CIF: Small: Gossiping, Intermittency, and Kalman Filtering

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This research considers large scale sparsely networked systems as they arise in large critical infrastructures, e.g., the power grid, or a large transportation system, or when monitoring large physical spaces instrumented with ad-hoc networks of sensors, as in environmental applications. The research will develop a completely novel distributed estimator to process the measurements collected by the networked sensors. A significant problem is the intermittent availability of the measurements because the underlying communication infrastructure among the sensors may exhibit random failures (e.g., due to packet loss in switched networks) or may communicate through a random protocol (gossip or variations.) Due to this intermittency, the estimator equations are random and the filter equations switch at random times between two modes of behaviors; in particular, the filter Riccati equation switches randomly between a linear algebraic matrix equation (Lyapounov) and a quadratic algebraic matrix equation. This research develops an appropriate distributed estimation algorithm, establishes under what structural conditions and for what rates of measurement collection does the filtering error stay bounded, and aims at determining the probability of the filtering error growing unbounded. To study these issues, this work develops new, potentially transformative, powerful methodologies that draw from the theory of random dynamical systems and (moderate) large deviations principle. The work will define appropriate conditions under which the sequences of the filter random covariances converge, determine their limiting invariant distributions, define appropriate rare events, and develop a moderate deviations principle that determines appropriate best rates at which the probability of rare events vanishes.