The Salted Kalman Filter: Kalman Filtering on Hybrid Dynamical Systems

TL;DR

The paper introduces the Salted Kalman Filter (SKF), an extension of Kalman filtering designed for hybrid dynamical systems, which improves state estimation accuracy during hybrid events by incorporating the saltation matrix.

Contribution

The paper develops the SKF, integrating the saltation matrix into Kalman filtering for hybrid systems, enhancing accuracy over naive methods and providing a new approach for hybrid state estimation.

Findings

SKF reduces mean squared error after hybrid events.

SKF outperforms naive Jacobian-based updates.

Particle filters outperform SKF with many particles.

Abstract

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 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.