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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Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 39, 930–945 (1993)
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)
Hornik, M., Stichcombe, M., White, H.: Multilayer feed forward networks are universal approximators. Neural Networks 3, 256–366 (1989)
Hrinca, I.: An Optimal Control Problem for the Lotka-Volterra System with Delay. Nonlinear Analysis, Theory, Methods, Applications 28, 247–262 (1997)
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)
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)
Kirk, D.E.: Optimal Control Theory: An Introduction. Dover Publications, Inc., Mineola (1989)
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)
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)
Makozov, Y.: Uniform approximation by neural networks. Journal of Approximation Theory 95, 215–228 (1998)
Padhi, R., Unnikrishnan, N., Wang, X., Balakrishnan, S.N.: Adaptive-critic based optimal control synthesis for distributed parameter systems. Automatica 37, 1223–1234 (2001)
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)
Polak, E.: Optimization Algorithms and Consistent Approximation. Springer, Heidelberg (1997)
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Process. Nauka, Moscow (1983) (in Russian)
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)
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)
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)
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)
Xia, Y., Feng, G.: A New Neural Network for Solving Nonlinear Projection Equations. Neural Network 20, 577–589 (2007)
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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
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