Transfer function identification by minimizing the adaptive vs. optimal filter state estimates mismatch

I. V. Semushin

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 27 (2023) no. 3, pp. 544-572.

Résumé

The article is concerned with a further development of the Active Principle of parametric system identification in the class of linear, time-invariant, completely observable models. As the identification target model, the optimal Kalman filter (OKF) is designated that is present, no more than conceptually, in the system’s discretely observed response to a training excitation of the white noise type. By modifying the physically given structure into the standard observable model in both the observed response and the Adaptive Kalman Filter (AKF), a so-called Generalized Residual (GR) is constructed equaling the mismatch between the adaptive and the optimal filter state estimates plus an AKF-independent noise component. By virtue of this modification, the GR mean square becomes a new model proximity criterion for these filters. Minimizing this criterion via conventional practical optimization methods produces exactly the same result (AKF = OKF) as would be obtained by minimizing the theoretical criterion being, unfortunately, inaccessible to any AKF numerical optimization methods. The article presents a detailed step-by-step procedure explaining the above solution in terms of a parameterized transfer function. For the sake of clarity and for stimulating real world applications of the approach, the article employs the transfer function model of a twisted-pair line in a typical xDSL system. The implementation challenges of theoretical provisions of the method are discussed. The issue of extending the proposed approach to the problems of identifying linear models for nonlinear systems is outlined in the directions for further research.

Détail

Keywords: LTI model, complete observability, Kalman filter, adaptive filter, indirect performance index, implementation challenges.

Bibliographie

[1] Pillonetto G., Ljung L., “Full Bayesian identification of linear dynamic systems using stable kernels”, Proc. Natl. Acad. Sci. USA, 120:18 (2023), e2218197120 | DOI

[2] Schoukens J., Vaes M., Pintelon R., “Linear system identification in a nonlinear setting: Nonparametric analysis of the nonlinear distortions and their impact on the best linear approximation”, IEEE Contr. Systems Magaz., 36:3 (2016), 38–69 | DOI | Zbl

[3] Ljung L., “Convergence analysis of parametric identification methods”, IEEE Trans. Auto. Contr., 23:5 (1978), 770–783 | DOI | Zbl

[4] Ljung L., “System identification”, Signal Analysis and Prediction, Applied and Numerical Harmonic Analysis, eds. A. Procházka, J. Uhlíř, P. W. J. Rayner, N. G. Kingsbury, Birkhäuser, Boston, MA, 1998, 163–173 | DOI

[5] Ljung L., System Identification: Theory for the User, Prentice Hall, Upper Saddle River, N.J., 1999, xxii+609 pp. | Zbl

[6] Gevers M., “System identification without Lennart Ljung: what would have been different?”, Forever Ljung in System Identification, eds. T. Glad, G. Hendeby, Studentlitteratur AB, Lund, Sweden, 2006, 61–85

[7] Schoukens J., Pintelon R., Rolain Y., “Time domain identification, frequency domain identification. Equivalencies! Differences?”, Proc. 2004 American Control Conf. (30 June 2004 – 02 July 2004), v. 1, Boston, MA, USA, 2004, 661–666 | DOI

[8] Oomen T., van Herpen R., Quist S., et al., “Connecting system identification and robust control for next-generation motion control and a wafer stage”, IEEE Trans. Contr. Syst. Technol., 22:1 (2014), 102–118 | DOI

[9] Dedene N., Pintelon R., Lataire P., “Estimation of a global synchronous machine model using a multiple-input multiple-ouput estimator”, IEEE Trans. Energy Conver., 18:1 (2003), 11–16 | DOI

[10] Wei Y., Mantooth A., “LLC resonant converter—frequency domain analysis or time domain analysis”, 2020 IEEE 9th Int. Power Electronics and Motion Control Conf. (IPEMC2020-ECCE Asia; 29 November–02 December, 2020), Nanjing, China, 2020, 552–557 | DOI

[11] Rivera D. E., Lee H., Mittelmann H. D., Braun M. W., “Constrained mulisine output signals for plant-friendly identification of chemical process systems”, J. Process Contr., 19:4 (2009), 623–635 | DOI

[12] Peeters B., Ventura C. E., “Comparative study of modal analysis techniques for bridge dynamic characteristics”, Mech. Syst. Signal Process., 17:5 (2003), 965–988 | DOI

[13] Westwick D. T., Kearney R. E., Identification of Nonlinear Physiological Systems, Wiley-IEEE Press, Piscataway, N.J., 2003, xii+261 pp. | DOI

[14] Semushin I., Tsyganova J., Kulikova M., et al., “Identification of human body daily temperature dynamics via minimum state prediction error method”, Proc. 2016 Europ. Control Conf. (ECC2016; June 29–July 1, 2016), Aalborg, Denmark, 2016, 2429–2434 | DOI

