Collaborative Research: Nonlinear Balancing: Reduced Models and Control

实时控制和工程设计需要对复杂工程系统进行快速、准确的计算机仿真。这笔赠款将支持研究，以促进非线性系统的平衡截断模型简化，这是一个产生可靠、准确和计算效率高的仿真器的数学框架。尽管在20世纪90年代已经奠定了理论基础，但到目前为止，还缺乏能够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使用现代高性能算法来克服这一障碍，这些算法利用了必须求解的方程的数学结构。例如，由此产生的模拟器将通过准确地实时估计卫星的大气阻力来改进卫星的控制和运行；通过允许大量设计迭代的低资源计算模型来推进飞机的设计；通过高效地模拟水流和水质来优化城市的供水网络，以便对泵站进行最佳调度。这将为社会带来更大的好处，改善民用基础设施，并有助于提高美国的工业竞争力。这笔赠款还将通过弗吉尼亚理工大学针对早期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。到目前为止，这个框架还没有应用于高维非线性系统的模型降阶，因为求解作为平衡法核心的Hamilton-Jacobi-Bellman(HJB)方程对于大系统来说仍然是不可行的。张量演算、非线性状态变换和多项式反馈定律的最新发展使这个问题的解决变得可行。该项目将开发一种基于可伸缩张量的方法来求解HJB方程，以获得平衡截断所需的能量函数的多项式展开，以及用于分析张化问题的条件的高性能算法和数值分析。此外，通过利用参数空间中的结构，将设计出有效的参数非线性平衡算法。此外，降阶非线性控制器的设计将采用同时降阶和控制的框架，这远远优于现有的先降阶再控制的框架。该项目还将开发一种关于这些控制器的健壮性及其应用于高维系统时的稳定特性的理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.