On general constrained extended Kalman filter formulated by condition equations: three algorithms

Abstract

Three constrained extended Kalman filters (CEKF) are developed by making use of condition equations which equations allows one to predict directly the residuals of all variables. The first one is a general CEKF algorithm in which it is supposed that all of the observation equations, system equations and constraints of a dynamic problem are non-linear functions. Although the constrained Kalman filter was already investigated by a few contributions, they assumed some restrictive conditions such as linearity of constraints and/or equations. Moreover, this generalization helps one to deal with problems which encounter with raw GPS data. In some problems, constrains can be expressed by a quadratic form. Hence, the second algorithm proposes a CEKF solution with quadratic constraints. In this algorithm, the constraints are not linearized. Eventually, in case of using refined GPS data in which quadratic constraints must be imposed to the state vector, the third algorithm is developed. In this algorithm one does not require to linearize any part of the dynamic model. Rigorous prediction of posterior dispersion (variance-covariance) matrix of the unknown parameters is the other attainment of this contribution. A numerical example demonstrates the efficiency of the proposed algorithm.