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.

翻译：近期，对抗性系统设计问题推动了逆贝叶斯滤波器的发展。例如，逆卡尔曼滤波器（I-KF）已被提出用于估计对手的卡尔曼滤波跟踪估计，从而预测对手的后续步骤。本文及其姊妹篇（第一部分）旨在通过提出逆扩展卡尔曼滤波器（I-EKF）解决非线性系统中的逆滤波问题。姊妹篇提出了I-EKF（含与不含未知输入）及I-KF（含未知输入）的理论框架。本文进一步将该理论推广至高度非线性模型，采用二阶、高斯和以及抖动前向EKF。具体而言，我们利用有界非线性方法导出了逆二阶EKF的理论稳定性保证。针对标准I-EKF中系统模型和前向滤波器须被防御方完全已知的局限性，我们提出基于再生核希尔伯特空间的EKF，通过观测数据学习未知系统动态，并可作为逆滤波器推断对手的估计值。数值实验以递归克拉美-罗下界为基准，展示了所提出滤波器的状态估计性能。