Exact Outlier Cancellation in Discrete-Time Observers via Stubborn Redesign

This paper addresses the problem of robust state estimation for nonlinear discrete-time systems subject to sporadic high-magnitude measurement outliers. Given a nominal convergent observer, we propose a "stubborn" redesign by saturating the output injection term with a dynamic threshold. In contrast to previous approaches that rely on scalar filters, we introduce a higher order augmentation based on a finite-memory buffer. Our redesign ensures a deadbeat recovery property, where the influence of an outlier on the saturation threshold vanishes exactly after a finite number of steps determined by the system's observability index. This allows for perfect rejection, at the steady state, of sporadic disturbances enjoying a suitable dwell-time property. In the absence of disturbances, we recover the global convergence properties of the nominal observer. The proposed redesign is illustrated through simulation on a single-link flexible-joint manipulator.