control problem is transcribed into nonlinear programming problem which is implemented with feed forward adaptive critic neural network to find optimal control and optimal trajectory. The proposed simulation methods is illustrated by the optimal control problem of nitrogen transformation cycle model with discrete time delay of nutrient uptake. Results show that adaptive critic based systematic approach are promising in obtaining the optimal control with discrete time delays in state and control variables subject to control and state constraints.

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References

  1. Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 39, 930–945 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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. Journal of Computational and Applied Mathematics 120, 85–108 (2000)

    Article  MathSciNet  Google Scholar 

  3. Hornik, M., Stichcombe, M., White, H.: Multilayer feed forward networks are universal approximators. Neural Networks 3, 256–366 (1989)

    Google Scholar 

  4. Hrinca, I.: An Optimal Control Problem for the Lotka-Volterra System with Delay. Nonlinear Analysis, Theory, Methods, Applications 28, 247–262 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gnecco, A.: A Comparison Between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization. Journal of Applied Mathematics 2012, article ID 806945 (2012)

    Google Scholar 

  6. Gollman, L., Kern, D., Mauer, H.: Optimal control problem with delays in state and control variables subject to mixed control-state constraints. Optim. Control Appl. Meth. 30, 341–365 (2009)

    Article  Google Scholar 

  7. Kirk, D.E.: Optimal Control Theory: An Introduction. Dover Publications, Inc., Mineola (1989)

    Google Scholar 

  8. Kmet, T.: Material recycling in a closed aquatic ecosystem. I. Nitrogen transformation cycle and preferential utilization of ammonium to nitrate by phytoplankton as an optimal control problem. Bull. Math. Biol. 58, 957–982 (1996)

    MATH  Google Scholar 

  9. Kmet, T.: Neural network solution of optimal control problem with control and state constraints. In: Honkela, T. (ed.) ICANN 2011, Part II. LNCS, vol. 6792, pp. 261–268. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  10. Makozov, Y.: Uniform approximation by neural networks. Journal of Approximation Theory 95, 215–228 (1998)

    Article  MathSciNet  Google Scholar 

  11. Padhi, R., Unnikrishnan, N., Wang, X., Balakrishnan, S.N.: Adaptive-critic based optimal control synthesis for distributed parameter systems. Automatica 37, 1223–1234 (2001)

    Article  MATH  Google Scholar 

  12. Padhi, R., Balakrishnan, S.N., Randoltph, T.: A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems. Neural Networks 19, 1648–1660 (2006)

    Article  MATH  Google Scholar 

  13. Polak, E.: Optimization Algorithms and Consistent Approximation. Springer, Heidelberg (1997)

    Google Scholar 

  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. Rumelhart, D.F., Hinton, G.E., Wiliams, R.J.: Learning internal representation by error propagation. In: Rumelhart, D.E., McClelland, D.E. (eds.) PDP Research Group: Parallel Distributed Processing: Foundation, vol. 1, pp. 318–362. The MIT Press, Cambridge (1987)

    Google Scholar 

  16. Sandberg, E.W.: Notes on uniform approximation of time-varying systems on finite time intervals. IEEE Transactions on Circuits and Systems-1: Fundamental Theory and Applications 45, 305–325 (1998)

    Google Scholar 

  17. Sun, D.Y., Huang, T.C.: A solutions of time-delayed optimal control problems by the use of modified line-up competition algorithm. Journal of the Taiwan Institute of Chemical Engineers 41, 54–64 (2010)

    Article  Google Scholar 

  18. Werbos, P.J.: Approximate dynamic programming for real-time control and neural modelling. In: White, D.A., Sofge, D.A. (eds.) Handbook of Intelligent Control: Neural Fuzzy, and Adaptive Approaches, pp. 493–525 (1992)

    Google Scholar 

  19. Xia, Y., Feng, G.: A New Neural Network for Solving Nonlinear Projection Equations. Neural Network 20, 577–589 (2007)

    Article  MATH  Google Scholar 

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Kmet, T., Kmetova, M. (2013). Adaptive Critic Neural Network Solution of Optimal Control Problems with Discrete Time Delays. In: Mladenov, V., Koprinkova-Hristova, P., Palm, G., Villa, A.E.P., Appollini, B., Kasabov, N. (eds) Artificial Neural Networks and Machine Learning – ICANN 2013. ICANN 2013. Lecture Notes in Computer Science, vol 8131. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40728-4_61

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  • DOI: https://doi.org/10.1007/978-3-642-40728-4_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40727-7

  • Online ISBN: 978-3-642-40728-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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