The present thesis explores the use of non-linear state errors to devise extended Kalman filters (EKFs). First we depart from the theory of invariant observers on Lie groups and propose a more general yet simpler framework allowing to obtain non-linear error variables having the novel unexpected property of being governed by a (partially) linear differential equation. This result is leveraged to ensure local stability of the invariant EKF (IEKF) under standard observability assumptions, when extended to this class of (non-invariant) systems. Real applications to some industrial problems in partnership with the company SAGEM illustrate the remarkable performance gap over the conventional EKF. A second route we investigate is to turn the noise on and consider the invariant errors as stochastic processes. Convergence in law of the error to a fixed probability distribution, independent of the initialization, is obtained if the error with noise turned off is globally convergent, which in turn allows to assess gains in advance that minimize the error's asymptotic dispersion. The last route consists in stepping back a little and exploring general EKFs (beyond the Lie group case) relying on a non-linear state error. Novel mathematical (global) properties are derived. In particular, these methods are shown to remedy the famous problem of false observability created by the EKF if applied to simultaneous localization and mapping (SLAM), which is a novel result.
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