CIF: Small: Gossiping, Intermittency, and Kalman Filtering

这项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，电力网或大型运输系统，或者当监测配备有自组织传感器网络的大型物理空间时，如在环境应用中。本研究将开发一种全新的分布式估计器来处理网络化传感器收集的测量数据。一个重要的问题是测量的间歇可用性，因为传感器之间的底层通信基础设施可能表现出随机故障（例如，由于交换网络中的分组丢失），或者可以通过随机协议（流言或变体）进行通信。由于这种不确定性，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。本研究发展一个适当的分布式估计算法，建立在什么样的结构条件下，什么样的测量收集率的过滤误差保持有界，并旨在确定的概率过滤误差增长无界。为了研究这些问题，这项工作开发了新的，潜在的变革，强大的方法，从随机动力系统理论和（适度）大偏差原则。这项工作将定义适当的条件下，过滤器的随机协方差序列收敛，确定其限制不变分布，定义适当的罕见事件，并制定一个适度的偏差原则，确定适当的最佳率在罕见事件的概率消失。

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.