Remote State Estimation with Smart Sensors over Markov Fading Channels

TL;DR

This paper establishes a precise stability condition for remote state estimation over Markov fading channels, introducing a novel approach and demonstrating improved performance with smart sensors compared to conventional methods.

Contribution

It provides a necessary and sufficient stability condition for remote estimation over Markov channels, using a new estimation-cycle based approach and element-wise matrix bounds.

Findings

The stability condition is more effective than existing ones.

The stability region can be convex or concave depending on channel transition probabilities.

Smart sensors outperform conventional sensors in estimation accuracy.

Abstract

We consider a fundamental remote state estimation problem of discrete-time linear time-invariant (LTI) systems. A smart sensor forwards its local state estimate to a remote estimator over a time-correlated $M$-state Markov fading channel, where the packet drop probability is time-varying and depends on the current fading channel state. We establish a necessary and sufficient condition for mean-square stability of the remote estimation error covariance as $\rho^2(\mathbf{A})\rho(\mathbf{DM})<1$, where $\rho(\cdot)$ denotes the spectral radius, $\mathbf{A}$ is the state transition matrix of the LTI system, $\mathbf{D}$ is a diagonal matrix containing the packet drop probabilities in different channel states, and $\mathbf{M}$ is the transition probability matrix of the Markov channel states. To derive this result, we propose a novel estimation-cycle based approach, and provide new element-wise bounds of matrix powers. The stability condition is verified by numerical results, and is shown more effective than existing sufficient conditions in the literature. We observe that the stability region in terms of the packet drop probabilities in different channel states can either be convex or concave depending on the transition probability matrix $\mathbf{M}$. Our numerical results suggest that the stability conditions for remote estimation may coincide for setups with a smart sensor and with a conventional one (which sends raw measurements to the remote estimator), though the smart sensor setup achieves a better estimation performance.