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Remote State Estimation over Unreliable Channels with Unreliable Feedback: Fundamental Limits

Touraj Soleymani, Mohamad Assaad, John S. Baras

TL;DR

This work addresses real-time remote state estimation over a pair of unreliable channels: a forward packet-erasure link and a lossy feedback path, modeled for a multi-dimensional Gauss–Markov source. It derives a globally optimal coding policy profile consisting of a symmetric threshold encoder and a linear decoder, with four encoder recursions and one decoder recursion that jointly minimize a causal tradeoff between transmission cost and mean-squared error, quantified by $\Phi(\epsilon,\delta)=\mathsf{E}\left[ \sum_{k=0}^{N} \alpha_k \delta_k + (x_k-\hat{x}_k)^T(x_k-\hat{x}_k) \right]$. The main contribution is a complete synthesis, including the recursive update rules for $\check{x}_k$, $\breve{e}_k$, and $R_k$, plus a Bayes-optimal linear estimator at the decoder, and a detailed complexity assessment. Numerical results reveal how the achievable performance region shifts with the backward-channel error rate $\rho$, highlighting the substantial degradation caused by unreliable feedback and providing insight for designing goal-oriented communication in real-time networked estimation systems.

Abstract

This article is concerned with networked estimation in a system composed of a source that is observed by a sensor, a remote monitor that needs to estimate the state of the source in real time, and a communication channel that connects the source to the monitor. The source is a partially observable dynamical process, and the communication channel is a packet-erasure channel with feedback. Our main objective is to obtain the fundamental performance limits of the underlying networked system in the sense of a causal tradeoff between the packet rate and the mean square error when both forward and backward channels are unreliable. We characterize an optimal coding policy profile consisting of a scheduling policy for the encoder and an estimation policy for the decoder. We complement our theoretical results with a numerical analysis, and compare the performance limits of the networked system in different communication regimes.

Remote State Estimation over Unreliable Channels with Unreliable Feedback: Fundamental Limits

TL;DR

This work addresses real-time remote state estimation over a pair of unreliable channels: a forward packet-erasure link and a lossy feedback path, modeled for a multi-dimensional Gauss–Markov source. It derives a globally optimal coding policy profile consisting of a symmetric threshold encoder and a linear decoder, with four encoder recursions and one decoder recursion that jointly minimize a causal tradeoff between transmission cost and mean-squared error, quantified by . The main contribution is a complete synthesis, including the recursive update rules for , , and , plus a Bayes-optimal linear estimator at the decoder, and a detailed complexity assessment. Numerical results reveal how the achievable performance region shifts with the backward-channel error rate , highlighting the substantial degradation caused by unreliable feedback and providing insight for designing goal-oriented communication in real-time networked estimation systems.

Abstract

This article is concerned with networked estimation in a system composed of a source that is observed by a sensor, a remote monitor that needs to estimate the state of the source in real time, and a communication channel that connects the source to the monitor. The source is a partially observable dynamical process, and the communication channel is a packet-erasure channel with feedback. Our main objective is to obtain the fundamental performance limits of the underlying networked system in the sense of a causal tradeoff between the packet rate and the mean square error when both forward and backward channels are unreliable. We characterize an optimal coding policy profile consisting of a scheduling policy for the encoder and an estimation policy for the decoder. We complement our theoretical results with a numerical analysis, and compare the performance limits of the networked system in different communication regimes.
Paper Structure (10 sections, 9 theorems, 65 equations, 2 figures)

This paper contains 10 sections, 9 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions and Outline
  4. Problem Statement
  5. Source Model
  6. Channel Model
  7. A Team Decision-Making Problem
  8. Main Result
  9. Numerical Analysis
  10. Conclusions

Key Result

Theorem 1

The tradeoff between the packet rate and the mean square error admits a globally optimal solution $(\epsilon^\star, \delta^\star)$ such that $\epsilon^\star$ is a symmetric threshold scheduling policy given by in conjunction with $\check{x}_k = \mathop{\mathrm{\mathsf{E}}}\nolimits[x_k | \mathcal{I}^e_k]$ for $k \in \mathbb{N}_{[1,N]}$, where $\chi_k(\breve{e}_k, R_k) = \lambda^c \breve{e}_k^T A_

Figures (2)

  • Figure 1: The causal tradeoff curves when $\lambda = 0.1$ and $\rho \in \{0.0, 0.3, 0.7, 1.0\}$.
  • Figure 2: The causal tradeoff curves when $\lambda = 0.3$ and $\rho \in \{0.0, 0.3, 0.7, 1.0\}$.

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1: Global optimality
  • Remark 4
  • Definition 2: Value function
  • Theorem 1
  • Remark 5
  • Lemma 1
  • Lemma 2
  • ...and 6 more