[15] Semoushin I. V., “Identifying parameters of linear stochastic differential equations from incomplete noisy measurements”, Recent Development in Theories and Numerics (International Conference on Inverse Problems; January 9–12, 2002), eds. H. Yiu-Chung, Y. Masahiro, C. Jin, L. June-Yub, World Scientific Publishing Co. Pte. Ltd., River Edge, NJ, 2003, 281–290 | DOI | Zbl

[16] Semushin I. V., Tsyganova J. V., “Reducing computational complexity for DBZF precoding in xDSL downlinks”, J. Phys.: Conf. Ser., 1096 (2018), 012159 | DOI

[17] Düngen M., Crosstalk Mitigation Techniques for Digital Subscriber Line Systems, Dissertation (Dr.-Ing.), Technische Universität Hamburg, Hamburg, 2016, 160 pp. | DOI

[18] Begović A., Škaljo N., Behlilović N., “A simple model of copper local loop for use in current DSL application”, Proc. 2015 23rd Telecommunications Forum Telfor (TELFOR; November 24–26, 2015), Belgrade, Serbia, 2015, 114–117 | DOI

[19] Begović A., Škaljo N., Behlilović N., “An example of modeling local loop transfer function in DSL environment”, Proc. ELMAR-2014 (September 10–12, 2014), Zadar, Croatia, 2014, 1–6 | DOI

[20] Rodrigues R. M., Sales C., Klautau A., et al., “Transfer function estimation of telephone lines from input impedance measurements”, IEEE Trans. Instrum. Meas., 61:1 (2012), 43–54 | DOI

[21] Foubert W., Neus C., Boets P., van Biesen L., “Modeling the series impedance of a quad cable for common mode DSL applications”, Proc. 2008 IEEE Instrumentation and Measurement Technology Conf. (IMTC 2008; May 12–15, 2008), Victoria, BC, Canada, 2008, 250–253 | DOI

[22] Neus C., Boets P., van Biesen L., “Transfer function estimation of digital subscriber lines with single ended line testing”, Proc. 2007 IEEE Instrumentation and Measurement Technology Conf. (IMTC 2007; May 01–03, 2007), Warsaw, Poland, 2007, 1–5 | DOI

[23] Bostoen T., Boets P., Zekri M., et al., “Estimation of the transfer function of a subscriber loop by means of a one-port scattering parameter measurement at the central office”, IEEE J. Sel. Areas Commun., 20:5 (2002), 936–948 | DOI

[24] Semushin I. V., “Adaptation in stoshastic dynamic systems—Survey and new results II”, Int. J. Commun. Netw. Syst. Sci., 4:4 (2011), 266–285 | DOI

[25] Stolle R., “Electromagnetic coupling of twisted pair cables”, IEEE J. Sel. Areas Commun., 20:5 (2002), 883–892 | DOI

[26] Maybeck P. S., Stochastic Models, Estination, and Control. Vol. 1, Mathematics in Science and Engineering, 141, Academic Press, New York, 1979, xix+423 pp. | DOI | Zbl

[27] Bierman G. J., Factorization Methods for Discrete Sequential Estimation, Mathematics in Science and Engineering, 128, Academic Press, New York, 1977, xvi+241 pp. | DOI | Zbl

[28] Sima V., Algorithms for Linear-Quadratic Optimization, Pure and Applied Mathematics, Marcel Dekker, Marcel Dekker, New York, 1996, vii+366 pp. | DOI | Zbl

[29] Fletcher R., Practical Methods of Optimization, John Wiley Sons, Chichester, 2000, xvii+436 pp. | DOI

[30] Potter J. E., Stern R. G., “Statistical filtering of space navigaton measurements”, Proc. 1963 AIAA Guigance and Control Conf. (AIAA 1963; August 12–14, 1963), Cambridge, MA, 1963, 1–5 | DOI

[31] Tsyganova J. V., Orthogonalized Array Methods for Parametric Identification of Discrete Linear Stochastic Systems, Diss. Dr. Sci. (Phys. Math.), Ulyanovsk State Univ., Ulyanovsk, 2017, 400 pp. (In Russian)

[32] Kulikova M. V., Methods of Calculation of the Logarithmic Likelihood Function and its Gradient in Kalman Filtering Algorithms, Diss. Cand. Sci. (Phys. Math.), Ulyanovsk State Univ., Ulyanovsk, 2005, 131 pp. (In Russian)

[33] Semushin I. V., Tsyganova J. V., “Numerically implementing the API based solution to learn a plant dynamics for stochastic control concurrently”, Proc. 2020 European Control Conference (ECC 2020; May 12–15, 2020), St. Petersburg, 2020, 1105–1110 | DOI

[34] Kulikova M. V., Semoushin I. V., “On the evaluation of log likelihood gradient for Gaussian signals”, Int. J. Appl. Math. Stat., 3:5 (2005), 1–14

[35] Tsyganova J. V., Kulikova M. V., “On modern array algorithms for optimal discrete filtering”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 11:4 (2018), 5–30 (In Russian) | DOI | Zbl

[36] Box G. E. P., “Some problems of statistics and everyday life”, J. Am. Stat. Assoc., 74:365 (1979), 1–4 | DOI