Interpolatory Model Reduction for the Control of Fluids

流体流量控制问题无处不在。它们出现在重要的应用中，如减阻(具有节省燃料或提高航程/速度的好处)、增强混合(以提高燃烧效率)、减少结构疲劳、改进凝固和压铸，以及在大型室内空气环境中进行高效冷却。然而，对提高基本流动问题的精度和复杂性的不断增长的需求导致了超大规模的动力系统，其模拟和控制对计算资源的需求是巨大的和不可管理的。该研究项目旨在开发新的计算技术和新的严格的数学框架来非常有效地解决大规模的流量控制问题。此外，该项目还将开发一门为期一年的关于模型简化和流量控制的研究生课程，并将为学生提供有价值的跨学科教育。目前的最新技术是通过使用适当的正交分解构建的简化模型来解决流量控制问题。然而，这些模型的局限性在于，它们只能保证对预先选定的输入范围准确。该项目将为大规模动力系统的有效分析和近似提供显著改进的工具。它还将直接应用于双线性和二次型非线性问题的模型降阶。双线性模型存在于换热器的控制问题中，具有二次非线性的非线性偏微分方程组包括Korteweg-de Vries(浅波)、Kuramoto-Sivashinsky(湍流火焰)和Landau-Lifshitz(固体物理中的磁场)方程。利用有理插值法，这项研究将产生新的算法，系统地对与(离散化的)流动方程相关的线性和非线性系统进行高保真的、在大多数情况下是最优的模型降阶。这些简化的模型将被用于设计最优反馈律。新的框架将提供主要的优势：首先，与目前的方法不同，所提出的控制设计将不需要针对特定输入轨迹或大型稠密矩阵方程的解进行昂贵的全阶、时间精确的仿真；计算努力在于计算稳态解和求解少量稀疏线性系统。其次，简化后的模型对大范围的输入曲线都是一致准确的，不依赖于特定的输入轨迹。第三，该方法将自然地创建尊重原始流的稳定性属性的简化模型。

Fluid flow control problems are ubiquitous. They arise in important applications such as drag reduction (with the benefits of saving fuel or improving range/speed), enhancing mixing (for more efficient combustion), reducing structural fatigue, improved solidification and die casting, and efficient cooling in large indoor-air environments. However, the ever-increasing need for improved accuracy and complexity of the underlying flow problems lead to very large-scale dynamical systems whose simulations and control make overwhelming and unmanageable demands on computational resources. This research project aims to develop novel computational techniques and a new rigorous mathematical framework to solve large-scale flow control problems very efficiently. In addition, the project will develop a year-long graduate course on Model Reduction and Flow Control and will provide students with valuable interdisciplinary education.The current state-of-the-art is to solve flow control problems by using reduced models constructed using the proper orthogonal decomposition. However, these models are limited in that they are only guaranteed to be accurate for a pre-selected range of inputs. This project will provide significantly improved tools for the efficient analysis and approximation of large-scale dynamical systems. It will also have direct application to model reduction for problems with bilinear and quadratic nonlinearities. Bilinear models arise in control problems for heat exchangers, and nonlinear partial differential equations with quadratic nonlinearities include the Korteweg-de Vries (shallow waves), the Kuramoto-Sivashinsky (turbulent flames), and the Landau-Lifshitz (magnetic fields in solid state physics) equations. Using rational interpolation, this research will lead to new algorithms to systematically perform high-fidelity, in most cases optimal, model reduction for linear and nonlinear systems associated with (discretized) flow equations. These reduced models will be used to design optimal feedback laws. The new framework will offer major advantages: First, unlike current approaches, the proposed control design will not require expensive full-order, time-accurate simulations for specific input trajectories or solutions of large dense matrix equations; the computational efforts lie in computing a steady-state solution and solving a modest number of sparse linear systems. Second, the reduced models will be uniformly accurate for a wide range of input profiles and will not depend on specific input trajectories. Third, the methodology will naturally create reduced models that respect the stability properties of the original flow.