Legendre Wavelets with Scaling in Time-delay Systems
AbstractThis research presents the integration, product, delay and inverse time operational matrices of Legendre wavelets with an arbitrary scaling parameter and illustrates how to design this parameter in order to improve their accuracy and capability in handling optimal control and analysis of time-delay systems. Using the presented Legendre wavelets, the piecewise delay operational matrix is derived to develop the applicability of Legendre wavelets in systems with piecewise constant time-delays or time-varying delays. With the aid of these matrices, the new Legendre wavelets method is applied on linear time-delay systems. The reliability and efficiency of the method are demonstrated by some numerical experiments.
A. Graps, An Introduction to Wavelets, IEEE computational science and engineering, vol. 2, no. 2, pp. 50–61, 1995.
E. Kreyszig, Advanced engineering mathematics, John Wiley & Sons, 2010.
K.B. Datta, B.M. Mohan, Orthogonal functions in systems and control, Advanced Series in Electrical and Computer Engineering, World Scientific Publishing Co., 1995.
L.C. Andrews, Special functions of mathematics for engineers, New York: McGraw-Hill, 1992.
I.Malmir, Optimal contro lof linear time-varying systems with state and input delays by Chebyshev wavelets, Statistics,Optimization& Information Computing, vol. 5, no. 4, pp. 302–324, 2017.
H.R. Sharif, M.A. Vali, M. Samavat, A.A. Gharavisi, A New Algorithm for Optimal Control of Time-Delay Systems, Applied Mathematical Sciences, vol. 5, no. 12, pp. 595–606, 2011
I. Malmir, Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models, Applied Mathematical Modelling, 2018. doi: https://doi.org/10.1016/j.apm.2018.12.009
I. Malmir, A novel wavelet-based optimal linear quadratic tracker for time-varying systems with multiple delays, arXiv preprint,arXiv: 1802.05618, 2018.
W.A. Al-Salam, On the product of two Legendre polynomials, Mathematica Scandinavica, vol. 4, pp. 239–242, 1957.
Y. Shen, W. Lin, Collocation method for the natural boundary mintegral equation, Applied Mathematics Letters, vol. 19, no. 11, pp. 1278–1285, 2006.
M. Razzaghi, S. Yousefi, Legendre wavelets direct method for variational problems, Mathematics and computers in simulation, vol. 53, no. 3, pp. 185–192, 2000.
M. Razzaghi, S. Oppenheimer, F. Ahmad, A Legendre wavelet method for the radiative transfer equation in remote sensing, Journal of electromagnetic waves and applications, vol. 16, no. 12, pp.1681–1693, 2002.
K. Maleknejad, M.T. Kajani, Y. Mahmoudi, Numerical solution of linear Fredholm and Volterra integral equations of the second kind by using Legendre wavelets, Kybernetes, vol. 32, no. 9/10, pp. 1530 -1539, 2003.
H. Jaddu, Optimal control of time-varying linear systems using wavelets, PhD, Japan Advanced Institute of Science and Technology, Ishikawa, Japan, 2006.
S. Ma, H. Gao, G. Zhang, L. Wu, Abel inversion using Legendre wavelets expansion, Journal of Quantitative Spectroscopy & Radiative Transfer, vol. 107, no. 1, pp. 61–71, 2007.
R. Ebrahimi, M.A. Vali, M. Samavat, A.A. Gharavisi, A computational method for solving optimal control of singular systems using the Legendre wavelets, ICGST Automatic Control and System Engineering Journal, vol. 9, no. 2, pp. 1–6, 2009.
N. Liu, E.B. Lin, Legendre wavelet method for numerical solutions of partial differential equations, Numerical Methods for Partial Differential Equations, vol. 26, no. 1, pp. 81–94, 2010.
E. A. Rawashdeh, Legendre wavelets method for fractional integro-differential equations, Applied Mathematical Sciences, vol. 5, no. 2, pp. 2467–74, 2011.
C. Bandt, M. Barnsley, R. Devaney, K.J. Falconer, V. Kannan, V. Kumar P.B. (Editors), Fractals, Wavelets, and their Applications: Contributions from the International Conference and Workshop on Fractals and Wavelets, Springer, vol. 92, pp. 491–494, 2014.
A. Setia, B. Prakash, A.S. Vatsala, Numerical Solution by Fourth Order Fractional Integro-Differential Equation by Using Legendre Wavelets, Neural, Parallel, and Scientific Computations, vol. 23, pp. 377–386, 2015.
K. Maleknejad, E. Saeedipoor, R. Dehbozorgi, Legendre Wavelets Direct Method for the Numerical Solution of Fredholm Integral Equation of the First Kind, Proceedings of the World Congress on Engineering, vol.1, 2016.
S.T. Mohyud-din, M.A. Iqbal, U. Khan U, X.J. Yang, MHD squeezing flow between two parallel disks with suction or injection via Legendre wavelet-quasilinearization technique, Engineering Computations, vol. 34, no. 3, pp. 892–901, 2017.
S. Yadav, S. Upadhyay, K.N. Rai, Legendre Wavelet Modified Petrov–Galerkin Method in Two-Dimensional Moving Boundary Problem, Zeitschrift für Naturforschung A., vol. 73, no. 1, pp. 23–34, 2017.
S. Balaji, G. Hariharan, A novel wavelet approximation method for the solution of nonlinear differential equations with variable coefficients arising in astrophysics, Astrophysics and Space Science, vol. 363, no. 1, pp. 1–16, 2018.
S.Singh,V.K.Patel,V.K.Singh,Application of wavelet collocation method for hyperbolic partial differential equations via matrices, Applied Mathematics and Computation, vol. 320, pp. 407–24, 2018.
M. Sohaib, S. Haq, S. Mukhtar, I. Khan, Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method, Results in physics, vol. 8, pp. 1204–1208, 2018.
M. Kumar, S. Upadhyay, K.N. Rai, A study of cryosurgery of lung cancer using Modified Legendre wavelet Galerkin method, Journal of thermal biology vol. 78, pp. 356–366, 2018.
Z. Lin, L. He, T. Wu, C. Xu, The parameter estimation of the multivariate matrix regression models, Statistics, Optimization & Information Computing, vol. 6, no. 2, pp. 286–291, 2018.
H.R. Marzban, M. Razzaghi, Solution of time-varying delay systems by hybrid functions, Mathematics and Computers in Simulation,vol. 64, no. 6, pp. 597–607, 2004.
F. Khellat, N. Vasegh, Suboptimal control of linear systems with delays in state and input by orthonormal basis, International Journal of Computer Mathematics, vol. 88, no. 4, pp. 781–794, 2011.
R. Mohammadzadeh, M. Lakestani, Analysis of time-varying delay systems by hybrid of block-pulse functions and biorthogonal multiscaling functions, International Journal of Control, vol. 88, no. 12, pp. 2444–2456, 2015.
H.R. Marzban, H. Pirmoradian, A novel approach for the numerical investigation of optimal control problems containing multiple delays, Optimal Control Applications and Methods, vol. 39, no. 1, pp. 302–325, 2018.
A. Galindro, D.F. M. Torres, A simple mathematical model for unemployment: a case study in Portugal with optimal control, Statistics, Optimization & Information Computing, vol. 6, no. 1, pp. 116–129, 2018.
J. N.C. Goncalves, H.S. Rodrigues, M.T. T. Monteiro, On the dynamics of a viral marketing model with optimal control using indirect and direct methods, Statistics, Optimization & Information Computing, vol. 6, no. 4, pp. 633–644, 2018.
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