Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials

Mahmood Dadkhah

doi:doi:10.11648/j.sjams.20150304.15

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Solving Linear Time Varying Systems by Orthonormal Bernstein Polynomials

Mahmood Dadkhah

Published in Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 4)

Received: 3 May 2015 Accepted: 22 May 2015 Published: 29 July 2015

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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), 2024. Published by Science Publishing Group

Keywords

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

References

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[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]	LÃ¡zaro 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.

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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

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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

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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

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@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}
}

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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
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T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
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PB  - Science Publishing Group
SN  - 2376-9513
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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
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Author Information

Mahmood Dadkhah

Department of Mathematics, PayameNoor University, Tehran, Iran

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Plain Text BibTeX RIS

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

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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