Synthesis of an algorithm for processing radar data with adaptation to a typical trajectory of an aircraft based on the scientific and methodological apparatus for reducing the Lagrange problem to an isoperimetric problem
Authors: Kostoglotov A.A. , Penkov A.S. 
Published: February 2025

The constant growth in the tactical and technical characteristics of modern aircraft creates the need to develop more advanced tracking systems, characterized by increased accuracy and speed. In practice, finding a compromise between these parameters is a very difficult task and requires scientific research to find new approaches to solving this problem related to the analysis of the physical characteristics of objects under observation.

The well-known optimal Kalman filter algorithm does not always have satisfactory accuracy in conditions of maneuvering an observed object, and the use of algorithms based on the interaction of several models for various types of maneuver leads to unacceptably high computational complexity.

One of the approaches to finding a compromise between accuracy and speed is the use of a dynamic motion model in the structure of a quasi-optimal algorithm for processing radar information. To synthesize the model, it is proposed to use the scientific and methodological apparatus of reducing the Lagrange problem to an isoperimetric one. Its key stage is to find conditions for the non-negativity of the asynchronous variation of the extended functional, which is the result of convolution of the accuracy criterion and the action integral. This makes it possible to obtain the necessary conditions for the extremum of the objective functional in the form of a boundary value problem obtained on the basis of reducing the Lagrange problem to an isoperimetric one using the Hamiltonian action integral.

The set of trajectories that are the solution to the boundary value problem determines the structure of the quasi-optimal motion model in terms of differential equations and can be successfully used in the construction of recurrent algorithms for adaptive estimation of aircraft motion parameters when moving along standard trajectories with a computational complexity corresponding to the classical Kalman filter algorithm.

The procedure for synthesizing a discrete processing algorithm in the work is proposed to be carried out using an approximate solution of the vector-matrix differential equation of the motion model based on the Taylor series expansion of the matrix exponential. Under conditions of a quasi-stationary approximation, this method makes it possible to determine the structure of the transition state matrix and the covariance matrix of the disturbance, which determine the quasi-optimal algorithm for processing radar information with adaptation to a typical trajectory.

The paper presents a numerical simulation of the process of estimating the altitude and vertical speed of an aircraft performing a landing approach along a typical descent trajectory. Over the entire trajectory under consideration, the gain of the proposed algorithm relative to the classical Kalman filter is on average 15% in height, 20% in vertical speed, and is achieved by adapting to a typical descent trajectory at equal computational costs.