Uncertainties unavoidably exist in modeling for nonlinear systems: state equation, measurement equation, and/or noises statistics might be uncertain. Such model mismatches render the performance of nominally optimal state estimators being deteriorated or even unsatisfactory. Therefore, robust filters that are insensitive to modeling uncertainties have to be designed. The challenge is to quantitatively describe the uncertainties and then design accordingly efficient robust filters. Since uncertainties in nominal models make prior state distributions and likelihood distributions uncertain as well, this article proposes a distributionally robust particle filtering framework for nonlinear systems subject to modeling uncertainties. Specifically, we use worst-case prior state distributions (near the nominal prior state distributions) to generate prior state particles and/or determine their weights. Likewise, worst-case likelihood distributions (near the nominal likelihood distributions) are used to evaluate the worst-case likelihoods of prior state particles at given measurements. The “worst-case” scenario is quantified by entropy of distributions, and maximum entropy distributions are found in balls centered at nominal distributions with radii defined by statistical similarity measures such as moments-based similarity, Wasserstein distance, and Kullback-Leibler divergence. We prove that Gaussian approximation filters (e.g., unscented/cubature/ensemble Kalman filter) are distributionally robust in the sense that they use maximum entropy prior state distributions and maximum entropy likelihood distributions. Moreover, we show that the distributionally robust particle filtering framework provides a likelihood evaluation method for general nonlinear measurement equation with non-additive and non-multiplicative measurement noises. At last, we discuss measurement outlier treatment strategies in the distributionally robust particle filtering framework.

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非线性系统的分布鲁棒状态估计

非线性系统建模中不可避免地存在不确定性：状态方程、测量方程和/或噪声统计可能是不确定的。这种模型不匹配导致名义上最优状态估计器的性能恶化甚至不令人满意。因此，必须设计对建模不确定性不敏感的鲁棒滤波器。挑战在于定量描述不确定性，然后设计相应的有效鲁棒滤波器。由于名义模型中的不确定性使得先验状态分布和似然分布也不确定，因此本文提出了一种分布鲁棒的粒子滤波框架，用于受建模不确定性影响的非线性系统。具体来说，我们使用最坏情况的先验状态分布（接近标称先验状态分布）来生成先验状态粒子和/或确定它们的权重。同样，最坏情况似然分布（接近标称似然分布）用于评估给定测量中先前状态粒子的最坏情况似然。“最坏情况”的情况是通过分布的熵来量化的，最大熵分布在以标称分布为中心的球中，其半径由统计相似性度量定义，例如基于矩的相似性、Wasserstein 距离和 Kullback-Leibler 散度。我们证明了高斯近似滤波器（例如，unscented/cubature/ensemble Kalman filter）在分布上是稳健的，因为它们使用最大熵先验状态分布和最大熵似然分布。此外，我们表明，分布式鲁棒粒子滤波框架为具有非加性和非乘性测量噪声的一般非线性测量方程提供了似然评估方法。最后，我们讨论了分布式鲁棒粒子滤波框架中的测量异常值处理策略。