A numerical method of nonlinear estimation based on difference equations

The article considers a new numerical method for estimating the parameters of nonlinear mathematical models based on difference equations describing the results of observations. The algorithm of the numerical method includes:
— the construction of a linear-parametric discrete model of the process under study in the form of difference equations, the coefficients of which are known to be associated with the parameters of a nonlinear mathematical model;
— the formation of a generalized regression model based on the difference equations;
— the calculation of the initial approximation estimate and the iterative procedure for refining the mean-square estimates of the coefficients of the generalized regression model;
— the calculation of the estimates of the parameters of the nonlinear mathematical model based on the mean-square estimates of the coefficients of the difference equations;
— evaluation of the error of the results of calculations based on the methods of statistical processing of experimental data.
Various approaches to the construction of systems of difference equations for mathematical models in the form of nonlinear functional dependencies are proposed. The relations underlying the iterative process of refining the coefficients of the generalized regression model constructed on the basis of difference equations are obtained. The procedure for estimating the error of the results of calculations of the parameters of nonlinear functional dependencies, which are known to be associated with the coefficients of the system of difference equations, is described. The application of the numerical method based on the difference equations is illustrated by the examples of estimation of the parameters of the mathematical model of the linear oscillator with attenuation, the model of free oscillations of the dissipative mechanical system with turbulent friction, as well as the parameters of the logistic trend described by the Verhulst (Pearl–Reed) function.