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Information Technologies and Control

The Journal of Institute of Information and Communication Technologies of Bulgarian Academy of Sciences

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1312-2622
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Orthogonal Polynomials Approximation and Balanced Truncation for a Lowpass Filter

K. Perev
Published Online: 2015-05-05 | DOI: https://doi.org/10.1515/itc-2015-0001

Abstract

This paper considers the problem of orthogonal polynomial approximation based balanced truncation for a lowpass filter. The proposed method combines the system properties of balanced truncation, the computational effectiveness of proper orthogonal decomposition and the approximation capability of the orthogonal polynomials approximation. Orthogonal polynomials series expansion of the reachability and observability gramians is used in order to avoid solving large-scale Lyapunov equations and thus, significantly reducing the computational effort for obtaining the balancing transformation. The proposed method is applied for model reduction of a lowpass analog filter. Different sets of orthonormal functions are obtained from Legendre, Laguerre and Chebyshev orthogonal polynomials and the corresponding reduced order models are compared. The approximation precision is measured by the relative mean square error between the outputs of the full order model and the obtained reduced order models.

Keywords : Legendre; Laguerre and Chebyshev orthogonal polynomials; balanced truncation; reachability and observability gramians; lowpass analog filter

References

  • 1. Abramowitz, M., I. Stegun, Edts. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. N.Y., Dover Publ., 1972.Google Scholar

  • 2. Amghayrir, A., N. Tanguy, P. Brehonnet, P. Vilbe, L. Calvez. Laguerre-based reduced-order Modeling. - IEEE Trans. Automat. Control, 50, 2005, No. 9, 1432-1435.Google Scholar

  • 3. Antoulas, A. Approximation of Large-scale Dynamical Systems. Philadelphia, SIAM Publ., 2005.Google Scholar

  • 4. Dahlquist, G., A. Bjoerck. Numerical Methods. Prentice Hall, Englewood Cliffs, 1974.Google Scholar

  • 5. Dastjerdi, H., F. Ghaini. Numerical Solution of Volterra - Fredholm Integral Equations by Moving Least Squares Method and Chebyshev Polynomials. - Appl. Math. Model., 36, 2012, 3283-3288.Google Scholar

  • 6. Eid, R., B. Lohmann. Moment Matching Model Order Reduction in Time Domain via Laguerre Series. Proceedings of the17th World Congress of IFAC, Seoul, 2008, 3198-3203.Google Scholar

  • 7. Eid, R., B. Salimbahrami, B. Lohmann. Equivalence of Laguerre - Based Model Order Reduction and Moment Matching. - IEEE Trans. Automat. Control, 52, 2007, No. 6, 1104-1108.Web of ScienceGoogle Scholar

  • 8. El-Kady, M., N. El-Sawy. Legendre Coefficients Method for Linear-Quadratic Optimal Control Problems by Using Genetic Algorthms. - Int. J. Appl. Math. Research, 2, 2013, No. 1, 140-150.Google Scholar

  • 9. Li, L., S. Billings. Continuous-time System Identification Using Shifted Chebyshev Polynomials. - Internat. J. Systems Sci., 32, 2001, No. 3, 303-306.Google Scholar

  • 10. Maleknejad, K., S. Sohrabi, Y. Rostami. Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Using Chebyshev Polynomials. - Appl. Math. Comput., 188, 2007, 123-128.Google Scholar

  • 11. Moore, B. Principal Component Analysis in Linear Systems: Controllability, Observability and Model Reduction. - IEEE Trans. Automat. Control, AC-26, 1981, No. 1, 17-32.CrossrefGoogle Scholar

  • 12. Nagurka, M., S. Wang. A Chebyshev-based State Representation for Linear Quadratic Optimal Control. - J. Dyn. Syst. Meas. Control, 115, 1993, 1-6.Google Scholar

  • 13. Paraskevopoulos, P. Legendre Series Approach to Identification and Analysis of Linear Systems. - IEEE Trans. Automat. Control, 30, 1985, No. 6, 585-589.Google Scholar

  • 14. Paraskevopoulos, P. Chebyshev Series Approach to System Identification, Analysis and Optimal Control. - J. Franklin Inst., 316, 1983, 135-157.Google Scholar

  • 15. Petkov, P., M. Konstantinov. Robust Control Systems. Analy- sis and Design with MATLAB, Sofia, ABC Technika, 2002 (in Bulgarian).Google Scholar

  • 16. Schetzen, M. The Volterra and Wiener Theories of Nonlinear Systems. R. Krieger Publ. Corp., Malabar, 1989.Google Scholar

  • 17. Shen, J. Efficient Spectral Galerkin Method I. Direct Solvers for the Second and Fourth Order Equations Using Legendre Polynomials. - SIAM J. Sci. Comput., 15, 1994, No. 6, 1489-1505.Google Scholar

  • 18. Sirovitch, L. Turbulence and the Dynamics of Coherent Structures. Part I-II. - Quart. Appl. Math., 45, 1987, 561-590.Google Scholar

  • 19. Slavov, T., N. Madjarov. Adaptive Multiple Model Algorithm for Control of Stochastic System with Orthogonal Models. - Inf. Techn. Control, 2006, No. 2, 22-32.Google Scholar

  • 20. Zhou, Q., E. Davis. A Simplified Algorithm for Balanced Realization of Laguerre Network Models. Proceedings of the 39th Conference on Decision and Control, Sydney, 2000, 4636-4640. Google Scholar

About the article

Received: 2014-10-30

Published Online: 2015-05-05

Published in Print: 2013-12-01


Citation Information: Information Technologies and Control, Volume 11, Issue 4, Pages 2–16, ISSN (Online) 1312-2622, DOI: https://doi.org/10.1515/itc-2015-0001.

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© 2015. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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