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.
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