Projection of the Nonlinear Differential Equation Riccati Into Volterra Polynomials Using the Finite Element Method

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