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 system satisfies a mild Lipschitz continuity or equivalently a robust exponential stability 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.
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