A Lyapunov-like Function for Full Information Estimation

Full information estimation (FIE) is a type of optimization-based state estimation related to moving horizon estimation (MHE). In FIE, all available measurements (as opposed to those within a fixed horizon) are used to estimate the system's state. Although we expect it to produce a better state estimate than MHE because it includes more information, recent results have shown MHE to be robustly stable in the presence of bounded disturbances, while a general proof for FIE remains elusive. Lyapunov functions and ISS Lyapunov functions are invaluable tools for robustness analysis in model predictive control (MPC). Here, we present a Lyapunov-like function for FIE and use it to show that FIE is asymptotically stable in the absence of disturbances. This function is, to our knowledge, the first of its kind for nonlinear optimization-based state estimation, and it construction requires both an assumption of nonlinear stabilizability and a new type of storage function, termed an i-IOSS Lyapunov function, related to the property of incremental input/output-to-state stability (i-IOSS), a common form of nonlinear detectability.