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Optimal Control of Delay Systems by Using a Hybrid Functions Approximation

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Abstract

In this paper, a new numerical method for solving the optimal control of linear time-varying delay systems with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions, consisting of block-pulse functions and Bernoulli polynomials, are presented. The operational matrices of integration, product, delay and the integration of the cross product of two hybrid functions of block-pulse and Bernoulli polynomials vectors are given. These matrices are then utilized to reduce the solution of the optimal control of delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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References

  1. Jamshidi, M., Wang, C.M.: A computational algorithm for large-scale nonlinear time-delay systems. IEEE Trans. Syst. Man Cybern. 14, 2–9 (1984)

    MathSciNet  MATH  Google Scholar 

  2. Kwakernaak, H., Sivan, R.: Linear Optimal Control Systems. Wiley-Interscience, New York (1972)

    MATH  Google Scholar 

  3. Khellat, F.: Optimal control of linear time-delayed systems by linear Legendre multiwavelets. J. Optim. Theory Appl. 143, 107–121 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kharatishvili, G.L.: The maximum principle in the theory of optimal process with time-lags. Dokl. Akad. Nauk SSSR 136, 39–42 (1961)

    Google Scholar 

  5. Inoue, K., Akashi, H., Ogino, K., Sawaragi, Y.: Sensitivity approaches to optimization of linear systems with time-delay. Automatica 17, 671–676 (1971)

    Article  MathSciNet  Google Scholar 

  6. Jamshidi, M., Razzaghi, M.: Optimization of linear systems with input time-delay. Kybernetika 11, 375–384 (1975)

    MathSciNet  MATH  Google Scholar 

  7. Malek-Zavarei, M., Jamshidi, M.: Time-Delay Systems: Analysis, Optimization and Applications. North-Holland, Amsterdam (1978)

    Google Scholar 

  8. Alekal, Y., Brunovsky, P., Chyung, D.H., Lee, E.B.: The quadratic problem for systems with time delays. IEEE Trans. Autom. Control 16, 673–687 (1971)

    Article  MathSciNet  Google Scholar 

  9. Delfour, M.C.: The linear quadratic control problem with delays in state and control variables: a state space approach. SIAM J. Control Optim. 24, 835–883 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eller, D.H., Aggarwal, J.K., Banks, H.T.: Optimal control of linear time-delay systems. IEEE Trans. Autom. Control 14, 678–687 (1969)

    Article  MathSciNet  Google Scholar 

  11. Uchida, K., Shimemura, E., Kubo, T., Abe, N.: The linear-quadratic optimal control approach to feedback control design for systems with delay. Automatica 24, 773–780 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Razzaghi, M., Elnagar, G.: Linear quadratic optimal control problems via shifted Legendre state parameterization. Int. J. Syst. Sci. 25, 393–399 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Razzaghi, M., Razzaghi, M.: Instabilities in the solution of a heat conduction problem using Taylor series and alternative approaches. J. Franklin Inst. 326, 683–690 (1989)

    Article  MATH  Google Scholar 

  14. Kajani, M.T., Vencheh, A.H.: Solving second kind integral equations with hybrid Chebyshev and block-pulse functions. Appl. Math. Comput. 163, 71–77 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Razzaghi, M., Marzban, H.R.: Direct method for variational problems via hybrid of block-pulse and Chebyshev functions. Math. Probl. Eng. 6, 85–97 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, X.T., Li, Y.M.: Numerical solutions of integro differential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials. Appl. Math. Comput. 209, 266–272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hsiao, C.H.: Hybrid function method for solving Fredholm and Volterra integral equations of the second kind. J. Comput. Appl. Math. 230, 59–68 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marzban, H.R., Razzaghi, M.: Hybrid functions approach for linearly constrained quadratic optimal control problems. Appl. Math. Model. 27, 471–485 (2003)

    Article  MATH  Google Scholar 

  19. Marzban, H.R., Razzaghi, M.: Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials. J. Franklin Inst. 341, 279–293 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Singh, V.K., Pandey, R.K., Singh, S.: A stable algorithm for Hankel transforms using hybrid of block-pulse and Legendre polynomials. Comput. Phys. Commun. 181, 1–10 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marzban, H.R., Razzaghi, M.: Analysis of time-delay systems via hybrid of block-pulse functions and Taylor series. J. Vib. Control 11, 1455–1468 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marzban, H.R., Razzaghi, M.: Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series. J. Sound Vib. 292, 954–963 (2006)

    Article  MathSciNet  Google Scholar 

  23. Costabile, F., Dellaccio, F., Gualtieri, M.I.: A new approach to Bernoulli polynomials. Rend. Mat., Serie VII. 26, 1–12 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Arfken, G.: Mathematical Methods for Physicists, 3rd edn. Academic Press, San Diego (1985)

    Google Scholar 

  25. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)

    MATH  Google Scholar 

  26. Lancaster, P.: Theory of Matrices. Academic Press, New York (1969)

    MATH  Google Scholar 

  27. Datta, K.B., Mohan, B.M.: Orthogonal Functions in Systems and Control. World Scientific, Singapore (1995)

    MATH  Google Scholar 

  28. Pananisamy, K.R., Rao, G.P.: Optimal control of linear systems with delays in state and control via Walsh functions. Proc. IEEE 130, 300–312 (1983)

    Article  Google Scholar 

  29. Banks, H.T., Burns, J.A.: Hereditary control problem: numerical methods based on averaging approximations. SIAM J. Control Optim. 16, 169–208 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lee, A.Y.: Numerical solution of time-delayed optimal control problems with terminal inequality constraints. Optim. Control Appl. Methods 14, 203–210 (1993)

    Article  MATH  Google Scholar 

  31. Teo, K.L., Wong, K.H., Clements, D.J.: Optimal control computation for linear time-lag systems with linear terminal constrains. J. Optim. Theory Appl. 44, 509–526 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Alzahra University, Tehran, Iran

    N. Haddadi & Y. Ordokhani

  2. Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS, 39762, USA

    M. Razzaghi

Authors
  1. N. Haddadi
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  2. Y. Ordokhani
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  3. M. Razzaghi
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Corresponding author

Correspondence to M. Razzaghi.

Additional information

Communicated by Ilio Galligani.

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Haddadi, N., Ordokhani, Y. & Razzaghi, M. Optimal Control of Delay Systems by Using a Hybrid Functions Approximation. J Optim Theory Appl 153, 338–356 (2012). https://doi.org/10.1007/s10957-011-9932-1

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  • DOI: https://doi.org/10.1007/s10957-011-9932-1

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