  Optimization-based state estimation is useful for handling of constrained
linear or nonlinear dynamical systems. It has an ideal form, known as full
information estimation (FIE) which uses all past measurements to perform state
estimation, and also a practical counterpart, known as moving-horizon
estimation (MHE) which uses most recent measurements of a limited length to
perform the estimation. Due to the theoretical ideal, conditions for robust
stability of FIE are relatively easier to establish than those for MHE, and
various sufficient conditions have been developed in literature. This work
reveals a generic link from robust stability of FIE to that of MHE, showing
that the former implies at least a weaker robust stability of MHE which
implements a long enough horizon. The implication strengthens to strict robust
stability of MHE if the corresponding FIE satisfies a mild Lipschitz continuity
condition. The revealed implications are then applied to derive new sufficient
conditions for robust stability of MHE, which further reveal an intrinsic
relation between the existence of a robustly stable FIE/MHE and the system
being incrementally input/output-to-state stable.
