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 Cram\'{e}r-Rao lower bound as a benchmark.
翻译:最近的反对抗系统设计问题促使了反巴伊西亚过滤器的发展,例如,最近设计了反Kalman过滤器(I-KF)来估计对手的Kalman过滤器跟踪估计结果,从而预测对手的未来步骤。本文和配套文件(Part I)的目的是通过提议一个反向的Kalman过滤器(I-EKF)来解决非线性系统中的反过滤问题。在一份配套文件(第一部分)中,我们开发了I-EKF(有和没有未知的投入)和I-KF(有未知的投入)的理论。在本文件中,我们为高度非线性模型开发了这一理论,这些模型采用二级、高斯和前向式EKFs。特别是,我们利用封闭的非线性Kalman过滤器(I-EKF)来为反线系统提供理论稳定性保障。为了解决标准的I-EKFs(系统模型和前方过滤器为维护者所熟知)的局限性,我们提议用较低的基内尔·希尔伯特空间过滤器空间观察模型,作为在不断测试的周期中学习不为核心的状态的系统。