  This paper discusses a general framework for designing robust state
estimators for a class of discrete-time nonlinear systems. We consider systems
that may be impacted by impulsive (sparse but otherwise arbitrary) measurement
noise sequences. We show that a family of state estimators, robust to this type
of undesired signal, can be obtained by minimizing a class of nonsmooth convex
functions at each time step. The resulting state observers are defined through
proximal operators. We obtain a nonlinear implicit dynamical system in term of
estimation error and prove, in the noise-free setting, that it vanishes
asymptotically when the minimized loss function and the to-beobserved system
enjoy appropriate properties. From a computational perspective, even though the
proposed observers can be implemented via efficient numerical procedures, they
do not admit closed-form expressions. The paper argues that by adopting
appropriate relaxations, simple and fast analytic expressions can be derived.
