Stochastic observers on Lie groups: a tutorial
Résumé
In this tutorial paper, we discuss the design of geometric
observers on Lie groups in the presence of noise. First we review Lie groups, and the
mathematical definition of noises on Lie groups, both in
discrete and continuous time. In particular, we discuss the It\^o-Stratonovich dilemma. Then, we review the recently introduced notion of group affine systems on
Lie groups. For those systems, we discuss how using the machinery of Harris chains, (almost) globally convergent deterministic observers might be shown to possess stochastic properties in the presence of noise. We also discuss the design of (invariant)
extended Kalman filters (IEKF), and we recall the main result, i.e., the Riccati equation
computed by the filter to tune its gains has the remarkable property
that the Jacobians (A,C) with respect to the system's dynamics and
output map are independent of the followed trajectory, whereas the noise
covariance matrices that appear in the Riccati equation may depend
on the followed trajectory. Owing to this partial independence, some
local deterministic convergence properties of the IEKF for group-affine systems on Lie
groups may be proved under standard observability conditions.
Origine : Fichiers produits par l'(les) auteur(s)
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