Identification of Input Signals in Integral Models of
One Class of Nonlinear Dynamic Systems
The problem of restoring input signals is one of the intense developing research areas and is the intersection of the mathematical modeling theory, the automatic control theory and the inverse problems theory. This paper focuses on solving the identification problem of the input signal that corresponds to a given (desired) output in the case of no feedback. An approach to the approximate solution of polynomial Volterra equations of the first kind of the Nth degree that arise when modeling nonlinear dynamics by the apparatus of Volterra integro-power series is described. These equations appear when a nonlinear dynamic process is modeled using the integro-power Volterra series.One class of nonlinear dynamical black box type systems is considered. Unlike a scalar input, the form of the integral model is complicated by the inclusion of terms that take into account the simultaneous change of individual components of the input signal vector. Integral models with constant Volterra kernels were considered earlier. This paper assumes the symmetric Volterra kernels are representable as the product of a finite number of continuous functions. The identification problem is solved using the Newton-Kantorovich method. A numerical solution of the corresponding linear integral Volterra equation of the first kind is proposed as an initial approximation. The obtained formulas for calculations are based on quadrature methods (right rectangles). The effectiveness of the proposed algorithms is illustrated for the reference dynamic system and confirmed by numerical results.
Svetlana Solodusha, Cand. Sci. (Phys.–Math.), Assoc. Prof., Leading Scientific Researcher of Melentiev Energy Systems Institute SB RAS, 130, Lermontov St., Irkutsk, 664033, Russian Federation, tel: (3952)500646 e-mail: email@example.com
Solodusha S.V. Identification of Input Signals in Integral Models of One Class of Nonlinear Dynamic Systems. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 230, pp. 73-82. https://doi.org/10.26516/1997-7670.2019.30.73
- Apartsyn A.S. Polilineynyye uravneniya Vol’terra I roda: elmenty teorii i chiisldennyye metody [Polilinear integral Volterra equations of the first kind: the elements of the theory and numeric methods]. The Bulletin of Irkutsk State University. Series Mathematics, 2007, vol. 1, no. 1, pp. 13-41. (in Russian)
- Apartsyn A.S. Studying the polynomial volterra equation of the first kind for solution stability. Automation and Remote Control, 2011, vol. 72, no. 6, pp. 1229–1236. https://doi.org/10.1134/S0005117911060099
- Cheng C.M., Peng Z.K., Zhang W.M., Meng G. Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review. Mechanical Systems and Signal Processing, 2017, vol. 87, pp. 430-364. https://doi.org/10.1016/j.ymssp.2016.10.029
- Kantorovich L.V., Akilov G.P. Funktsional’nyy analiz v normirovannykh prostranstvakh [Functional Analysis in Normed Spaces]. Moscow, Fizmatlit Publ., 1959. (in Russian)
- Kleiman E.G. Identification of input signals in dynamical systems. Automation and Remote Control, 1999, vol. 60, no. 12, pp. 1675–1685.
- Solodusha S.V. To the numerical solution of one class of systems of the Volterra polynomial equations of the first kind. Num. Anal. Appl., 2018, vol. 11, no. 1, pp. 89–97. https://doi.org/10.1134/S1995423918010093
- Solodusha S.V. Automatic control systems modeling by Volterra polynomials. Model. Anal. Inform. Sist., 2012, vol. 19, no. 1, pp. 60–68. (in Russian)
- Spravochnik po teorii avtomaticheskogo upravleniya [Handbook of Theory of Automatic Control]. Ed. A.A. Krasovskiy. Moscow, Nauka Publ., 1987. (in Russian)
- Verlan’ A.F., Sizikov V.S. Integral’nyye uravneniya: metody, algoritmy, programmy [Integral equations: methods, algorithms, programs]. Kiev, Naukova Dumka Publ., 1986. (in Russian)
- Volterra V. A Theory of Functionals, Integral and Integro-Differential Equations. New York, Dover Publ., 1959.