Optimization of Pseudospectra

伪谱的优化

• 批准号: 0412049
• 负责人: Michael Overton
• 金额: $ 38.43万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-15 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0412049&HistoricalAwards=false
• 关键词: Optimization; Pseudospectra

Optimization problems involving eigenvalues arise in manyapplications. In recent years, attention has focused onsemidefinite programs, which are linear optimization problems inthe space of real symmetric matrices, with positive semidefiniteconstraints. This project focuses on optimization problems in thelarger space of square matrices, not generally symmetric. In thiscontext, it is well known that eigenvalues, which provideinformation about asymptotic behavior of dynamical systems, arenot as useful as pseudospectra, which provide a more robustmeasure of system behavior. This project focuses on thecomputation, analysis and, especially, numerical optimization ofpseudospectral functions. A secondary goal is to advance thedevelopment of algorithms for general nonsmooth, nonconvexoptimization problems.Applications of this work arise in many contexts. A good examplethat we have used is a model of a Boeing 767 at a flutter condition.Flutter occurs when the plane flies so fast that the interactionof the aerodynamic (wind) forces with the structural forces inthe airplane combine to generate an instability which could becatastrophic. Normally, a plane like this does not fly so fast butit is important to know what would happen if it did, and whether asimple controller could bring the plane under control. Eigenvalues tell us about stability in the long term, but pseudospectra are far more informative, telling us about stability in the short term. Optimizing the pseudospectra over the parameters of the controller may result in the design of a controller that is simple yet effective in avoiding the instability and preventing catastrophe.

许多应用中都会出现涉及特征值的优化问题。近年来，人们的注意力集中在半定规划上，它是实对称矩阵空间中具有正半定约束的线性优化问题。 该项目重点关注较大方阵空间中的优化问题，而不是一般对称的方阵。在这种情况下，众所周知，提供有关动力系统渐近行为信息的特征值不如伪谱有用，后者提供了对系统行为的更稳健的测量。该项目重点关注伪谱函数的计算、分析，特别是数值优化。第二个目标是推进一般非光滑、非凸优化问题的算法的开发。这项工作的应用出现在许多情况下。 我们使用的一个很好的例子是颤振条件下的波音 767 模型。当飞机飞得如此快以至于空气动力（风）力与飞机中的结构力的相互作用结合起来产生可能是灾难性的不稳定性时，就会发生颤振。 通常情况下，这样的飞机不会飞得那么快，但重要的是要知道如果飞得那么快会发生什么，以及一个简单的控制器是否可以控制飞机。 特征值告诉我们长期的稳定性，但伪谱的信息量要大得多，告诉我们短期的稳定性。 根据控制器的参数优化伪谱可以导致控制器的设计简单但有效地避免不稳定性和防止灾难。