Remote State Estimation with Smart Sensors over Markov Fading Channels

TL;DR

A novel estimation-cycle based approach is proposed and new elementwise bounds of matrix powers are provided and the stability region in terms of the packet drop probabilities in different channel states can either be convex or nonconvex depending on the transition probability matrix of the Markov channel states.

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 $ρ^2(\mathbf{A})ρ(\mathbf{DM})<1$, where $ρ(\cdot)$ 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.

Figures (8)

• Figure 1: Remote state estimation system.
• Figure 2: Illustration of estimation cycles, where red and green circles denote failed and successful transmissions, respectively, and big circles denote the beginning of estimation cycles.
• Figure 3: The necessary and sufficient stability region of Theorem \ref{['theorem:main']} (i.e., the solid line bounded area) and the sufficient stability region Quevedo (i.e., the dashed line bounded area).
• Figure 4: The original process $\mathbf{x}_t \triangleq [\mathbf{x}_t(1),\mathbf{x}_t(2),\mathbf{x}_t(3),\mathbf{x}_t(4)]$ of the system \ref{['eq:pendubot']} and the remote estimation $\mathbf{\hat{x}}_t \triangleq [\mathbf{\hat{x}}_t(1),\mathbf{\hat{x}}_t(2),\mathbf{\hat{x}}_t(3),\mathbf{\hat{x}}_t(4)]$.
• Figure 5: Average estimation MSE versus packet drop probabilities for the smart-sensor-based and conventional-sensor-based cases.

Theorems & Definitions (29)

• Example 1
• Definition 1: Mean-Square Stability
• Theorem 1
• Remark 1
• Corollary 1: Special Case I
• Remark 2
• Corollary 2: Special Case II
• Remark 3
• Proposition 1: Asymptotic upper bound of the estimation-error function
• Proposition 2: Asymptotic lower bound of the estimation-error function