Collaborative Research: Nonlinear Balancing: Reduced Models and Control

复杂工程系统的快速、准确的计算机仿真是实时控制和工程设计的需要。这笔拨款将支持将推进非线性系统平衡截断模型缩减的研究，这是一个数学框架，可以产生可靠、准确和计算效率高的模拟器。尽管在20世纪90年代就已经奠定了理论基础，但到目前为止，还缺乏适用于当今复杂工程系统所需的高维计算实现。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，这些算法利用了必须解决的方程的数学结构。例如，由此产生的模拟器将通过对大气卫星阻力的精确实时估计来促进卫星的控制和操作；通过允许大量设计迭代的低资源计算模型推进飞机设计；并通过有效地模拟水流和水质来优化我们城市的供水网络，从而优化泵站的调度。这将为社会带来更大的利益，改善民用基础设施，并有助于提高美国的工业竞争力。这笔拨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，以及通过本科生研究机会，支持科学、技术、工程和数学（STEM）劳动力培训。本研究旨在通过非线性平衡截断的概念，为复杂的高维多项式非线性系统开发一类新的降阶模型和控制器。迄今为止，该框架尚未应用于高维非线性系统的模型约简，因为求解平衡方法的核心Hamilton-Jacobi-Bellman （HJB）方程对于大型系统仍然是不可行的。最近在张量演算、非线性状态变换和多项式反馈律方面的发展使得解决这个问题变得可行。该项目将开发一种基于可扩展张量的方法来解决HJB方程，以获得平衡截断所需的能量函数的多项式展开式，以及高性能算法和数值分析来分析张量化问题的条件。利用参数空间中的结构，设计有效的参数非线性平衡算法。此外，将使用同时约简和控制框架设计降阶非线性控制器，这远远优于现有的先约简后控制框架。该项目还将开发这些控制器的鲁棒性理论，以及它们在应用于高维系统时的稳定特性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.