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http://dx.doi.org/10.5302/J.ICROS.2013.13.1911

Numerical Method for the Analysis of Bilinear Systems via Legendre Wavelets  

Kim, Beomsoo (Mechanical System Engineering, Gyeongsang National University)
Publication Information
Journal of Institute of Control, Robotics and Systems / v.19, no.9, 2013 , pp. 827-833 More about this Journal
Abstract
In this paper, an efficient computational method is presented for state space analysis of bilinear systems via Legendre wavelets. The differential matrix equation is converted to a generalized Sylvester matrix equation by using Legendre wavelets as a basis. First, an explicit expression for the inverse of the integral operational matrix of the Legendre wavelets is presented. Then using it, we propose a preorder traversal algorithm to solve the generalized Sylvester matrix equation, which greatly reduces the computation time. Finally the efficiency of the proposed method is discussed using numerical examples.
Keywords
bilinear systems; Legendre wavelets; Sylvester matrix equation; preorder traversal algorithm;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 R. R. Mohler, Nonlinear Systems, Volume II: Applications to Bilinear Control, Englewood-Cliffs, New Jersey: Prentice-Hall, 1991.
2 K. K. B. Datta, Orthongonal Functions in Systems and Control, World Scientific, 1995.
3 H. Akca, M. H. Al-Lail, and V. Covachev, "Survey on wavelet transform and application in ODE and wavelet networks," Advances in Dynamical Systems and Applications, vol. 1, no. 2, pp. 129-162, 2006.
4 C. Chen and C. Hsiao, "Haar wavelet method for solving lumped and distributed-parameter systems," Control Theory and Applications, IEE Proceedings-, pp. 87-94, 1997.
5 B. S. Kim and I. J. Shim, "Study for state analysis of linear systems using haar wavelet," Journal of Institute of Control, Robotics and Systems (in Korean), vol. 14, no. 10, pp. 977-982, 2008.   과학기술학회마을   DOI   ScienceOn
6 J. Juang, "Generalized bilinear system identification," The Journal of the Astronautical Sciences, vol. 57, no. 1-2, pp. 261-273, 2009.   DOI
7 X. Wang and Y. Jiang, "On model reduction of K-power bilinear systems," Int. J. Syst. Sci., pp. 1-13, 2013.
8 C. Hsiao and W. Wang, "State analysis and parameter estimation of bilinear systems via Haar wavelets," Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, vol. 47, no. 2, pp. 246-250, 2000.   DOI   ScienceOn
9 C. Hwang and M. Chen, "Analysis and parameter identification of bilinear systems via shifted Legendre polynomials," Int J Control, vol. 44, no. 2, pp. 351-362, 1986.   DOI   ScienceOn
10 J. Shim and D. S. Ahn, "A study on the analysis and state estimation of bilinear systems via orthogonal functions," The Transactions of KIEE (in Korean), vol. 39, no. 6, pp. 598-606, 1990.   과학기술학회마을
11 B. Cheng and N. Hsu, "Analysis and parameter estimation of bilinear systems via block-pulse functions," Int J Control, vol. 36, no. 1, pp. 53-65, 1982.   DOI   ScienceOn
12 M. Razzaghi and S. Yousefi, "The Legendre wavelets operational matrix of integration," Int. J. Syst. Sci., vol. 32, no. 4, pp. 495-502, 2001.   DOI   ScienceOn
13 K. S. Miller, "On the inverse of the sum of matrices," Mathematics Magazine, vol. 54, no. 2, pp. 67-72, 1981.   DOI
14 B. Kim, I. Shim, M. Lim, and Y. Kim, "Combined preorder and postorder traversal algorithm for the analysis of singular systems by Haar wavelets," Mathematical Problems in Engineering, vol. 2008, 2008.
15 F. Khellat and S. Yousefi, "The linear Legendre mother wavelets operational matrix of integration and its application," Journal of the Franklin Institute, vol. 343, no. 2, pp. 181-190, 2006.   DOI   ScienceOn
16 K. B. Petersen and M. S. Pedersen, The Matrix Cookbook, Technical University of Denmark, 2006.
17 W. Ledermann, "A note on skew-symmetric determinants," Proc. Edinburgh Math. Soc, pp. 335-338, 1993.
18 H. Jaddu, "Optimal control of time-varying linear systems using wavelets," School of Information Science, Japan Advanced Institute of Science and Technology, Submitted for Publication, 1998.
19 H. Gould and J. Quaintance, "Double fun with double factorials," Mathematics Magazine, vol. 85, no. 3, pp. 177-192, 2012.   DOI
20 B. S. Kim, "An efficient computational method for linear time-invariant systems via legendre wavelet," (to appear in) Journal of Institute of Control, Robotics and Systems (in Korean), vol. 19, no. 7, pp.577-582, 2013.   DOI   ScienceOn