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

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)资助的。系统和控制的许多未来应用将与网络上的多个代理(传感器和执行器)的(可能)分布式估计和控制问题有关。例如分布式传感器网络、分布式自主代理的控制、冲突避免、分布式电力系统等。研究此类系统的核心是研究随机Lyapunov和Riccati递归的行为(类似于传统的LTI系统，其中确定性Lyapunov和Riccati递归和方程扮演着重要的角色)。不幸的是，到目前为止，用于分析这类系统的工具严重缺乏，表面上是因为递归既是非线性的，也是随机的，因此如果想要准确地分析它们，就很难处理。本文提出的方法是利用大型随机矩阵理论中的工具，当系统的状态数n较大时，求出随机Riccati递推中矩阵的渐近特征分布。在许多情况下，特征分布包含关于系统整体行为的足够信息。从特征分析中可以推断出稳定性。特征值的平均值简单地与迹的平均值(即系统的均方误差)有关，而特征分布的支持集反映了系统的最佳和最差情况的性能。此外，这种方法的一般原理是识别和展示系统的通用行为，前提是这种行为确实存在。在这里，“普遍”指的是不依赖于系统的微观细节(损耗发生在哪里、网络的确切拓扑结构或基本分布是什么)的行为，而是取决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)取代高维矩阵值的非线性和随机递归，更易于分析和计算。该项目将包括课程开发和招募女性和少数民族学生进行研究。它还将通过加州理工大学的SURF和MURF计划利用本科生和未被充分代表的少数族裔学生研究人员。

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.