Nonlinear dynamic system state optimal filtering by observations with random delays

A mathematical model of a nonlinear dynamic observation system with a discrete time which allows taking into account the dependence of the time of receiving observations on the state of the observed object is proposed. The model implements the assumption that the time between the moment when the measurement of the state is formed and the moment when the measured state is received by the observer depends randomly on the position of the moving object. Such an assumption source is the process of observation by stationary means of an autonomous underwater apparatus in which the time of obtaining up-to-date data depends on the unknown distance between the object and the observer. Unlike deterministic delays formed by the known state of the observation environment, to account for the dependence of time delays on the unknown state of the object of observation, it is required to use random functions to describe them. The main result of the study of the proposed model is the solution of the optimal filtering problem. For this purpose, recurrent Bayesian relations describing the evolution of the a posteriori probability density are obtained. The difficulties of using a semifinished filter for practical purposes are discussed. The proposed model is illustrated by a practical example of the task of tracking a moving underwater object based on the results of measurements performed by typical acoustic sensors. It is assumed that the object moves under the water in a plane with a known average speed, constantly performs chaotic maneuvers, and is observed by two independent complexes of acoustic sensors measuring the distances to the object and the guiding cosines. The complexity of determining the position of such an object is illustrated by a simple filter using the geometric properties of the measured quantities and the least squares method.