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Kornet M. E., Shishkina A. V. About non-parametric identification of infinitely fast systems with delay. Modern technologies. System analysis. Modeling, 2018, Vol. 59, No. 3, pp. 16–23. DOI: 10.26731/1813-9108.2018.3(59).16-23.

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Исследуется задача непараметрической идентификации линейных динамических объектов. В отличие от параметрической идентификации, рассматривается ситуация, когда порядок уравнения, описывающий динамический объект, не задан с точностью до параметров. Более того, задача идентификации рассматривается в условиях нормального функционирования объекта, в отличие от ранее известного подхода к непараметрической идентификации, основанного на подаче на вход объекта функции Хевисайда и дальнейшем применении интеграла Дюамеля. В условиях нормального функционирования на вход объекта подают сигнал произвольного вида. При этом на выходе объекта наблюдается соответствующий отклик. Следует заметить, что измерения входной и выходной переменных осуществляются со случайными помехами. В итоге имеем реализацию (выборку) входных-выходных переменных. Поскольку линейная динамическая система описывается интегралом Дюамеля, то при известных входных и выходных переменных объекта могут быть найдены соответствующие значения весо-вой функции. Подобная реализация в дальнейшем использует непараметрическую оценку весовой функции в виде непараметрической оценки Надарая – Ватсона. Подставляя ее в интеграл Дюамеля, получаем непараметрическую модель линейной динамической системы неизвестного порядка.

The article investigates a problem of nonparametric identification of linear dynamic objects. Unlike parametric identification, the authors consider the situation when the order of the equation describing the dynamic object is not determined by the parameters. Moreover, the identification problem is considered under normal conditions of operation of the object, in contrast to the previously known approach to nonparametric identification, based on the representation of the Heaviside function at the input of the object and the further application of the Duhamel integral. In normal operation, an arbitrary signal is input to the object input. In this case, the corresponding response is observed at the output of the object. It should be noted that measurements of input and output variables are performed with random interference. As a result, we have an implementation (sample) of input and output variables. Since a linear dynamical system is described by the Duhamel integral, the corresponding values of the weight function can be found with known input and output variables of the object. This is achieved by discrete recording of the latter. Having such an implementation, we use a nonparametric estimation of the weight function in the form of a nonparametric Nadaraya-Watson estimate. Substituting this into the Duhamel integral, we thus obtain a nonparametric model of a linear dynamical system of unknown order.

The paper also presents an interesting case of constructing a nonparametric model, when delta-shaped functions are introduced into the input. It was important to find out how the delta-shaped function can differ from the delta function. The estimation of the weight function, and in this case, was determined in the class of nonparametric Nadaraya-Watson estimates. The proposed nonparametric models were investigated in sufficient detail using statistical modeling. In general, nonparametric models have shown quite high efficiency in terms of the accuracy of the forecast of the nonparametric model with respect to the actually measured output of the object. Naturally, the accuracy of nonparametric models is somewhat reduced due to the growing influence of interference from measuring input / output variables or the discreteness of their measurement.

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