  Optimization-based state estimation is useful for nonlinear or constrained
dynamic systems for which few general methods with established properties are
available. The two fundamental forms are moving horizon estimation (MHE) which
uses the nearest measurements within a moving time horizon, and its theoretical
ideal, full information estimation (FIE) which uses all measurements up to the
time of estimation. Despite extensive studies, the stability analyses of FIE
and MHE for discrete-time nonlinear systems with bounded process and
measurement disturbances, remain an open challenge. This work aims to provide a
systematic solution for the challenge. First, we prove that FIE is robustly
globally asymptotically stable (RGAS) if the cost function admits a property
mimicking the incremental input/output-to-state stability (i-IOSS) of the
system and has a sufficient sensitivity to the uncertainty in the initial
state. Second, we establish an explicit link from the RGAS of FIE to that of
MHE, and use it to show that MHE is RGAS under enhanced conditions if the
moving horizon is long enough to suppress the propagation of uncertainties. The
theoretical results imply flexible MHE designs with assured robust stability
for a broad class of i-IOSS systems. Numerical experiments on linear and
nonlinear systems are used to illustrate the designs and support the findings.
