Outlier analysis in set-based estimation for nonlinear systems using convex relaxations

Set-based estimation for nonlinear systems is a useful tool to handle sparse and uncertain data. The tool provides outer bounds on feasible parameter sets and reachable states, as well as provable inconsistency certificates for entire parameter regions. In case of errors in the data such as outliers or incorrect a priori assumptions on variable uncertainties, set-based approaches can, however, lead to poor estimates or even rejection of a consistent model. We present a set-based approach to systematically identify outliers or incorrect variable uncertainty assumptions. The basic idea is to detect outliers by quantifying the influence they have on the inconsistency of an underlying feasibility problem. The results build on a set-based estimation framework that employs convex relaxations. Specifically we derive model consistency measures and sensitivity measures that combine the sensitivity information stored in the Lagrange dual variables. An algorithm is developed that iteratively detects outliers that contribute most to inconsistency. The algorithm terminates once the data and model are no longer proved inconsistent. The approach is illustrated by an example.