Abstract
The special characteristics of dynamic systems put forward new requirements for the creation of stable algorithms for describing the dynamic characteristics of these systems. Spectral methods for dynamic characteristics analyzing make it possible to create models that can be used to solve problems of identification and diagnostics of technical systems, for example, data channels, power lines, electric or hydraulic drives. Impulse response functions, correlation and autocorrelation functions are used as dynamic characteristics of systems. Spectral models are determined on the basis of the well-known Fourier integral in the basis of functions, the justification of which is also very important. The phased implementation of the transformation procedures, the normalization of the Chebyshev–Legendre polynomials made it possible to synthesize the transformed generalized orthonormal Chebyshev–Legendre functions that retain their properties on the argument interval [0, ∞]. These functions can be used to approximate the impulse response functions of dynamic systems. The research of the properties of the synthesized orthonormal functions made it possible to establish their recurrence formulas, which form the basis of computational procedures in spectral mathematical models. The obtained results allow ensuring the uniqueness of mathematical models, their connection with other operator models (for example, Laplace), stability in determining the parameters of models, the implementation of computational procedures and create universal algorithms for identification and diagnostics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Ljung, System Identification: Theory for the User, 2nd edn. (Prentice-Hall, Englewood Cliffs, NJ, 1999)
T. Chen, L. Ljung, Regularized system identification using orthonormal basis functions. European Control Conference, ECC 2015, 1291–1296 (2015). https://doi.org/10.1109/ECC.2015.7330716
P. Heuberger, P. Van Den Hof, B. Wahlberg, Modelling and identification with rational orthogonal basis functions, in Modelling and Identification with Rational Orthogonal Basis Functions (2005), pp. 1–397. https://doi.org/10.1007/1-84628-178-4
G. Pillonetto, G. De Nicolao, A new kernel-based approach for linear system identification. Automatica 46(1), 81–93 (2010). https://doi.org/10.1016/j.automatica.2009.10.031
J.S. Bendat, A.G. Piersol, Engineering Applications of Correlation and Spectral Analysis (Wiley-Interscience, New York, 1980)
G. Rogers, Power System Oscillations (Kluwer, 2000)
P. Korba, Real-time monitoring of electromechanical oscillations in power systems: First findings). IET Gener. Transmission Distrib. 1(1), 80–88 (2007). https://doi.org/10.1049/iet-gtd:20050243
D. Łuczak, K. Nowopolski, Identification of multi-mass mechanical systems in electrical drives, in Proceedings of the 16th International Conference on Mechatronics, Mechatronika (2014), pp. 275–282. https://doi.org/10.1109/MECHATRONIKA.2014.7018271
A.B. Sadridinov, Analysis of energy performance of heading sets of equipment at a coal mine. Gornye nauki i tekhnologii = Mining Sci. Technol. (Russia) 5(4), 367–375 (2020) https://doi.org/10.17073/2500-0632-2020-4-367-375
F.P. Shkrabets, Electrıc supply of underground consumers of deep energy-ıntensıve mınes. Gornye nauki i tekhnologii = Mining Sci. Technol. (Russia) 3, 25–46 (2017). https://doi.org/10.17073/2500-0632-2017-3-25-42
H. Lütkepohl, Impulse Response Function (The New Palgrave Dictionary of Economics, 2008)
A. Hatemi-J, Asymmetric generalized impulse responses with an application in finance. Econ. Model. 36, 18–22 (2014). https://doi.org/10.1016/j.econmod.2013.09.014
P. Eykhoff, System Identification: Parameter and State Estimation (Wiley-Interscience, London, 1974)
P.K. Suetin, Classical Orthogonal Polynomials (Nauka, Moscow, 1979). ((in Russian))
G. Szegö, Orthogonal Polynomials, 4th ed. (American Mathematical Society Colloquium Publication, American Mathematical Society, Providence, RI, 1975), p. 23
V.L. Petrov, Identification of models of electromechanical systems using analysis methods in bases of continuous orthonormal functions. Mekhatronika, avtomatizatsiia, upravlenie 10, 29–36 (2003). ((in Russian))
V.L. Petrov, Federal training and guideline association on applied geology, mining, oil and gas production and geodesy—a new stage of government, academic community and industry cooperation. Gornyi Zhurnal 9, 115–119 (2016). https://doi.org/10.17580/gzh.2016.09.23
V.L. Petrov, Training of mineral dressing engineers at Russian Universities. Tsvetnye Metally 7, 14–19 (2017). https://doi.org/10.17580/tsm.2017.07.02
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Petrov, V.L. (2022). Synthesis and Research of Orthonormal Functions Based on Chebyshev–Legendre Polynomials. In: Karuppusamy, P., Perikos, I., García Márquez, F.P. (eds) Ubiquitous Intelligent Systems. Smart Innovation, Systems and Technologies, vol 243. Springer, Singapore. https://doi.org/10.1007/978-981-16-3675-2_15
Download citation
DOI: https://doi.org/10.1007/978-981-16-3675-2_15
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-16-3674-5
Online ISBN: 978-981-16-3675-2
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)