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快速准确的计算机模拟。该赠款将支持将降低非线性系统的平衡截断模型的研究，这是一个数学框架，以产生可靠，准确和计算上有效的模拟器。尽管在1990年代奠定了理论基础，但迄今为止缺乏扩展到当今复杂工程系统所需的高维度的计算实现。这项研究将通过开发和采用现代高性能算法来克服这一障碍，从而利用必须解决的方程式的数学结构。例如，由此产生的模拟器将通过准确的大气卫星阻力实时估算来推动卫星的控制和操作；通过低资源的计算模型推进飞机的设计，该模型允许大量设计迭代；并通过有效模拟水流和水质来优化我们的城市水网络，以便最佳地安排泵站。这将为社会带来更大的利益，改善民用基础设施，并为美国的工业竞争力做出贡献。该赠款还将通过弗吉尼亚理工大学的研讨会来支持科学，技术，工程和数学（STEM）劳动力培训，该研讨会针对早期研究人员，以及通过本科研究机会。本研究旨在通过非线化概念的概念来开发一种新的降低级阶模型，并为复杂的高维非利用系统的复杂高维非线性系统开发。迄今为止，自从解决汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程以来，该框架尚未应用于模型的降低，这些方程是平衡方法的核心，对于大型系统而言，它仍然是不可避免的。最近，张量计算，非线性状态转换和多项式反馈定律的发展使得解决此问题可行。该项目将开发一种基于可扩展的张量的方法来求解HJB方程，以获得平衡截断所需的能量函数的多项式扩展，以及高性能算法和数值分析，以分析张开问题的条件。此外，将通过在参数空间中利用结构来设计参数非线性平衡的有效算法。此外，还将使用简单的还原框架和控制框架设计降低的非线性控制器，该框架远远超过了现有的减少控制框架。该项目还将为这些控制器的鲁棒性开发理论，并在应用于高维系统时稳定属性。本奖反映了NSF的坚定任务，并通过基金会的智力优点和更广泛的影响来评估NSF的坚定任务，并被认为值得通过评估。