Numerical Differentiation in the Measurement Model

Abstract

The problems of increasing the accuracy of numerical differentiation in the measurement model are investigated. We consider the measurements of target characteristics of an object that are not directly measurable, and also the reduction of measurements using the example of determining derivatives and initial parameters in the Cauchy problem. We clarify the relationship between the initial parameters of the differential dependencies and particular solutions of the differential equation of the Cauchy problem at the measurement points of the input quantities of the measurement model. Using the example of determining the derivatives in the Cauchy problem, we note the efficiency of the Lagrangian approximation of the functions of the input and output quantities of the measurement model. It is shown that the maximum accuracy of the approximation of the studied characteristics of the object is attainable using the theory of inverse problems. The following scientific results were obtained: assuming the form of the differential equation for the Cauchy problem, particular solutions of the equation are found; using a polynomial approximation, we compute the function of the measurement model input parameters measured by the sensors; we derive formulas for computing the derivatives of the function of input quantities; by the method of measurement reduction, an approximation grid is determined that minimizes the influence of sensor error. We propose a criterion for estimating the efficiency of solving the measurement reduction problem. Formulas are obtained for estimating the level of error in the derivatives of the function of the input quantities, taking into account the given sensor error. It was shown that the experimental results are consistent with the theoretical ones. The applications for the research results include information-measuring systems for monitoring the status of complex technical objects.