CPS: Small: Random Matrix Recursions and Estimation and Control over Lossy Networks

CPS：小：随机矩阵递归以及有损网络的估计和控制

• 批准号: 0932428
• 负责人: Babak Hassibi
• 金额: $ 50.81万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0932428&HistoricalAwards=false
• 关键词: CPS; Small; Random; Matrix; Recursions

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Many of the future applications of systems and control that will pertain to cyber-physical systems are those related to problems of (possibly) distributed estimation and control of multiple agents (both sensors and actuators) over networks. Examples include areas such as distributed sensor networks, control of distributed autonomous agents, collision avoidance, distributed power systems, etc. Central to the study of such systems is the study of the behavior of random Lyapunov and Riccati recursions (the analogy is to traditional LTI systems where deterministic Lyapunov and Riccati recursions and equations play a prominent role). Unfortunately, to date, the tools for analyzing such systems are woefully lacking, ostensibly because the recursions are both nonlinear and random, and hence intractable if one wants to analyze them exactly. The methodology proposed in this work is to exploit tools from the theory of large random matrices to find the asymptotic eigendistribution of the matrices in the random Riccati recursions when the number of states in the system, n, is large. In many cases, the eigendistribution contains sufficient information about the overall behavior of the system. Stability can be inferred from the eigenanalysis. The mean of the eigenvalues is simply related to the mean of the trace (i.e., the mean-square-error of the system), whereas the support set of the eigendistribution says something about best- and worst-case performances of the system. Furthermore, a general philosophy of this approach is to identify and exhibit the universal behavior of the system, provided such a behavior does exist. Here, "universal" means behavior that does not depend on the microscopic details of the system (where losses occur, what the exact topology of the network or underlying distributions are), but rather on some simple macroscopic properties. A main idea of the approach is to replace a high-dimensional matrix-valued nonlinear and stochastic recursion by a scalar-valued deterministic functional recursion (involving an appropriate transform of the eigendistribution), which is much more amenable to analysis and computation.The project will include course development and the recruitment of women and minority students to research. It will also make use of undergraduateand underrepresented minority student researchers through Caltech's SURF and MURF programs.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。与网络物理系统相关的系统和控制的许多未来应用都与（可能）分布式估计和问题相关。通过网络控制多个代理（传感器和执行器）。例子包括分布式传感器网络、分布式自治代理的控制、碰撞避免、分布式电力系统等领域。此类系统研究的核心是研究随机 Lyapunov 和 Riccati 递归的行为（类似于传统的 LTI）确定性 Lyapunov 和 Riccati 递归和方程发挥重要作用的系统）。不幸的是，迄今为止，用于分析此类系统的工具严重缺乏，表面上是因为递归既是非线性的又是随机的，因此如果想要准确地分析它们就很棘手。 这项工作提出的方法是利用大型随机矩阵理论中的工具来查找当系统中的状态数 n 很大时随机 Riccati 递归中矩阵的渐近特征分布。在许多情况下，特征分布包含有关系统整体行为的足够信息。稳定性可以从特征分析中推断出来。特征值的平均值仅与迹线的平均值（即系统的均方误差）相关，而特征分布的支持集说明了系统的最佳和最差情况性能。 此外，这种方法的一般原理是识别并展示系统的普遍行为，前提是这种行为确实存在。 在这里，“通用”意味着行为不依赖于系统的微观细节（损失发生在哪里，网络的确切拓扑或底层分布是什么），而是依赖于一些简单的宏观属性。 该方法的主要思想是用标量值确定性函数递归（涉及特征分布的适当变换）代替高维矩阵值非线性和随机递归，这更易于分析和计算。将包括课程开发以及招募女性和少数族裔学生进行研究。 它还将通过加州理工学院的 SURF 和 MURF 项目利用本科生和代表性不足的少数族裔学生研究人员。