APPLICATION OF MATHEMATICAL MODEL OF CUBIC SPLINE INTERPOLATION IN AUTOMATION OF AIR NAVIGATION SYSTEM

Abstract

The principles of application of the mathematical model of cube spline interpolation in the automation of the aeronautical system are considered in the article. It is emphasized that, in contrast to the discrete equation describing state and measurement in a discrete model, the evolution of a random process and time measurements in a stochastic differential model can be described by a differential equation of state and a discrete equation of measurement. It is noted that stochastic differential equations have two methods of analysis: strong solutions and weak solutions, and only some types of stochastic differential equations belong to closed solutions with a strong solution; while a weak solution is a distributed solution, ie a probability in a continuous functional space. It is emphasized that in stochastic differential equations the Brownian motion is only the driving force of microscopic random fluctuations, and not the appearance in the property of the macroscopic mean. The weak solution of the stochastic differential equation is determined by the transfer function. The cubic spline algorithm is substantiated, which is a one-dimensional case, the probability density is segmented to approximate the a priori probability density of the system state, it directly solves the probability density function of the system state to more clearly understand the different possibilities of the state. By constructing the conditions of cubic spline interpolation, in the paper this function is used to approximate the solution of the direct Kolmogorov equation to transform the problem into a solution of a system of ordinary differential equations with respect to the coefficients in the piecewise function. It is mathematically proved that the optimal solution for state estimation can be obtained by the Bayesian formula, but first of all it is necessary to know the preliminary probability density function of the system state and investigate the expediency of using the cubic spline method to solve it. An algorithm for interpolating a cubic spline is proposed, which transforms partial differential equations of the state probability density in time into a solution of the ordinary differential of the piecewise equation and solves the state of the system in the one-dimensional space of functions.