Mathematical and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations

Abstract

This paper selects a set of reference points in the form of an arithmetic progression for planning an experiment to evaluate the parameters of systems of differential equations. This choice makes it possible to construct estimates of the parameters of a system of first-order differential equations based on the reversibility of the observation matrix, as well as estimates of the parameters of a system of second-order differential equations describing vibrations in a mechanical system by switching to a system of first-order differential equations. In turn, the reversibility of the observation matrix used in parameter estimation is established using the Vandermonde formula. A volumetric computational experiment has been carried out showing how, with an increase in the number of observations in the vicinity of reference points and with a decrease in the step of arithmetic progression, the accuracy of estimates of the parameters of the analyzed system increases. Among the estimated parameters, the most important are the oscillation frequencies of a conservative mechanical system, which establish its proximity to resonance, and therefore, determine the stability and reliability of the system.

Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

“…This paper presents the conditions under which the estimates of the function and its derivative at the reference point converge to exact values with an increase in the number of observations in the vicinity of the reference point/points. Then, the accuracy of the estimation of the parameter of the differential equation and its consistency are determined by the accuracy of the solution of the algebraic equation, which includes estimates of the function and derivative at the reference point/points (see [20] (Theorem 4)).…”

“…The reference point number is usually equal to the number of unknown parameters [20]. In this paper, this approach allows us to construct fast running algorithms to estimate unknown parameters and apply them to estimating the parameters of partial differential equations and their solutions.…”