CIF: Small: Gossiping, Intermittency, and Kalman Filtering

本研究考虑了大规模稀疏网络系统，因为它们出现在大型关键基础设施中，例如电网，或大型运输系统，或在监测大型物理空间时使用自组织传感器网络，如在环境应用中。该研究将开发一种全新的分布式估计器来处理由网络传感器收集的测量数据。一个重要的问题是测量的间歇性可用性，因为传感器之间的底层通信基础设施可能会出现随机故障（例如，由于交换网络中的数据包丢失）或可能通过随机协议（八卦或变化）进行通信。由于这种间歇性，估计器方程是随机的，滤波器方程在两种行为模式之间随机切换；特别是，滤波器Riccati方程在线性代数矩阵方程（Lyapounov）和二次代数矩阵方程之间随机切换。本研究开发了一种合适的分布式估计算法，确定了在何种结构条件和何种测量采集速率下滤波误差保持有界，旨在确定滤波误差无界增长的概率。为了研究这些问题，这项工作从随机动力系统理论和（中等）大偏差原理中发展出新的、可能具有变革性的、强大的方法。该工作将定义滤波器随机协方差序列收敛的适当条件，确定其极限不变分布，定义适当的罕见事件，并发展中等偏差原则，以确定罕见事件概率消失的适当最佳速率。

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.