  We analyze the convergence aspects of the invariant extended Kalman filter
(IEKF), when the latter is used as a deterministic non-linear observer on Lie
groups, for continuous-time systems with discrete observations. One of the main
features of invariant observers for left-invariant systems on Lie groups is
that the estimation error is autonomous. In this paper we first generalize this
result by characterizing the (much broader) class of systems for which this
property holds. Then, we leverage the result to prove for those systems the
local stability of the IEKF around any trajectory, under the standard
conditions of the linear case. One mobile robotics example and one inertial
navigation example illustrate the interest of the approach. Simulations
evidence the fact that the EKF is capable of diverging in some challenging
situations, where the IEKF with identical tuning keeps converging.
