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Approximating the Solution of Optimal Control Problems by Fuzzy Systems

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Abstract

In this paper, the ability of fuzzy systems is used to estimate the solution of crisp optimal control problems. To solve an optimal control problem, first the well-known Euler–Lagrange conditions are obtained and then, the solution of these conditions is approximated by defining a trial solution based on fuzzy systems. The parameters of fuzzy systems are adjusted by an optimization algorithm. Numerical examples and comparisons with exact solutions reveal the capability and accuracy of proposed method.

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References

  1. Rubio JE (1986) Control and optimization, the linear treatment of nonlinear problems. Manchester University Press, Manchester

    Google Scholar 

  2. Geering Hans P (2007) Optimal control with engineering applications. Springer, Berlin

    MATH  Google Scholar 

  3. Kirk DE (1970) Optimal control theory, an introduction. Dover Publications, Mineola, NY

    Google Scholar 

  4. Krabs W, Pickl S (2010) An optimal control problem in cancer chemotherapy. Appl Math Comput 217:1117–1124

    MathSciNet  MATH  Google Scholar 

  5. Kirk DE (2004) Optimal control theory-an introduction. Dover Pub. Inc., New York

    Google Scholar 

  6. Lewis F, Syrmos VL (1995) Optimal control. Wiley, New York

    Google Scholar 

  7. Modares H, Naghibi MB (2011) Sistani solving nonlinear optimal control problems using a hybrid IPSOSQP algorithm. Eng Appl Artif Intell 24:476–484

    Article  Google Scholar 

  8. Hilscher RS, Zeidan V (2012) HamiltonJacobi theory over time scales and applications to linear-quadratic problems. Nonlinear Anal 75:932–950

    Article  MathSciNet  MATH  Google Scholar 

  9. Berkani S, Manseur F, Maidi A (2012) Optimal control based on the variational iteration method. Comput Math Appl 64:604–610

    Article  MathSciNet  MATH  Google Scholar 

  10. Garg D, Patterson M, Hagera WW, Raoa AV, Bensonb DA, Huntington GT (2010) A unified framework for the numerical solution of optimal control problems using pseudo-spectral methods. Automatica 46:1843–1851

    Article  MathSciNet  Google Scholar 

  11. Clever D, Lang J, Ulbrich S, Ziems JC (2010) Combination of an adaptive multilevel SQP method and a space-time adaptive PDAE solver for optimal control problems. Procedia Comput Sci 1:1435–1443

    Article  Google Scholar 

  12. de Oliveira VA, Silva GN, Rojas-Medar MA (2009) KT-invexity in optimal control problems. Nonlinear Anal 71:4790–4797

    Article  MathSciNet  MATH  Google Scholar 

  13. Gerdts M (2008) A non-smooth Newton’s method for control-state constrained optimal control problems. Math Comput Simul 79:925–936

    Article  MathSciNet  MATH  Google Scholar 

  14. Buldaev AS (2008) Perturbation methods in optimal control problems. Ecol Modell 216:157–159

    Article  MathSciNet  Google Scholar 

  15. Salama AA (2006) Numerical methods based on extended one-step methods for solving optimal control problems. Appl Math Comput 183:243–250

    MathSciNet  MATH  Google Scholar 

  16. Zhou YY, Yangb XQ, Teob KL (2006) The existence results for optimal control problems governed by a variational inequality. J Math Anal Appl 321:595–608

    Article  MathSciNet  Google Scholar 

  17. Chryssoverghi I, Coletsos I, Kokkinis B (2006) Discretization methods for optimal control problems with state constraints. J Comput Appl Math 191:1–31

    Article  MathSciNet  MATH  Google Scholar 

  18. England R, Gomez S, Lamourc R (2005) Expressing optimal control problems as differential algebraic equations. Comput Chem Eng 29:1720–1730

    Article  Google Scholar 

  19. Rodriguez A (2004) On the local stability of the solution to optimal control problems. J Econ Dyn Control 28:2475–2484

    Article  MathSciNet  MATH  Google Scholar 

  20. Nik H Saberi, Effati S, Shirazian M (2012) An approximate-analytical solution for the Hamilton-Jacobi-Bellman equation via homotopy perturbation method. Appl Math Modell 36:5614–5623

    Article  MathSciNet  MATH  Google Scholar 

  21. Cheng T, Sun H, Qu Z, Lewis FL (2009) Neural network solution for suboptimal control of non-holonomic chained form system. Trans Inst Meas Control 31(6):475–494

    Article  Google Scholar 

  22. Cheng T, Lewis FL (2007) Neural network solution for finite-horizon H-infinity constrained optimal control of nonlinear systems. J Control Theory Appl 5:1–11

    Article  MathSciNet  Google Scholar 

  23. Cheng T, Lewis FL, Abu-Khalaf M (2007) Fixed-final-time-constrained optimal control of nonlinear systems using neural network HJB approach. IEEE Trans Neural Netw 18:1725–1737

    Article  Google Scholar 

  24. Vrabie D, Lewis F (2009) Neural network approach to continuous-time direct adaptive optimal control for partially unknown nonlinear systems. Neural Netw 22:237–246

    Article  Google Scholar 

  25. Effati S, Pakdaman M (2013) Optimal control problem via neural networks. Neural Comput Appl 23:2093–2100

    Article  Google Scholar 

  26. Li-Xin W (1997) A course in fuzzy systems and control. Prentice Hall PTR, Upper Saddle River

    MATH  Google Scholar 

  27. Shirazian M, Effati S (2012) Solving a class of nonlinear optimal control problems via He’s variational iteration method. Int J Control Autom Syst 10:249–256

    Article  Google Scholar 

  28. Maidi A, Pierre J (2012) Corriou optimal control of nonlinear chemical processes using the variational iteration method. In: 8th IFAC symposium on advanced control of chemical processes. Furama Riverfront, Singapore, July 10–13, pp. 898–903

  29. Burghes DN, Graham A (2004) Control and optimal control theories with applications. Woodhead Publishing, Chichester

    Book  MATH  Google Scholar 

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Acknowledgments

The authors are thankful to the unknown referees for their valuable and informative comments that have significantly improved the quality of the paper. They also appreciate the Editors and Editor in Chief for their valuable suggestions.

Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

    Morteza Pakdaman & Sohrab Effati

  2. Center of Excellence on Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran

    Morteza Pakdaman & Sohrab Effati

Authors
  1. Morteza Pakdaman
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  2. Sohrab Effati
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Correspondence to Morteza Pakdaman.

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Pakdaman, M., Effati, S. Approximating the Solution of Optimal Control Problems by Fuzzy Systems. Neural Process Lett 43, 667–686 (2016). https://doi.org/10.1007/s11063-015-9440-7

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  • DOI: https://doi.org/10.1007/s11063-015-9440-7

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