DOI QR코드

DOI QR Code

Numerical Method for the Analysis of Bilinear Systems via Legendre Wavelets

르장드르 웨이블릿을 이용한 쌍일차 시스템 수치 해석

  • Kim, Beomsoo (Mechanical System Engineering, Gyeongsang National University)
  • 김범수 (경상대학교 기계시스템공학과)
  • Received : 2013.04.18
  • Accepted : 2013.07.15
  • Published : 2013.09.01

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.

References

  1. K. K. B. Datta, Orthongonal Functions in Systems and Control, World Scientific, 1995.
  2. 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.
  3. 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.
  4. 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. https://doi.org/10.5302/J.ICROS.2008.14.10.977
  5. 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. https://doi.org/10.5302/J.ICROS.2013.13.1908
  6. R. R. Mohler, Nonlinear Systems, Volume II: Applications to Bilinear Control, Englewood-Cliffs, New Jersey: Prentice-Hall, 1991.
  7. J. Juang, "Generalized bilinear system identification," The Journal of the Astronautical Sciences, vol. 57, no. 1-2, pp. 261-273, 2009. https://doi.org/10.1007/BF03321504
  8. X. Wang and Y. Jiang, "On model reduction of K-power bilinear systems," Int. J. Syst. Sci., pp. 1-13, 2013.
  9. 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. https://doi.org/10.1109/81.828579
  10. 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. https://doi.org/10.1080/00207178608933604
  11. 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.
  12. 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. https://doi.org/10.1080/00207178208932874
  13. 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.
  14. 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. https://doi.org/10.1016/j.jfranklin.2005.11.002
  15. M. Razzaghi and S. Yousefi, "The Legendre wavelets operational matrix of integration," Int. J. Syst. Sci., vol. 32, no. 4, pp. 495-502, 2001. https://doi.org/10.1080/00207720120227
  16. K. S. Miller, "On the inverse of the sum of matrices," Mathematics Magazine, vol. 54, no. 2, pp. 67-72, 1981. https://doi.org/10.2307/2690437
  17. K. B. Petersen and M. S. Pedersen, The Matrix Cookbook, Technical University of Denmark, 2006.
  18. W. Ledermann, "A note on skew-symmetric determinants," Proc. Edinburgh Math. Soc, pp. 335-338, 1993.
  19. 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.
  20. H. Gould and J. Quaintance, "Double fun with double factorials," Mathematics Magazine, vol. 85, no. 3, pp. 177-192, 2012. https://doi.org/10.4169/math.mag.85.3.177