Numerical Linear Algebra and Control

本课题研究光滑矩阵值函数的存在唯一性及光滑变奇异值分解的计算方法。光滑奇异值分解在微分代数方程中有广泛的应用，包括广义系统的时变最优控制问题。用实解析函数进行的初步工作将扩展到更大的光滑函数类别，并且正在研究更有效的跟踪全部或部分光滑变化奇异值分解的数值方法。该项目还开发了计算可控性半径、规则半径和计算控制中感兴趣的其他距离的数值方法。它正在开发高效、可靠、数值稳定的方法。这里的成功将推广到一种估计从一般矩阵铅笔到非一般铅笔的代数变化和一般条件估计的距离的方法。

This project studies existence, uniqueness and methods for calculating a smoothly varying singular value decomposition of a smooth matrix valued function. The smooth singular value decomposition has applications in differential algebraic equations including time dependent optimal control problems for descriptor systems. Preliminary work with real analytic functions is to be extended to a larger class of smooth functions and more efficient numerical methods for tracking all of or part of a smoothly varying singular value decomposition are being investigated. This project is also developing numerical methods for calculating the controllability radius, regularity radius, and other distances of interest in computational control. It is developing efficient, provably reliable, numerically stable methods. Success here will generalize to a means of estimating the distance from a generic matrix pencil to an algebraic variety of nongeneric pencils and to general condition estimators.