This paper considers the problem of state observation for nonlinear
dynamics. While model-based observer synthesis is difficult due to the
need of solving partial differential equations, this work proposes an
efficient model-free, data-driven approach based on online learning.
Specifically, by considering the observer dynamics as a Chen-Fliess
series, the estimation of its coefficients has a least squares
formulation. Since the series converges only locally, the coefficients
are recursively updated, resulting in an online optimization scheme
driven by instantaneous gradients. When the state trajectories are
available, the online least squares guarantees an ultimate upper bound
of average observation error proportional to the average variation of
states. In the realistic situations where the states cannot be measured,
the immersed linear dynamics based on the Kazantzis-Kravaris/Luenberger
structure is assigned, followed by online kernel principal component
analysis for dimensionality reduction. The proposed approach is
demonstrated by a limit cycle dynamics and a chaotic system.