| Peer-Reviewed

Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials

Received: 3 May 2015     Accepted: 22 May 2015     Published: 29 July 2015
Views:       Downloads:
Abstract

In this paper we present a method to find the solution of time-varying systems using orthonormal Bernstein polynomials. The method is based upon expanding various time functions in the system as their truncated orthonormal Bernstein polynomials. Operational matrix of integration is presented and is utilized to reduce the solution of time-varying systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Published in Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 4)
DOI 10.11648/j.sjams.20150304.15
Page(s) 194-198
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Orthonormal Bernstein Polynomials, Time Varying System, Operational Matrix, Linear Systems

References
[1] A. Deb, A. Dasgupta, G. Sarkar, A new set of orthogonal functions and its application to the analysis of dynamic systems, J. Frank. Inst. 343, 1-26, 2006.
[2] F. Khellat, S. A. Yousefi, The linear Legendre wavelets operational matrix of integration and its application, J. Frank. Inst. 343, 181-190, 2006.
[3] F. Marcellan, and W.V. Assche, Orthogonal Polynomials and Special Functions (a Computation and Appli- cations), Springer-Verlag Berlin Heidelberg, 2006.
[4] H. R. Marzban and M. Shahsiah, Solution of piecewise constant delay systems using hybrid of block- pulse and Chebyshev polynomials, Optim. Contr. Appl. Met., Vol. 32, pp. 647-659, 2011.
[5] Kung, F. C. and Lee, H., Solution and parameter estimation of linear time-invariant delay systems using Laguerre polynomial expansion, Journal on Dynamic Systems, Measurement, and Control, 297-301, 1983.
[6] Lázaro I, Anzurez J, Roman M, Parameter estimation of linear systems based on walsh series, The Electronics, Robotics and Automotive Mechanics Conference, 355-360, 2009.
[7] M. H. Farahi, M. Dadkhah, Solving Nonlinear Time Delay Control Systems by Fourier series, Int. Journal of Engineering Research and Applications, Vol. 4, Issue 6 , 217-226, 2014.
[8] M. I. Bhatti and P. Bracken, Solutions of differential equations in a Bernstein polynomial basis, Journal of Computational and Applied Mathematics, 205, 272 -280, 2007.
[9] M. Sezer, and A.A. Dascioglu, Taylor polynomial solutions of general linear differential-difference equations with variable coefficients, Appl. Math.Comput.174, 1526-1538, 2006.
[10] M. Shaban and S. Kazem and J. A. Rad, A modification of the homotopy analysis method based on Chebyshev operational matrices, Math. Comput. Model, in press(2013)
[11] Richard A. Bernatz. Fourier series and numerical methods for partial defferential equations. John Wiley & Sons, Inc., New York, 2010.
[12] S. A. Youseï, M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int. J. Syst. Sci. 41, 709-716, 2010.
[13] Wang, X., T, Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials Applied Mathematics and Computation, 184, 849-856, 2007.
[14] Z.H. Jiang, W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, Springer-Verlag, 1992.
Cite This Article
  • APA Style

    Mahmood Dadkhah. (2015). Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials. Science Journal of Applied Mathematics and Statistics, 3(4), 194-198. https://doi.org/10.11648/j.sjams.20150304.15

    Copy | Download

    ACS Style

    Mahmood Dadkhah. Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials. Sci. J. Appl. Math. Stat. 2015, 3(4), 194-198. doi: 10.11648/j.sjams.20150304.15

    Copy | Download

    AMA Style

    Mahmood Dadkhah. Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials. Sci J Appl Math Stat. 2015;3(4):194-198. doi: 10.11648/j.sjams.20150304.15

    Copy | Download

  • @article{10.11648/j.sjams.20150304.15,
      author = {Mahmood Dadkhah},
      title = {Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {3},
      number = {4},
      pages = {194-198},
      doi = {10.11648/j.sjams.20150304.15},
      url = {https://doi.org/10.11648/j.sjams.20150304.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20150304.15},
      abstract = {In this paper we present a method to find the solution of time-varying systems using orthonormal Bernstein polynomials. The method is based upon expanding various time functions in the system as their truncated orthonormal Bernstein polynomials. Operational matrix of integration is presented and is utilized to reduce the solution of time-varying systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.},
     year = {2015}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials
    AU  - Mahmood Dadkhah
    Y1  - 2015/07/29
    PY  - 2015
    N1  - https://doi.org/10.11648/j.sjams.20150304.15
    DO  - 10.11648/j.sjams.20150304.15
    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
    SP  - 194
    EP  - 198
    PB  - Science Publishing Group
    SN  - 2376-9513
    UR  - https://doi.org/10.11648/j.sjams.20150304.15
    AB  - In this paper we present a method to find the solution of time-varying systems using orthonormal Bernstein polynomials. The method is based upon expanding various time functions in the system as their truncated orthonormal Bernstein polynomials. Operational matrix of integration is presented and is utilized to reduce the solution of time-varying systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
    VL  - 3
    IS  - 4
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, PayameNoor University, Tehran, Iran

  • Sections