Spectral Value Sets: Theory, Algorithms and Applications

谱值集：理论、算法和应用

• 批准号: 1317205
• 负责人: Michael Overton
• 金额: $ 43.17万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1317205&HistoricalAwards=false
• 关键词: Spectral; Value; Sets; Theory; Algorithms

Spectral value sets arise in the modeling of linear dynamical systems with uncertain feedback. They are important because they model uncertainty inherent in the feedback which is assumed to depend linearly on the output. They are parametrized by a parameter E which bounds the norm of the uncertainty in question. Theoretical aspects of the project include analysis of properties of specific extremal points of spectral value sets for a given value of E and at points of coalescence of the spectral value set components for critical values of E, both in terms of local geometry and algebraic measures. Algorithmic aspects include the development and analysis of fast methods to compute (1) maximizers of the real part or modulus over a given spectral value set for fixed E and (2) the complex stability radius (or its reciprocal, the H-infinity norm) defined as the largest value of E such that the associated spectral value set lies inside the stability region (the left half-plane or the unit disk). They also include developing methods to design controllers for open-loop plants that result in closed-loop systems with desired stability and optimality properties, such as locally maximizing the stability radius (minimizing the H-infinity norm). Since these functions are not concave or convex and their optimizers are typically at points where they are not differentiable, methods for nonsmooth, nonconvex minimization are needed, including methods that can handle constraints efficiently.The mathematical properties of spectral value sets and associated algorithms to compute their extremal values have both theoretical and practical importance. The broader goal of the project is to bring the tools of algorithms for optimization over spectral value sets and related problems to a wide community of scientists and engineers, for use in many different kinds of applications. The investigator's open-source software is already in use in a variety of applications, including the design of aircraft controllers, a proton exchange membrane fuel cell system, power systems, observer-based fault detection and minimally invasive surgery. All of these systems require controllers to work effectively: a complex system such as an airplane or a power plant requires automatic controllers to function safely and effectively, in addition to skilled operators who know how to use such systems. However, current methods are limited to small or moderate-sized systems, which cannot model real physical systems accurately. The new methods will allow the design of controllers for much larger systems than was previously possible, including control of discretized systems of partial differential equations, which have applications throughout the natural sciences and engineering.

谱值集是在不确定反馈线性动力系统建模中出现的。 它们很重要，因为它们模拟了反馈中固有的不确定性，假设反馈线性依赖于输出。 它们由参数E参数化，该参数E限制了所讨论的不确定性的范数。该项目的理论方面包括分析特定极值点的光谱值集的属性为一个给定的E值和在点的合并的光谱值集组件的临界值E，无论是在局部几何和代数措施。数学方面包括快速方法的开发和分析，以计算（1）固定E的给定谱值集上的真实的部分或模的最大化者，以及（2）定义为E的最大值的复稳定半径（或其倒数，H-无穷范数），使得相关谱值集位于稳定区域（左半平面或单位圆盘）内。 它们还包括开发方法来设计开环工厂的控制器，从而使闭环系统具有所需的稳定性和最优性，例如局部最大化稳定半径（最小化H ∞范数）。由于这些函数既不是凹的也不是凸的，并且它们的优化器通常在它们不可微的点上，因此需要非光滑、非凸的最小化方法，包括能够有效处理约束的方法。谱值集的数学性质及其极值计算算法具有重要的理论和实际意义。该项目更广泛的目标是将优化光谱值集和相关问题的算法工具带给广大的科学家和工程师，用于许多不同类型的应用。 研究人员的开源软件已经在各种应用中使用，包括飞机控制器的设计，质子交换膜燃料电池系统，电力系统，基于神经网络的故障检测和微创手术。所有这些系统都需要控制器有效地工作：一个复杂的系统，如飞机或发电厂，除了需要熟练的操作员知道如何使用这些系统外，还需要自动控制器安全有效地工作。 然而，目前的方法仅限于小型或中型的系统，不能准确地模拟真实的物理系统。 新方法将允许设计比以前可能的大得多的系统的控制器，包括偏微分方程的离散系统的控制，其在整个自然科学和工程中具有应用。