Collaborative Research: Nonlinear Balancing: Reduced Models and Control

合作研究：非线性平衡：简化模型和控制

• 批准号: 2130727
• 负责人: Boris Kramer
• 金额: $ 35.91万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-01-01 至 2024-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2130727&HistoricalAwards=false
• 关键词: Collaborative; Research; Nonlinear; Balancing; Reduced

Fast and accurate computer simulation of complex engineering systems is required for real-time control and engineering design. This grant will support research that will advance balanced truncation model reduction for nonlinear systems, a mathematical framework to produce reliable, accurate, and computationally efficient simulators. Despite the theoretical foundations having been laid in the 1990s, computational implementations that scale to the high dimensionality needed for today’s complex engineering systems are lacking to date. This research will overcome this barrier by developing and employing modern high-performance algorithms that exploit the mathematical structure of the equations that have to be solved. The resulting simulators will, for instance, advance the control and operation of satellites through accurate real-time estimation of atmospheric satellite drag; advance the design of aircraft through low-resource computational models that allow for a large number of design iterations; and optimize our cities’ water networks through efficiently simulating water flows and water quality so that pump stations can be scheduled optimally. This will result in greater benefits to society, improvements of civil infrastructure, and contribute to the industrial competitiveness of the United States. This grant will also support science, technology, engineering and mathematics (STEM) workforce training through a workshop at Virginia Tech that targets early-career researchers, as well as through undergraduate research opportunities.This research seeks to develop a new class of reduced-order models and controllers for complex high-dimensional polynomial nonlinear systems via the concept of nonlinear balanced truncation. To date, this framework has not been applied to model reduction for high-dimensional nonlinear systems since solving the Hamilton-Jacobi-Bellman (HJB) equations, which are at the core of the balancing approach, remained infeasible for large-scale systems. Very recent developments in tensor calculus, nonlinear state transformations, and polynomial feedback laws now make the solution to this problem feasible. This project will develop a scalable tensor-based approach to solve the HJB equations to obtain polynomial expansions of the energy functions required for balanced truncation, as well as high-performance algorithms and numerical analysis to analyze the conditioning of the tensorized problems. Moreover, efficient algorithms for parametric nonlinear balancing will be designed by exploiting the structure in parameter space. Additionally, reduced-order nonlinear controllers will be designed using a simultaneous reduction and control framework, which is far superior to the existing reduce-then-control framework. The project will also develop a theory for the robustness of these controllers, and their stabilizing properties when applied to the high-dimensional systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

实时控制和工程设计需要对复杂工程系统进行快速、准确的计算机模拟。这笔拨款将支持推进非线性系统平衡截断模型简化的研究，这是一种产生可靠、准确和计算高效的模拟器的数学框架。尽管理论基础已在 20 世纪 90 年代奠定，但迄今为止仍缺乏可扩展到当今复杂工程系统所需的高维的计算实现。这项研究将通过开发和采用利用现代高性能算法来克服这一障碍。例如，由此产生的模拟器将通过准确实时估计大气卫星阻力来推进卫星的控制和操作；通过低资源计算模型推进飞机的设计。进行大量的设计迭代；并通过有效模拟水流和水质来优化我们的城市供水网络，以便优化规划泵站，这将为社会带来更大的利益，改善民用基础设施，并为人类健康做出贡献。美国的产业竞争力。这笔赠款还将通过弗吉尼亚理工大学针对早期职业研究人员的研讨会以及本科生研究机会来支持科学、技术、工程和数学 (STEM) 劳动力培训。这项研究旨在开发一类新的降阶通过非线性平衡截断的概念构建复杂高维多项式非线性系统的模型和控制器迄今为止，自从求解 Hamilton-Jacobi-Bellman (HJB) 方程以来，该框架尚未应用于高维非线性系统的模型简化。作为平衡方法的核心，对于大规模系统来说仍然不可行。张量微积分、非线性状态变换和多项式反馈定律的最新发展现在使得该问题的解决方案变得可行。该项目将开发一种可扩展的张量。基于方法来求解 HJB 方程以获得平衡截断所需的能量函数的多项式展开，以及高性能算法和数值分析来分析张量化问题的条件。参数非线性平衡将通过利用参数空间中的结构来设计，此外，降阶非线性控制器将使用同时降阶和控制框架来设计，该框架也将远远优于现有的降阶控制框架。开发一种关于这些控制器的鲁棒性及其应用于高维系统时的稳定性能的理论。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。