Moderate Deviations of the Random Riccati Equation

TL;DR

This paper analyzes the behavior of the invariant distributions of the random Riccati equation under intermittent observations, establishing a moderate deviations principle and explicitly identifying the rate function for controllable and observable systems.

Contribution

It proves that for controllable and observable systems, the invariant distributions satisfy a moderate deviations principle as observation probability approaches one, with an explicitly identified rate function.

Findings

Invariant distributions converge as observation probability approaches one.

Moderate deviations principle holds with a explicitly identified rate function.

Results apply to controllable and observable systems.

Abstract

We characterize the invariant filtering measures resulting from Kalman filtering with intermittent observations (\cite{Bruno}), where the observation arrival is modeled as a Bernoulli process. In \cite{Riccati-weakconv}, it was shown that there exists a $\overline{\gamma}^{\{\scriptsize{sb}}}>0$ such that for every observation packet arrival probability $\overline{\gamma}$, $\overline{\gamma}>\overline{\gamma}^{\{\scriptsize{sb}}}>0$, the sequence of random conditional error covariance matrices converges in distribution to a unique invariant distribution $\mathbb{\mu}^{\overline{\gamma}}$ (independent of the filter initialization.) In this paper, we prove that, for controllable and observable systems, $\overline{\gamma}^{\{\scriptsize{sb}}}=0$ and that, as $\overline{\gamma}\uparrow 1$, the family $\{\mathbb{\mu}^{\overline{\gamma}}\}_{\overline{\gamma}>0}$ of invariant distributions satisfies a moderate deviations principle (MDP) with a good rate function $I$. The rate function $I$ is explicitly identified. In particular, our results show: