Rapid advances in designing cognitive and counter-adversarial systems have motivated the development of inverse Bayesian filters. In this setting, a cognitive `adversary' tracks its target of interest via a stochastic framework such as a Kalman filter (KF). The target or `defender' then employs another inverse stochastic filter to infer the forward filter estimates of the defender computed by the adversary. For linear systems, inverse Kalman filter (I-KF) has been recently shown to be effective in these counter-adversarial applications. In the paper, contrary to prior works, we focus on non-linear system dynamics and formulate the inverse unscented KF (I-UKF) to estimate the defender's state with reduced linearization errors. We then generalize this framework to an unknown system model by proposing reproducing kernel Hilbert space-based UKF (RKHS-UKF) to learn the system dynamics and estimate the state based on its observations. Our theoretical analyses to guarantee the stochastic stability of I-UKF and RKHS-UKF in the mean-squared sense shows that, provided the forward filters are stable, the inverse filters are also stable under mild system-level conditions. Our numerical experiments for several different applications demonstrate the state estimation performance of the proposed filters using recursive Cram\'{e}r-Rao lower bound as a benchmark.
翻译:随着认知和反对抗系统设计的快速发展,推动了逆向贝叶斯滤波器的发展。在这种情况下,一个认知“对手”通过随机框架(例如Kalman滤波器(KF))跟踪其感兴趣的目标。然后,目标或“防御者”使用另一个逆随机滤波器来推断由对手计算的防御者的前向滤波器估计值。针对线性系统,最近已经证明了逆卡尔曼滤波器(I-KF)在这些反对抗应用中是有效的。在本文中,与以前的工作相反,我们关注非线性系统动态,并制定逆向无味Kalman滤波器(I-UKF)以估计降低线性化误差的防御者状态。然后,通过提出基于重现核希尔伯特空间的UKF(RKHS-UKF)来将这一框架推广到未知系统模型中,从而学习系统动态并根据其观测估计状态。我们的理论分析表明,为了保证I-UKF和RKHS-UKF在均方意义下的随机稳定性,前提是前向滤波器稳定,在轻微的系统级条件下,逆滤波器也是稳定的。我们针对几种不同应用的数值实验展示了所提出滤波器的状态估计性能,并以递归Cramér-Rao下限作为基准。