Robust identification of nonlinear objects with the help of an evolving radial basis network
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ROBUST IDENTIFICATION OF NONLINEAR OBJECTS WITH THE HELP OF AN EVOLVING RADIAL BASIS NETWORK O. G. Rudenko,a† O. O. Bezsonov,a‡ and S. O. Rudenkoa†
UDC 519.71
Abstract. The problem of neural network-based robust identification of nonlinear dynamic objects in the presence of non-Gaussian noise is considered. To solve this problem, a radial basis network was chosen whose structure is specified and training is provided with the help of a genetic algorithm. The simulation results are presented that confirm the efficiency of the proposed approach. Keywords: neural network, training, identification, evolutionary algorithm, robustness. INTRODUCTION The problem of obtaining mathematical models that describe real objects and adequately represent their properties is not only of interest in itself but also is an integral part of the problem of optimization of functioning definite objects (their control, behavior prediction, etc.). The main difficulties in obtaining a high-quality solution to an identification problem are conditioned by the nonlinearity and nonstationarity of characteristics of objects being investigated, presence of various noises, and absence of sufficient a priori information on the objects themselves and their functioning conditions. Whereas the theory of identification of linear stationary objects is developed rather thoroughly, nonlinear objects are mostly identified subjectively using mainly the approximation of nonlinearities by various series (Volterra, Hammerstein, Wiener, etc.) or polynomials. However, these classical models are nonparametric, which considerably complicates the solution of identification problems. Difficulties connected with the identification of nonlinear dynamic objects by traditional methods have led to the appearance and development of an alternative neural network-based approach to the solution of this problem. Since, from the mathematical viewpoint, the identification problem is the problem of approximation (or recovery) of some nonlinear function that is complicated in general form, to solve the problem, artificial neural networks (ANNs) are used that are formed by neurons with nonlinear activation functions and are good approximators. It should be noted that, in investigating nonlinear objects with the help of ANNs, a fundamental role is played by objects of the form NARMAX (Nonlinear Auto-Regressive Moving Average with eXogeneous inputs) or NARX (Nonlinear Auto-Regressive eXogeneous with inputs) models that are, respectively, of the form [1–3] y( k ) = f [ y( k - 1),... , y( k - K y ), u( k - 1),... , u( k - K u ), x ( k - 1),... , x ( k - K x )] + x ( k ) ,
(1)
y( k ) = f [ y( k - 1),... , y( k - K y ), u( k - 1),... , u( k - K u )] + x ( k ) ,
(2)
where y( i ) and u( i ) are output and input signals, respectively, K y , K u , and K x are orders of lag for the output and input signals of the object and noise, respectively, f [•] is a nonlinear function, and x( k ) is noise. For models (1) and (2), the identification problem consists of obtaining an estimate for the f
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