Inverse Extended Kalman Filter -- Part II: Highly Non-Linear and Uncertain Systems

TL;DR

This paper advances inverse filtering techniques for highly non-linear systems by developing an inverse extended Kalman filter with stability guarantees and a kernel-based approach to learn unknown dynamics, enhancing adversarial state estimation.

Contribution

It introduces a theory for inverse EKF in highly non-linear models, including stability analysis and a kernel-based method to handle unknown system dynamics.

Findings

The inverse second-order EKF is shown to be stable under bounded non-linearity.

Kernel-based EKF effectively learns unknown system dynamics from observations.

Numerical experiments demonstrate the proposed filters approach the recursive Cramér-Rao lower bound.

Abstract

Counter-adversarial system design problems have lately motivated the development of inverse Bayesian filters. For example, inverse Kalman filter (I-KF) has been recently formulated to estimate the adversary's Kalman-filter-tracked estimates and hence, predict the adversary's future steps. The purpose of this paper and the companion paper (Part I) is to address the inverse filtering problem in non-linear systems by proposing an inverse extended Kalman filter (I-EKF). The companion paper proposed the theory of I-EKF (with and without unknown inputs) and I-KF (with unknown inputs). In this paper, we develop this theory for highly non-linear models, which employ second-order, Gaussian sum, and dithered forward EKFs. In particular, we derive theoretical stability guarantees for the inverse second-order EKF using the bounded non-linearity approach. To address the limitation of the standard I-EKFs that the system model and forward filter are perfectly known to the defender, we propose reproducing kernel Hilbert space-based EKF to learn the unknown system dynamics based on its observations, which can be employed as an inverse filter to infer the adversary's estimate. Numerical experiments demonstrate the state estimation performance of the proposed filters using recursive Cram\'{e}r-Rao lower bound as a benchmark.