In this paper, several inverse Kalman filtering problems are addressed, where unknown parameters and/or inputs in a filtering model are reconstructed from observations of the posterior estimates that can be noisy or incomplete. In particular, duality in inverse filtering and inverse optimal control is studied. It is shown that identifiability and solvability of the inverse Kalman filtering is closely related to that of an inverse linear quadratic regulator (LQR). Covariance matrices of model uncertainties are estimated by solving a well-posed inverse LQR problem. Identifiability of the considered inverse filtering models is established and least squares estimators are designed to be statistically consistent. In addition, algorithms are proposed to reconstruct the unknown sensor parameters as well as raw sensor measurements. Effectiveness and efficiency of the proposed methods are illustrated by numerical simulations.

中文翻译：

---

离散时间系统的逆卡尔曼滤波问题

在本文中，解决了几个逆卡尔曼滤波问题，其中滤波模型中的未知参数和/或输入是根据对可能有噪声或不完整的后验估计的观察来重建的。特别是，研究了逆滤波和逆最优控制中的对偶性。结果表明，逆卡尔曼滤波的可识别性和可解性与逆线性二次调节器（LQR）密切相关。通过解决适定逆 LQR 问题来估计模型不确定性的协方差矩阵。建立了所考虑的逆滤波模型的可识别性，并且最小二乘估计器被设计为统计上一致的。此外，还提出了重建未知传感器参数以及原始传感器测量结果的算法。通过数值模拟说明了所提出方法的有效性和效率。