Remote State Estimation Over Unreliable Channels With Unreliable Feedback: Strategies and Limits

In this article, we establish a comprehensive theoretical framework for remote estimation in a networked 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. We consider a novel communication model that captures implicit information. Our main objective is to identify the optimal strategies and the fundamental performance limits of the underlying 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 characterise an optimal coding policy profile consisting of a scheduling policy for an encoder and an estimation policy for a decoder, collocated with the source and the monitor, respectively. We derive the recursive equations that must be solved online by the encoder and the decoder. In addition, we prove that the value function, originally defined over an expanding information set, admits a lower-dimensional representation depending only on two variables. We discuss the structural properties of the optimal policies, and analyse the computational complexity of an algorithm proposed for their computation. We then examine a range of special cases derived from our main theoretical results. We complement the theoretical results with a numerical analysis, and compare the performance of different remote estimation tasks in various operating regimes.