Spectral Value Sets: Theory, Algorithms and Applications

在具有不确定反馈的线性动态系统的建模中，会出现谱值集。它们之所以重要，是因为它们模拟了反馈中固有的不确定性，这种不确定性被认为是线性依赖于输出的。它们由一个参数E来参数化，该参数限定了所讨论的不确定性的范数。该项目的理论方面包括分析给定值E的谱值集的特定极值点的性质，以及E的临界值的谱值集分量在局部几何和代数度量方面的合并点的性质。算法方面包括快速方法的开发和分析，以计算(1)固定E的给定谱值集合上的实部或模的极大值，以及(2)被定义为E的最大值的复稳定性半径(或其倒数，H-无穷大范数)，使得相关的谱值集合位于稳定区域(左半平面或单位圆盘)内。它们还包括开发方法来设计开环对象的控制器，从而使闭环系统具有期望的稳定性和最优性，例如局部最大化稳定性半径(最小化H无穷范数)。由于这些函数不是凹的或凸的，而且它们的优化器通常位于它们不可微的点上，因此需要非光滑、非凸极小化的方法，包括能够有效地处理约束的方法。谱值集的数学性质及其极值计算的相关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谱值集合和相关问题的算法工具带给广泛的科学家和工程师社区，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、基于观察者的故障检测和微创手术。所有这些系统都需要控制器有效工作：飞机或发电厂等复杂系统除了需要知道如何使用此类系统的熟练操作员外，还需要自动控制器安全有效地运行。然而，目前的方法仅限于中小型系统，不能准确地对真实的物理系统进行建模。新方法将允许为比以前可能的更大的系统设计控制器，包括对偏微分方程组的离散系统的控制，这些系统在自然科学和工程中都有应用。

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.