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

Abstract

Recent counter-adversarial system design problems have 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). In a companion paper (Part I), we developed 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 Cramér-Rao lower bound as a benchmark.

Inverse Cubature and Quadrature Kalman filters

“…The defender also knows the attacker's forward filter. The case of the unknown system model may be handled using techniques proposed in our previous work [12]. We omit details of these methods in this paper because of the paucity of space.…”

“…(evaluated similarly as in I-CKF). Denote ẑk and Σ z k as in (12). For the m-point I-QKF, we denote the quadrature points in the n z -dimensional state space by {ζ j , ω j } 1≤j≤m nz such that I-QKF generates a set of m nz quadrature points as s j,k = ẑk + Σ z k ζ j for all j = 1, 2, .…”

“…In this regard, our recent work proposed inverse extended KF (I-EKF) for non-linear system settings in [9,10]. However, even EKF performs poorly in case of severe non-linearities and modeling errors [11], for which our follow-up work [12] introduced inverses of several EKF variants such as high-order and dithered EKFs.…”

“…Similar to the inverse cognition framework in [5,8], we assume perfect system information and propose inverse CKF (I-CKF) and inverse QKF (I-QKF). Our prior work [12] also addressed the case when the system model is not known. We derive the stability conditions for the proposed I-CKF and I-QKF, Our theoretical analyses show that the forward filter's stability is sufficient to guarantee the same for the inverse filter under mild conditions imposed on the system.…”