Comparative Analysis of Quasi-Linear Kalman-Type Algorithms in Estimating a Markov Sequence with Nonlinearities in the System and Measurement Equations

Abstract

The so-called Kalman type algorithms (KTA) are considered, among them quasi-linear KTAs introduced as a separate class, the features of which are the Gaussian approximation of the a posteriori probability density function (p.d.f.) at each step and the procedure for processing the current measurement based on the ideology of a linear optimal algorithm. The unified structure of such algorithms and their features are discussed. Two groups are distinguished the quasi-linear KTA: the first is algorithms using Taylor series expansion of the nonlinear functions, and the second is the so-called linear regression KTA. The methods of their designing are considered, and the common features are described. Detailed attention is paid to the following KTAs: the extended Kalman filter (EKF), polynomial filters of the second and third order (PF2 and PF3), as representatives of the first group, and the Unscented and Cubature Kalman filters (UKF and CKF), as representatives the second one. Their comparative analysis is carried out using the estimation problem of a scalar Markov sequence in the presence of nonlinearities in the shaping filter and in the measurement equations. For all the studied algorithms, the formulas are given in a form convenient for comparison. Based on these formulas, possible causes of a decrease in accuracy and a violation of the consistency properties are identified. Using the previously proposed procedure based on the method of statistical tests, predictive simulation was carried out, which made it possible to confirm the conclusions obtained previously on the basis of an analysis of the formulas for the algorithms being compared. The simulation also allowed to compare the computational complexity of the compared algorithms. The results of the study may be useful to developers involved in the processing of measurement information when choosing a filtering algorithm for solving specific practical estimation problems.