Modeling of Nonlinear Dynamic Systems with Volterra Polynomials: Elements of Theory and Applications

Modeling of Nonlinear Dynamic Systems with Volterra Polynomials: Elements of Theory and Applications

A. S. Apartsyn (Energy Systems Institute, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia), S. V. Solodusha (Energy Systems Institute, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia) and V. A. Spiryaev (Energy Systems Institute, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia)
Copyright: © 2013 |Pages: 28
DOI: 10.4018/ijeoe.2013100102
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Abstract

The paper presents a review of the studies that were conducted at Energy Systems Institute (ESI) SB RAS in the field of mathematical modeling of nonlinear input-output dynamic systems with Volterra polynomials. The first part presents an original approach to identification of the Volterra kernels. The approach is based on setting special multi-parameter families of piecewise constant test input signals. It also includes a description of the respective software; presents illustrative calculations on the example of a reference dynamic system as well as results of computer modeling of real heat exchange processes. The second part of the review is devoted to the Volterra polynomial equations of the first kind. Studies of such equations were pioneered and have been carried out in the past decade by the laboratory of ill-posed problems at ESI SB RAS. A special focus in the paper is made on the importance of the Lambert function for the theory of these equations.
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Introduction

In the theory of mathematical modeling of nonlinear dynamic systems the universal technique of Volterra functional series is well known (Volterra, 1982). It presents a response of the input-output system to an external disturbance (for simplicity consider and to be scalar time functions) as the integro-power series:

(1)

In (1) the functions , that are called Volterra kernels, are identified on the basis of information about system responses to certain families of test input signals.

Application of the finite segment of series (1):

(2) to modeling of nonlinear dynamic objects of various nature is based on continual analogs of the classical Weierstrass theorem about approximation of continuous function by polynomial, i.e. Frechet theorem (Frechet, 1910) and its different generalizations. For example, according to Baesler and Daugavert (1990) for any Volterra initial () mapping – compact in and any there is such that for all the equality:
holds, where has the form (2).

The problem of applying the Volterra series to modeling of technical systems (in particular electric power and heat supply systems) was considered in the monographs (Pupkov, Kapalin & Yushchenko, 1976; Deitch, 1979; Venikov & Sukhanov, 1982; Pupkov & Shmykova, 1982; Danilov, Matkhakov & Filippov, 1990). The international sources to be emphasized are the profound monographs (Rugh, 1981; Doyle III, Pearson & Ogunnaike, 2002; Ogunfunmi, 2007) that provide an extensive bibliography.

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