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Vol: 54(68) No: 4 / December 2009 Systems Modeling Based on Orthogonal Rational Functions Bratislav Danković Department of Automatics, University of Niš, Faculty of Electrical Engineering, A. Medvedeva 14, 18000 Niš, Republic of Serbia, phone: +38118529363, e-mail: bratislav.dankovic@elfak.ni.ac.rs Dragan Antić Department of Automatics, University of Niš, Faculty of Electrical Engineering, A. Medvedeva 14, 18000 Niš, Republic of Serbia, e-mail: dragan.antic@elfak.ni.ac.rs Zoran Jovanović Department of Automatics, University of Niš, Faculty of Electrical Engineering, A. Medvedeva 14, 18000 Niš, Republic of Serbia, e-mail: zoran.jovanovic@elfak.ni.ac.rs Saša Nikolić Department of Automatics, University of Niš, Faculty of Electrical Engineering, A. Medvedeva 14, 18000 Niš, Republic of Serbia, e-mail: sasa.s.nikolic@elfak.ni.ac.rs Marko Milojković Department of Automatics, University of Niš, Faculty of Electrical Engineering, A. Medvedeva 14, 18000 Niš, Republic of Serbia, e-mail: marko.milojkovic@elfak.ni.ac.rs Keywords: orthogonal function, orthogonal filter, systems modeling, genetic algorithm. Abstract This paper deals with a possibility of modeling continuous dynamical systems, via orthogonal rational functions. Recent results in the field of orthogonal functions were used to improve accuracy of modeling. Legendre orthogonal filters were designed on the basis of orthogonal rational functions and then adjustable models were formed. Models parameters were optimized using genetic algorithm. As a case study, an experimental simple hydraulic system was considered. Simulations were performed to approve theoretical results and demonstrate that the method described in the paper is very suitable for modeling continuous systems in the sense of model accuracy and modeling algorithm speed. References [1] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc., Colloq. Pub. vol. 23, Providence, 1975. [2] P. Heuberger, P. Van den Hof and B. Wahlberg, Modelling and Identification with Rational Orthogonal Basis Functions, Springer-Verlag, London, 2005. [3] M. M. Djrbashian, “Orthogonal systems of rational functions on the circle unit with given set of poles”, Dokl. Akad. Nauk SSSR vol. 147, pp. 1278 - 1281, 1962. [4] M. M. Djrbashian, “Orthogonal systems of rational functions on the circle”, Akad. Nauk. Armayan. vol. 1, pp. 3 - 24, 1966. [5] Ya. L. Geronimus, Polynomials orthogonal on a circle and interval, Fiz. Mat. Lit. , Moscow, 1958. [6] G. V. Milovanović, B. Danković and S. Rančić, “Some Müntz orthogonal systems”, J. Comp. Appl. Math., vol. 99, pp. 299 - 310, 1998. [7] S. B. Marinković, B. Danković, M. S. Stanković and P. M. Rajković, “Orthogonality of some sequences of the rational functions and the Müntz polynomials”, J. Comp. Appl. Math., vol. 163, pp. 419 - 427, 2004. [8] P. B. Borwein, T. Erdelyi and J. Zhang, “Müntz systems and orthogonal Müntz-Legendre polynomials”, Trans. Amer. Math. Soc. vol. 342, pp. 523 - 542, 1994. [9] B. Danković, G. V. Milovanović and S. Rančić, “Malmquist and Müntz orthogonal systems and applications”, in Iner Product Spaces and Applications, T. M. Rassias Eds., Addison-Wesley Longman. Harlow, pp. 22 - 41, 1997. [10] P. C. McCarthy, J. E. Sayre and B. L. R. Shawyer, “Generalized Legendre polynomials”, J. Math. Anal. Appl., vol. 177, pp. 530 - 537, 1993. [11] A. H. Gray, “Passive cascaded lattice digital filters”, IEEE Trans. Circ. Sys., no. 5, pp. 337 - 344, 1980. [12] B. Danković, S. Nikolić, M. Milojković and Z. Jovanović, “A class of almost orthogonal filters”, J. Circ. Sys. Comp., vol. 18, no. 5, pp. 923 - 931, 2009. [13] B. Danković, D. Antić, Z. Jovanović, S. Nikolić and M. Milojković, “Systems modeling based on Legendre polynomials”, 5th International Conference on Applied Computational Intelligence and Informatics SACI 2009, Timisoara, Romania, pp. 241 – 246, 2009. [14] A. H. Gray and J. D. Markel, “A normalized digital filter structure”, IEEE Trans. Acous. Sig. Proc., vol. 23, pp. 268 - 277, 1975. [15] L. J. Karam and J. H. McClellan, “Complex Chebyshev approximation for FIR filter design”, IEEE Trans. Circ. Sys. II, vol. 42, pp. 207 - 216, 1995. [16] J. H. Holland, Adaptation in natural and artificial systems, University of Michigan Press, Ann Arbor, 1975. [17] D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning, Addison-Wesley, Reading, 1989. [18] D. Antić and M. Milojković, “Nonlinear system control by using genetic algorithms and fuzzy sliding mode”, TEHNIKA, Elektrotehnika, vol. 56, pp. 9 - 16, 2007. [19] B. Danković, D. Antić, Z. Jovanović and M. Milojković, “Genetic algorithms applied in parameter optimization of casacade connected systems”, ICEST, Ohrid, Macedonia, pp. 557 - 560, 2007. [20] B. Danković, D. Antić and Z. Jovanović, Process Control, Faculty of Electrical Engineering, Nis, 1996. |