PROJECTION OF THE NONLINEAR DIFFERENTIAL EQUATION RICCATI INTO VOLTERRA POLYNOMIALS USING THE FINITE ELEMENT METHOD

Abstract

This paper concerns the problems of modeling nonlinear causal systems. The aim of the paper
is to transform the nonlinear Riccati differential equation into operator form. The brief review
of approaches to modeling nonlinear dynamic systems is provided. Problems of projection the
original equation into differential equations with Volterra kernels and solving the resulting equations
are solved for Volterra series model. A short description of the method of projection into
hyperspace using the Frechet functional derivative is given. The result of projection is differential
equations with solutions in the form of Volterra kernels is shown. The linear kernel is a solution to
an ordinary differential equation, and kernels higher than first order are found by solving partial
differential equations with respect to time domain variables. The model with only the first two
kernels of the series is considered. Attention is paid to the equation with a bilinear kernel. Search
of such kernel by analytical methods is more complicated compared to the equation with a linear
kernel, which is why this work attempts to calculate it using a numerical method. The detailed
description of the developed algorithm for calculating the bilinear kernel using the finite element
method is given. Using this method, the general operator model will the semi-analytic structure in
the form of a sum of convolutions with the analytical linear kernel and the finite element bilinear
kernel. An operator model for the weakly nonlinear system has been developed. The simulation
modeling was done for verify the operator model. The computational experiment consisted of obtaining
the transient response by test signal in the form of the Heaviside function. The responses of
the linearized and proposed operator model were calculated using discrete convolution. The obtained
characteristics were compared with the fourth-order Runge-Kutta solution as a reference
solution of the basic equation. The developed operator model gives a response closer to the reference
response, which is confirmed by the results of residual calculations.