Identification of linear dynamic systems of~fractional order with errors in variables based on~an~augmented system of equations

D. V. Ivanov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 3, pp. 508-518.

Résumé

Equations with derivatives and fractional order differences are widely used to describe various processes and phenomena. Currently, methods of identification of systems described by equations with fractional order differences are actively developing. The paper is devoted to the identification of discrete dynamical systems described by equations with fractional order differences with errors in variables. The problems of identifying systems with errors in variables are often ill-conditioned. The paper proposes an algorithm that uses the representation of a normal biased system as an augmented equivalent system. This representation allows to reduce the number of conditionality of the problem to be solved. Test examples have shown that the proposed algorithm has a higher accuracy than the algorithms based on the decomposition of Cholesky and the minimization of the generalized Rayleigh quotient.

Détail

Keywords: fractional difference, total least square, errors-in-variables, ill-conditioning.

Bibliographie

[1] Podlubny I., Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press, San Diego, 1999, xxiv+340 pp. | DOI | Zbl

[2] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006, xx+523 pp. | DOI | Zbl

[3] Uchaikin V. V., “Heredity and nonlocality”, Fractional Derivatives for Physicists and Engineers. Background and Theory, v. 1, Nonlinear Physical Science, Springer, Berlin, 2013, 3–58 | DOI

[4] Rabotnov Yu. N., Elements of Hereditary Solid Mechanics, Mir Publ., 1980, 388 pp. | Zbl

[5] Slonimsky G. L., “On the laws of deformation of real materials. I. The theories of Maxwell and Boltzmann”, Acta Physicochim. URSS, 12 (1940), 99–128 | Zbl

[6] Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, World Scientific, Hackensack, NJ, 2010, xx+347 pp. | DOI | Zbl

[7] Koeller R. C., “Applications of fractional calculus to the theory of viscoelasticity”, J. Appl. Mech., 51:2 (1984), 299–307 | DOI | Zbl

[8] Boikov I. V., Krivulin N. P., “Parametric identification of hereditary systems with distributed parameters”, University Proceedings. Volga Region. Engineering Sciences, 26:2 (2013), 120–129 (In Russian)

[9] Boykov I. V., Krivulin N. P., “Recovery of the parameters of linear systems described by differential equations with variable coefficients”, Meas. Tech., 56:4 (2013), 359–356 | DOI

[10] Cois O., Oustaloup A., Battaglia E., Battaglia J.-L., “Non integer model from modal decomposition for time domain system identification”, IFAC Proc. Vol., 33:15 (2000), 989–994 | DOI

[11] Cois O., Oustaloup A., Poinot T., Battaglia J.-L., “Fractional state variable filter for system identification by fractional model”, 2001 European Control Conference (ECC) (4–7 Sept. 2001, Porto, Portugal), 2001, 2481–2486 | DOI

[12] Malti R., Aoun M., Sabatier J., Oustaloup A., “Tutorial on system identification using fractional differentiation models”, IFAC Proc. Vol., 39:1 (2006), 606–611 | DOI

[13] Ogorodnikov E. N., Radchenko V. P., Ungarova L. G., “Mathematical modeling of hereditary elastically deformable body on the basis of structural models and fractional integro-differentiation Riemann–Liouville apparatus”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:1 (2016), 167–194 (In Russian) | DOI | Zbl

[14] Ungarova L. G., “The use of linear fractional analogues of rheological models in the problem of approximating the experimental data on the stretch polyvinylchloride elastron”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:4 (2016), 691–706 (In Russian) | DOI | Zbl

[15] Lewandowski R., Chorążyczewski B., “Identification of the parameters of the Kelvin–Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers”, Comp. Struct., 88:1–2 (2009), 1–17 | DOI

[16] Shabani R., Jahani K., Di Paola M., Sadeghi M. N., “Frequency domain identification of the fractional Kelvin–Voigt's parameters for viscoelastic materials”, Mech. Mater., 137 (2019), 103099 | DOI

[17] Söderström T., Errors-in-Variables Methods in System Identification, Communications and Control Engineering, Springer, Cham, Switzerland, 2018, xxvii+485 pp. | DOI | Zbl

[18] Chetoui M., Malti R., Thomassin M., Aoun M., Najar S., Oustaloup A., Abdelkrim M.N., “EIV methods for system identification with fractional models”, IFAC Proceedings Volumes, 45:16 (2012), 1641–1646 | DOI

[19] Chetoui M., Thomassin M., Malti R., Aoun M., Najar S., Abdelkrim M. N., Oustaloup A., “New consistent methods for order and coefficient estimation of continuous-time errors-in-variables fractional models”, Comp. Math. Appl., 66:5 (2013), 860–872 | DOI | Zbl

[20] Ivanov D. V., “Identification discrete fractional order linear dynamic systems with errors-in-variables”, East-West Design and Test Symposium (EWDTS 2013) (27–30 Sept. 2013, Rostov on Don, Russia), 2013, 374–377 | DOI

[21] Ivanov D. V., “Estimation of parameters of linear fractional order ARX systems with noise in the input signal”, Tomsk State University Journal of Control and Computer Science, 2014, no. 2(27), 30–38 (In Russian)

[22] Ivanov D. V., Ivanov A. V., “Identification fractional linear dynamic systems with fractional errors-in-variables”, J. Phys.: Conf. Ser., 803 (2017), 012058 | DOI

[23] Zhdanov A. I., Shamarov P. A., “A direct projection method in the problem of complete least squares”, Autom. Remote Control, 61:4 (2000), 610–620 | MR | Zbl

[24] Granger C. W. J., Joyeux R., “An introduction to long-memory time series models and fractional differencing”, J. Time Series Anal., 1:1 (1980), 15–29 | DOI | Zbl

[25] Hosking J. R. M., “Fractional differencing”, Biometrika, 68:1 (1981), 165–176 | DOI | Zbl

[26] Sierociuk D., Dzieliński A., “Fractional Kalman filter algorithm for the states parameters and order of fractional system estimation”, Int. J. Appl. Math. Comput. Sci., 16:1 (2006), 129–140 | Zbl

[27] Djouambai A., Voda A., Charef A., “Recursive prediction error identification of fractional order models”, Commun. Nonlinear Sci. Numer. Simulat., 17:6 (2012), 2517–2524 | DOI

[28] Dzieliński A., Sierociuk D., “Some applications of fractional order calculus”, Bull. Polish Acad. Sci. Tech. Sci., 58:4 (2010), 583–592 | DOI | Zbl

[29] Ivanov D. V., Ivanov A. V., Sandler I., Chertykovtseva N. V., “Identification of the heating model plastic injection molding machines”, Zhurnal SVMO, 19:3 (2017), 82–89 (In Russian) | DOI

[30] Ostalczyk P., Discrete Fractional Calculus. Applications in Control and Image Processing, Series in Computer Vision, 4, World Scientific, Hackensack, NJ, 2016, xxxi+361 pp. | DOI | Zbl

[31] Mozyrska D., Wyrwas M., “Stability of discrete fractional linear systems with positive orders”, IFAC-PapersOnLine, 50:1 (2017), 8115–8120 | DOI

[32] Wilkinson J. H., The Algebraic Eigenvalue Problem, Monographs on Numerical Analysis. Oxford Science Publications, Clarendon Press, Oxford, 1988, xviii+682 pp. | Zbl

[33] Zhdanov A. I., “The solution of ill-posed stochastic linear algebraic equations by the maximum likelihood regularization method”, U.S.S.R. Comput. Math. Math. Phys., 28:5 (1988), 93–96 | DOI | MR | Zbl