Approximation of Gaussian basis functions in the problem of adaptive control of nonlinear objects
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CYBERNETICS APPROXIMATION OF GAUSSIAN BASIS FUNCTIONS IN THE PROBLEM OF ADAPTIVE CONTROL OF NONLINEAR OBJECTS O. G. Rudenko,a† A. A. Bezsonov,a‡ A. S. Liashenko,a and R. A. Sunnab
UDC 519.71
Abstract. An approach to the development of a neurocontroller for controlling nonlinear dynamical objects on the basis of radial-basis function neural networks is considered. Piecewise-linear approximation of Gaussian basis functions is proposed to simplify the solution of the problem being considered. Simulation results show that the method allows one to reduce the time of construction of an object model and calculation of its control signal. Keywords: neural network, learning, training, nonlinear object, identification, adaptive control, basis function.
Introduction. The classical control theory is based on the theory of linear systems. Elements of this theory are also frequently used in investigating nonlinear objects, but good results can be obtained only when the nonlinearity of an object is insignificant or when the object being investigated is characterized by large time constants and stability in the open position. The absence of sufficiently complete information on properties of objects being investigated and their operating conditions have conditioned the development of an alternative neural network approach to the synthesis of control systems for nonlinear objects that is based on the use of artificial neural networks (ANNs) [1, 2]. This approach, as well as the traditional one, requires the construction of a model (a neural network) of the object being investigated as the basis for the synthesis of a neural network regulator. Among many existing ANNs for the solution of problems of control of nonlinear dynamic objects, radial-basis networks (RBNs) have gained rather widespread use [3–11]. This is conditioned, first of all, by the simplicity of their architectures, good approximating properties, and the presence of efficient optimization algorithms, in particular, recurrent algorithms providing fast training of networks of this type. An RBN approximates the nonlinear operator of the object being investigated by some system of basis functions (BFs) realized by neurons, namely, nonlinear functions F ( x, m ) dependent only on the (radial) distance r = || x - m || , where õ is a vector of independent variables and m is a vector of constant parameters (centers of BFs). Weights and parameters of these functions are determined as a result of training such networks that consists of the minimization of some training criterion. Though there are many various BFs at the present time, the Gaussian BF (GBF) has gained the greatest acceptance since it is optimal (as is shown in [11]) in the case of normally distributed input signals and the use of a quadratic training criterion. However, the computation of these BFs is connected with considerable time expenditures, which leads, on the one hand, to the deceleration of training (identification) processes and, on the other hand, to control lags. The objective of this work is the i
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