A Random Dynamical Systems Approach to Filtering in Large-scale Networks

The paper studies the problem of filtering a discrete-time linear system observed by a network of sensors. The sensors share a common communication medium to the estimator and transmission is bit and power budgeted. Under the assumption of conditional Gaussianity of the signal process at the estimator (which may be ensured by observation packet acknowledgements), the conditional prediction error covariance of the optimum mean-squared error filter is shown to evolve according to a random dynamical system (RDS) on the space of non-negative definite matrices. Our RDS formalism does not depend on the particular medium access protocol (randomized) and, under a minimal distributed observability assumption, we show that the sequence of random conditional prediction error covariance matrices converges in distribution to a unique invariant distribution (independent of the initial filter state), i.e., the conditional error process is shown to be ergodic. Under broad assumptions on the medium access protocol, we show that the conditional error covariance sequence satisfies a Markov-Feller property, leading to an explicit characterization of the support of its invariant measure. The methodology adopted in this work is sufficiently general to envision this application to sample path analysis of more general hybrid or switched systems, where existing analysis is mostly moment-based.

💡 Research Summary

The paper addresses optimal state estimation for a discrete‑time linear dynamical system whose measurements are collected by a large number of sensors sharing a common, bandwidth‑ and power‑constrained communication medium. Each sensor i observes the state through y⁽ⁱ⁾ₖ = Cᵢ xₖ + v⁽ⁱ⁾ₖ, where the process and measurement noises are Gaussian with covariances Q and Rᵢ, respectively. Because the medium is shared, only a random subset of sensors can successfully transmit their packets at each time step; the transmission schedule is governed by a generic randomized medium‑access protocol (e.g., ALOHA, CSMA). A crucial modeling assumption is that the estimator receives an acknowledgment (ACK) for every attempted transmission, so it knows exactly which sensors’ measurements arrived. This acknowledgment guarantees that, conditioned on the information available at the estimator, the state remains Gaussian, and the optimal MMSE estimator retains the Kalman‑filter structure, albeit with a time‑varying observation matrix that depends on the random set of received packets.

The authors show that the conditional prediction error covariance matrix Pₖ = E