Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

Karimov, Artur; Nepomuceno, Erivelton G.; Tutueva, Aleksandra; Butusov, Denis

doi:10.3390/math8020300

Open AccessArticle

Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

¹

Youth Research Institute, Saint Petersburg Electrotechnical University “LETI”, Saint Petersburg 197376, Russia

²

Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São, João del-Rei, São João del-Rei MG 36307-352, Brazil

³

Department of Computer Aided Design, Saint Petersburg Electrotechnical University “LETI”, Saint Petersburg 197376, Russia

^*

Author to whom correspondence should be addressed.

Mathematics 2020, 8(2), 300; https://doi.org/10.3390/math8020300

Submission received: 30 December 2019 / Revised: 17 February 2020 / Accepted: 19 February 2020 / Published: 24 February 2020

(This article belongs to the Section C2: Dynamical Systems)

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Abstract

The identification of partially observed continuous nonlinear systems from noisy and incomplete data series is an actual problem in many branches of science, for example, biology, chemistry, physics, and others. Two stages are needed to reconstruct a partially observed dynamical system. First, one should reconstruct the entire phase space to restore unobserved state variables. For this purpose, the integration or differentiation of the observed data series can be performed. Then, a fast-algebraic method can be used to obtain a nonlinear system in the form of a polynomial dynamical system. In this paper, we extend the algebraic method proposed by Kera and Hasegawa to Laurent polynomials which contain negative powers of variables, unlike ordinary polynomials. We provide a theoretical basis and experimental evidence that the integration of a data series can give more accurate results than the widely used differentiation. With this technique, we reconstruct Lorenz attractor from a one-dimensional data series and B. Muthuswamy’s circuit equations from a three-dimensional data series.

Keywords: nonlinear systems; nonlinear identification; system reconstruction; Buchberger–Möller algorithm; Laurent polynomials; nonlinear regression; memristor; chaotic system

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MDPI and ACS Style

Karimov, A.; Nepomuceno, E.G.; Tutueva, A.; Butusov, D. Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding. Mathematics 2020, 8, 300. https://doi.org/10.3390/math8020300

AMA Style

Karimov A, Nepomuceno EG, Tutueva A, Butusov D. Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding. Mathematics. 2020; 8(2):300. https://doi.org/10.3390/math8020300

Chicago/Turabian Style

Karimov, Artur, Erivelton G. Nepomuceno, Aleksandra Tutueva, and Denis Butusov. 2020. "Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding" Mathematics 8, no. 2: 300. https://doi.org/10.3390/math8020300

APA Style

Karimov, A., Nepomuceno, E. G., Tutueva, A., & Butusov, D. (2020). Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding. Mathematics, 8(2), 300. https://doi.org/10.3390/math8020300

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Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

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