AI 中文总结
针对非线性卡尔曼滤波中单一中心近似导致增益不准和协方差过度自信的问题，提出探针集方法，通过多中心局部近似选择最优增益并重校准协方差，在保持精度的同时显著降低协方差不一致性和罕见大误差。
AI 中文摘要
卡尔曼滤波器（KF）将增益选择为状态-测量互协方差与创新协方差逆的乘积。在非线性系统中，预测分布通常改变形状，这些协方差需要近似。传统的非线性KF围绕单一中心（即预测状态）进行近似。当单一局部近似不能代表整个不确定区域内的测量几何时，它们可能产生不准确的增益和过度自信的协方差估计。为解决此问题，我们引入了探针集，即滤波器重复其局部近似的小型协方差缩放的状态集合。然后，我们选择使这些局部近似报告的平均协方差最小化的增益。相同的构造用于我们先前提出的协方差重校准步骤，以提高协方差一致性。该构造适用于不同的KF变体，无需特定的近似规则。对于二次测量，我们推导了条件，在这些条件下，当预测不确定性被充分低估时，探测产生的归一化真实误差方差低于相应的单中心增益。我们比较了四种KF变体与多个基线，针对每个系统进行了300个随机设置。结果表明，探测保持了典型的准确性，同时显著减少了协方差不一致性和主导每次运行误差均方根的罕见大误差轨迹。代码可在该https URL获取。
英文摘要
Kalman filters (KFs) choose the gain as the product of the state--measurement cross-covariance and the inverse of the innovation covariance. In nonlinear systems, the predictive distribution generally changes shape, and these covariances require approximation. Conventional nonlinear KFs approximate them around a single center, the predicted state. They can produce inaccurate gains and overconfident covariance estimates when one local approximation does not represent the measurement geometry across the uncertainty region. To address this issue, we introduce probe sets, small covariance-scaled collections of states at which the filter repeats its local approximation. We then select the gain that minimizes the average covariance reported by these local approximations. The same construction is used in our previously proposed covariance recalibration step to improve covariance consistency. The construction applies to different KF variants without requiring a specific approximation rule. For quadratic measurements, we derive conditions under which probing yields lower normalized true error variance than the corresponding single-center gain when prediction uncertainty is sufficiently understated. We compare four KF variants with multiple baselines across 300 randomized setups for each of two systems. The results show that probing preserves typical accuracy while substantially reducing covariance inconsistency and the rare large-error trajectories that dominate the root mean square of per-run errors. The code is available at https://github.com/Shida-Jiang/Probe_KF.
