Numerical method of estimation of parameters

V. E. Zoteev ; E. D. Stukalova ; E. V. Bashkinova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 3, pp. 556-580.

Résumé

The main problem of mathematical simulation is the problem of nonlinear estimation of parameters of the different physical systems. The article contains new numerical method of parameters estimation of the nonlinear differential operator of the second order with the dissipative force, proportional to $n$-motion speed level assessment. Mean square estimation of coefficients of the generalized regression model constructed taking into account the difference equations describing results of measurements of a pulse response of system is the cornerstone of the numerical method. Two landmark procedure of differentiated estimation of parameters of dynamic process realized in a method allow to provide high adequacy of the constructed model to data of an experiment. Application of the developed numerical method allows to increase significantly (several times) the accuracy of estimates of parameters of the nonlinear differential operator in comparison with the known methods due to elimination of the offset in estimates caused by use of approximation in case of simulation of an envelope of vibration amplitudes.

Détail

Keywords: nonlinear differential operator, difference equations, generalized regression model, nonlinear regression, mean square estimation.
Mots-clés : dissipative force proportional

Bibliographie

[1] Bozhko A. E., Golub N. M., Dinamiko-energeticheskie sviazi kolebatel'nykh sistem [Dynamic-energy relations of oscillatory systems], Nauk. dumka, Kiev, 1980, 188 pp. (In Russian)

[2] Panovko A. G., Osnovy prikladnoi teorii kolebanii i udara [Fundamentals of applied theory of vibrations and shock], Mashinostroenie, Leningrad, 1976, 320 pp. (In Russian)

[3] Panovko Ya. G., Vnutrennee trenie pri kolebaniiakh uprugikh sistem [Internal Friction in Vibrations of Elastic Systems], Fizmatgiz, Moscow, 1960, 194 pp. (In Russian)

[4] Pisarenko G. S., Yakovlev A. P., Matveev V. V., Vibropogloshchaiushchie svoistva konstruktsionnykh materialov: Spravochnik [Vibration-Absorbing Properties of Structural Materials: A Handbook], Nauk. dumka, Kiev, 1971, 376 pp. (In Russian)

[5] Markov S. I., Minaev V. M., Artamonov B. I., Identifikatsiia kolebatel'nykh sistem avtomaticheskogo regulirovaniia [Identification of oscillatory automatic control systems], Energiia, Leningrad, 1975, 96 pp. (In Russian)

[6] Shteinberg Sh. E., Identifikatsiia v sistemakh upravleniia [Identification in control systems], Energoatomizdat, Moscow, 1987, 80 pp. (In Russian)

[7] Zoteev V. E., Parametricheskaia identifikatsiia dissipativnykh mekhanicheskikh sistem na osnove raznostnykh uravnenii [Parametric identification of dissipative mechanical systems based on difference equations], Mashinostroenie, Moscow, 2009, 344 pp. (In Russian)

[8] Bendat J., Piersol A., Engineering Application of Correlation and Spectral Analysis, Wiley-Interscience, New York, 1980, xiv+302 pp.

[9] Yavlenskii K. N., Yavlenskii A. K., Vibrodiagnostika i prognozirovanie kachestva mekhanicheskikh sistem [Vibration diagnostics and prediction of quality mechanical systems], Mashinostroenie, Leningrad, 1983, 239 pp. (In Russian)

[10] Vibratsii v tekhnike, Spravochnik. V 6 t, Mashinostroenie, M.; т. 1, 1978, 352 с.; т. 2, 1979, 351 с.; т. 5, 1981, 496 с. [Vibratsii v tekhnike [Vibrations in Engineering], Handbook in 6 Vols, Mashinostroenie, Moscow (In Russian)]; v. 1, 1978, 352 pp.; v. 2, 1979, 351 pp.; v. 5, 1981, 496 pp.

[11] Jenkins G. M., Watts D. G., Spectral analysis and its applications, Holden-Day series in time series analysis, Holden-Day, San Francisco, 1968, xviii+525 pp. | Zbl

[12] Dobrynin S. A., Fel'dman M. S., Firsov G. I., Metody avtomatizirovannogo issledovaniia vibratsii mashin: Spravochnik [Methods of Automated Vibration Research of Machines: A Handbook], Mashinostroenie, Moscow, 1987, 224 pp. (In Russian)

[13] Marple S. Lawrence, Jr., Digital Spectral Analysis: With Applications, Prentice-Hall Series in Signal Processing, Prentice-Hall, New York, 1987, xx+492 pp.

[14] Dech G., Rukovodstvo k prakticheskomu primeneniiu preobrazovaniia Laplasa i z-preobrazovaniia [Guide to the practical application of the Laplace transform and Z-transform], Nauka, Moscow, 1971, 288 pp. (In Russian)

[15] Nayfeh A. H., Introduction to perturbation techniques, John Wiley Sons, New York, 1993, xiv+519 pp.

[16] Pisarenko G. S., Matveev V. A., Yakovlev A. P., Metody opredeleniia kharakteristik kolebanii uprugikh sistem [Methods of determining the characteristics of vibration damping in elastic systems], Nauk. dumka, Kiev, 1976, 88 pp. (In Russian)

[17] Draper N. R., Smith H., Applied Regression Analysis, Wiley Series in Probability and Statistics, John Wiley Sons, New York, 1998, xix+716 | DOI

[18] Demidenko E. Z., Lineinaia i nelineinaia regressii [Linear and Nonlinear Regression], Finansy i statistika, Moscow, 1981, 302 pp. (In Russian)

[19] Marquardt D. W., “An Algorithm for Least-Squares Estimation of Nonlinear Parameters”, J. Soc. Indust. Appl. Math., 11:2 (1963), 431–441 | DOI | Zbl

[20] Hartley H. O., Booker A., “Nonlinear Least Squares Estimation”, Ann. Math. Statist, 36:2 (1965), 638–650 | DOI | Zbl

[21] Zoteev V. E., “Construction of difference equations for increasing the accuracy of parametric identification of vibrational systems with a weak general nonlinearity”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2000, no. 9, 169–173 (In Russian) | DOI

[22] Zoteev V. E., “Development and research in digital linear models of dissipative systems wavering”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 1999, no. 7, 170–177 (In Russian) | DOI

[23] Zoteev V. E., “Parametrical identification of linear dynamical system on the basis of stochastic difference equations”, Matem. Mod., 20:9 (2008), 120–128 (In Russian) | MR | Zbl