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。许多未来的系统和控制应用将涉及到网络物理系统是那些与（可能）分布式估计和控制问题的多个代理（传感器和执行器）在网络上。例子包括领域，如分布式传感器网络，控制分布式自治代理，避免碰撞，分布式电力系统等。中央对这些系统的研究是随机李雅普诺夫和Riccati递归的行为的研究（类比是传统的LTI系统，其中确定性李雅普诺夫和Riccati递归和方程发挥了突出的作用）。不幸的是，到目前为止，用于分析这类系统的工具非常缺乏，表面上是因为递归是非线性和随机的，因此如果想要精确地分析它们，就很难。 在这项工作中提出的方法是利用大随机矩阵理论的工具，找到随机Riccati递归矩阵的渐近本征分布时，在系统中的状态数，n，是大的。在许多情况下，本征分布包含了关于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。特征值的平均值简单地与迹的平均值相关（即，系统的均方误差），而本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和展示系统的普遍行为，只要这种行为确实存在。 在这里，“普适”意味着行为不依赖于系统的微观细节（损失发生在哪里，网络的确切拓扑结构或底层分布是什么），而是依赖于一些简单的宏观性质。 该方法的一个主要思想是用标量值确定性函数递归（涉及本征分布的适当变换）取代高维矩阵值非线性和随机递归，后者更易于分析和计算，该项目将包括课程开发和招募妇女和少数民族学生进行研究。 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民族学生研究人员。

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.