Optimal control of linear delay systems ‎involving ‎piecewise ‎constant ‎delay function using hybrid Chebyshev-block pulse method

Document Type : Original Paper

Author

Department of Mathematics, Faculty of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran

Abstract

In this paper, the optimal control of delay systems with piecewise constant delay function is investigated. Using Chebyshev hybrid functions, an approximate method is proposed to obtain the optimal solution to the control problem of linear delay systems. In order to present an approximate method, integral, product of multiplication and delay operational matrices of the hybrid functions have been introduced and used to solve the problem. The optimal control problem is transformed into an optimization problem with the help of the operational matrices and then solving it, an approximate solution to the original problem is obtained. Efficiency and accuracy of the proposed method are illustrated with two examples of the optimal control problem. ‎‎

Keywords

Main Subjects

References

[1] Betts J. T., Survey of numerical methods for trajectory optimization. Journal of Guidance, Control,
and Dynamics, 21 (1998), 193–207.
[2] Canuto C. Hussaini M. Y. Quarteroni A., and Zang T. A., Spectral Methods: Fundamentals in Single
Domains. Scientific Computation, Berlin: Springer, 2006.
[3] Chen W. L., and Jeng B. S., Analysis of piecewise constant delay systems via block-pulse functions.
International Journal of Systems Science, 12 (1981), 625–633.
[4] Chen W.-L., and Meng C.-H., A general procedure of solving the linear delay system via block pulse
functions. Computers and Electrical Engineering, 9(3-4) (1982), 153–166.
[5] Erneux T., Applied Delay Differential Equations. vol.3 of Surveys and Tutorials in the Applied
Mathematical Sciences. New York: Springer, 2009.
[6] Lancaster P., Theory of Matrices. New York, Academic Press, 1969.
[7] Marzban H. R., and Hoseini S. M., Solution of linear optimal control problems with time delay using a
composite Chebyshev finite difference method. Optimal Control Applications and Methods, 34 (2013),
253–274.
[8] Marzban H. R., and Hoseini S. M., Numerical treatment of non-linear optimal control problems involving
piecewise constant delay. IMA Journal of Mathematical Control and Information, 33 (2016),
1103–1134.
[9] Marzban H. R., and Hoseini S. M., An efficient discretization scheme for solving nonlinear optimal
control problems with multiple time delays. Optimal Control Applications and Methods, 37 (2016),
682–707.
[10] Marzban H. R., and Shahsiah M., Solution of piecewise constant delay systems using hybrid of blockpulse
and Chebyshev polynomials. Optimal Control Applications and Methods, 32 (2011), 647–659.
[11] Razzaghi M., and Marzban H. R., Direct method for variational problems via hybrid of block-pulse
and Chebyshev functions. Mathematical Problems in Engineering, 6 (2000), 85–97.
[12] Razzaghi M., and Marzban H. R., A hybrid domain analysis for systems with delays in state and
control. Mathematical Problems in Engineering, 7 (2001), 337–353.
Journal of Advanced Mathematical Modeling
Volume 12, Issue 2
June 2022
Pages 185-200

Files

History

  • Receive Date: 24 January 2022
  • Revise Date: 28 April 2022
  • Accept Date: 06 May 2022

Share

How to cite

Statistics