Many state estimation and control algorithms require knowledge of how probability distributions propagate through dynamical systems. However, despite hybrid dynamical systems becoming increasingly important in many fields, there has been little work on utilizing the knowledge of how probability distributions map through hybrid transitions. Here, we make use of a propagation law that employs the saltation matrix (a first-order update to the sensitivity equation) to create the Salted Kalman Filter (SKF), a natural extension of the Kalman Filter and Extended Kalman Filter to hybrid dynamical systems. Away from hybrid events, the SKF is a standard Kalman filter. When a hybrid event occurs, the saltation matrix plays an analogous role as that of the system dynamics, subsequently inducing a discrete modification to both the prediction and update steps. The SKF outperforms a naive variational update – the Jacobian of the reset map – by having a reduced mean squared error in state estimation, especially immediately after a hybrid transition event. Compared against a hybrid particle filter, the particle filter outperforms the SKF in mean squared error only when a large number of particles are used, likely due to a more accurate accounting of the split distribution near a hybrid transition.

许多状态估计和控制算法需要了解概率分布如何通过动态系统传播。然而，尽管混合动力系统在许多领域变得越来越重要，但很少有关于利用概率分布如何通过混合转换映射的知识的工作。在这里，我们利用传播定律，该定律采用跃迁矩阵（灵敏度方程的一阶更新）来创建加盐卡尔曼滤波器 (SKF)，这是卡尔曼滤波器和扩展卡尔曼滤波器对混合动力系统的自然扩展. 除了混合事件，SKF 是一个标准的卡尔曼滤波器。当混合事件发生时，跃迁矩阵起到与系统动力学类似的作用，随后对预测和更新步骤进行离散修改。SKF 通过降低状态估计中的均方误差（尤其是在混合转换事件之后立即出现）而优于朴素变分更新（重置映射的雅可比行列式）。与混合粒子滤波器相比，粒子滤波器仅在使用大量粒子时在均方误差方面优于 SKF，这可能是由于更准确地考虑了混合过渡附近的分裂分布。