MSE Tail Analysis for Remote State Estimation of Linear Systems Over Multiantenna Random Access Channels

In this article, we consider the Kalman filter (KF)-based state estimation in the multisensor networked control systems. The sensors communicate plant state observations to a remote state estimator over a multiantenna random access network, resulting in random fading in measurements and rank-deficient observation updates in KF. In contrast to the conventional literature where average state estimation mean-square-error (MSE) is adopted as the estimation performance measure, we analyze the state estimation MSE tail distribution which guarantees fine-grained characterizations of the MSE sample path. We developed various sufficient conditions for the well behaved tail distribution of the MSE. Specifically, we propose the observable and unobservable cone decomposition of the transformed coordinate system induced by the singular value decomposition of the measurement matrix, which enables us to overcome the limitation of MSE analysis with rank deficient measurement matrix. In the noiseless plant case, we establish the sufficient condition such that the state estimation MSE is a supermartingale in high SNR regime. Via martingale convergence theorems, we show that the state estimation MSE is almost surely stable, which results in outlier-free MSE sample path. In the noisy plant case, we show that the state estimation MSE is uniformly upper bounded by a Markov chain in high SNR regime. We establish a sufficient condition for the existence of the steady-state distribution of the Markov chain. Via analyzing the stochastic fixed point equation associated with the stationary distribution of the Markov chain, we provide the closed-form expression for the MSE tail probability. We further established a power law that provides closed-form characterizations of the tradeoff between the communication resource and the MSE tail decay rate, where more communication resource consumption leads to faster exponential decaying MSE tail. We also recover the results in the existing literature when applying our developed theory to the scalar systems.