Analysis of the function that forms an algebraic system in parametric identification

Problems formulation for modern industrial systems is completely or partially reduced to the problem of parametric identification of dynamic objects. Success of solving such problems largely depends on the availability and volume of a priori information, such as input and output signals of the control object measured with noise. However, the solution of identification problem also requires derivatives of measured signals, and the obtaining of its values by numerical differentiation is an ill-posed problem. This paper considers the parametric identification problem, in which parameters estimation of a mathematical model of a linear dynamic object from experimentally obtained values of input and output signals is reduced to solving a linear algebraic system formed by integral convolution operators with analytically given forming functions. An approach to solve the numerical differentiation problem by using integration by parts in a system of linear algebraic equations forming is proposed. However, for this operation to be correct from the point of view of identification it is necessary that the forming functions should satisfy certain requirements of behavior in both the time and frequency domains. Thus, key task in the formation of a linear algebraic equation system is the choice of linearly independent functions that form this system. The paper proposes such a forming function. A detailed analysis of its properties and properties of its derivatives is presented. Experimental results obtained illustrate the correctness of using an operation of integration by parts instead of numerical differentiation of measured signals relating to the identification problem. The aim of the work is to study features of the formation of an algebraic system of equations, to analyze properties of the functions that form this system in detail, including the impact of correcting parameters of these functions and also to produce recommendations for their choice in parametric identification.