Intermittent Kalman Filtering: Eigenvalue Cycles and Nonuniform Sampling

TL;DR

This paper analyzes the stability of Kalman filters under intermittent observations, introducing eigenvalue cycles to characterize critical erasure probabilities and showing that nonuniform sampling can improve stability conditions.

Contribution

It introduces eigenvalue cycles to characterize stability thresholds and demonstrates how nonuniform sampling can nearly eliminate the critical erasure probability barrier.

Findings

Eigenvalue cycles determine the critical erasure probability.

Nonuniform sampling can almost surely break eigenvalue cycles.

Stability conditions depend on system eigenvalues and sampling methods.

Abstract

We consider Kalman filtering problems when the observations are intermittently erased or lost. It was known that the estimates are mean-square unstable when the erasure probability is larger than a certain critical value, and stable otherwise. But the characterization of the critical erasure probability has been open for years. We introduce a new concept of \textit{eigenvalue cycles} which captures periodicity of systems, and characterize the critical erasure probability based on this. It is also proved that eigenvalue cycles can be easily broken if the original physical system is considered to be continuous-time --- randomly-dithered nonuniform sampling of the observations makes the critical erasure probability almost surely $\frac{1}{|\lambda_{max}|^2}$.