Inverse Extended Kalman Filter

Recent advances in counter-adversarial systems have garnered significant research interest in inverse filtering from a Bayesian perspective. For example, interest in estimating the adversary's Kalman filter tracked estimate with the purpose of predicting the adversary's future steps has led to recent formulations of inverse Kalman filter (I-KF). In this context of inverse filtering, we address the key challenges of nonlinear process dynamics and unknown input to the forward filter by proposing inverse extended Kalman filter (I-EKF). We derive I-EKF with and without an unknown input by considering nonlinearity in both forward and inverse state-space models. In the process, I-KF-with-unknown-input is also obtained. We then provide theoretical stability guarantees using both bounded nonlinearity and unknown matrix approaches. We further generalize these formulations and results to the case of higher-order, Gaussian-sum, and dithered I-EKFs. Numerical experiments validate our methods for various proposed inverse filters using the recursive Cram\'er-Rao lower bound as a benchmark.