Research Initiation Award: A Unified Framework for Optimal Synthesis in Linear Systems

这项研究的目的是完成一种统一的控制分析和设计方法，包括一维(一个自变量，通常是时间)和n-d(n个自变量)系统，用于降阶控制器，以实现线性、不确定对象模型的鲁棒性能。为了得到不过于保守的n-d系统的综合方法，我们还将开发改进的分析技术(验证稳定性、计算范数等)。对于n-d系统。这种方法意义重大，因为它将用一个统一的理论将当前控制研究的许多不同领域联系在一起。这将导致算法增加控制工程师在初始系统配置中的作用，因为组件质量和组件建模中的成本/收益权衡可以事先分析。整个方法是基于矩阵的线性分式变换和一个研究它们的数学框架，称为结构奇异值。结构奇异值是分析矩阵摄动问题的一种数学工具。

The objective of this research is to complete a unified control analysis and design methodology, encompassing both 1-d (one independent variable, usually time) and n-d (n independent variables) systems, for reduced order controllers that achieve robust performance for linear, uncertain plant models. In order to derive synthesis methods for n-d systems which are not overly conservative, we will also develop improved analysis techniques (verifying stability, computing norms, etc.) for n-d systems. This approach is significant, because it will tie together many different areas of current control research with a unified theory. It will lead to algorithms that will increase the role that a control engineer can have in initial system configuration, since the cost/benefit tradeoffs in component quality and in component modelling can be analyzed apriori. The entire methodology is based on linear fractional transformations of matrices and a mathematical framework to study them called the structured singular value. The structured singular value is a mathematical tool useful in analyzing matrix perturbation problems.