Efficient Algorithms for Optimal Control of Time-Periodic and Nonlinear Systems

流体经常表现出周期性运动，要么是由于外部周期性力量(例如，月球潮汐)，要么是由于与环境相互作用而自然产生的内力(例如，通过轻微打开的车窗的空气抖动，或者由于河流水流与桥柱相互作用而形成的涡流)。对这种流动的控制非常重要，因为在许多情况下，流动概况的微小变化要么会产生巨大的好处，要么会产生灾难性的代价。例如，将循环流型稳定并调整到给定的频率在风电场和波能转换器的设计中很有用，而在其他情况下，完全消除振荡运动可能有助于减少施加在关键支撑结构上的疲劳载荷。该项目通过开发新的模拟和控制数学算法，将潜在的周期性系统行为整合到核心建模框架中，专门解决振荡现象的建模和控制问题，从而有望提高精度和降低计算成本。这项研究将使人们更好地理解具有周期性行为的系统的数学模型，并开发新的模拟和建模工具，这将直接影响到生物学(例如循环系统和呼吸系统)和能源(例如风力涡轮机和电网动力学)的广泛应用。长期以来，弗洛奎变换一直是解决这类问题的理论工具，它允许改变系统表示，有效地将周期性的时间依赖从内部动力学转移到输入/输出端口。直到最近，使用这种方法来解决中小型问题才是可行的。这项研究的最初重点将是开发可伸缩的、数值有效的大规模Floquet变换算法，使时变周期流动动力学的高保真简化模型成为可能。通过将算子分裂方法与二次系统的最优模型降阶方法相结合，将开发和分析与输入无关的非线性动力学模型降阶的新能力。本文提出的模型降阶框架以适当的代价提供了更高的数值效率和更高的精度。这种方法的基础是将动力学的精确表示集成到降阶模型中，更好地尊重基本最优控制问题的性质。将开发强大的计算工具，帮助模拟和模拟大范围的振荡动力学，并提供给科学和工程界。这一奖项反映了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。

Fluids often exhibit cyclical motion, either due to external periodic forces (e.g., lunar tides) or due to internal forces that naturally arise through interaction with the environment (e.g., air buffeting through a slightly opened car window or the eddies that develop from a river current interacting with a bridge pillar). The control of such flows is of great interest since in many cases small changes in a flow profile can produce either dramatic benefits or catastrophic costs. Stabilizing and tuning cyclic flow patterns to a given frequency can be useful in the design of wind farms and wave energy converters, for example, while in other circumstances eliminating oscillatory motion altogether may help reduce fatigue loads placed on critical support structures. This project specifically addresses the modeling and control of oscillatory phenomena through the development of new mathematical algorithms for simulation and control that integrate underlying periodic system behavior into the core modeling framework, thus promising both improved accuracy and reduced computational costs. This research will result in improved understanding of mathematical models of systems with periodic behavior as well as the development of new simulation and modeling tools, which will have an immediate bearing on a wide range of applications found in biology (e.g., circulatory and respiratory systems) and energy (e.g., wind turbines and power grid dynamics).Simulation and control of periodic flow structures require methods specialized to the task. The Floquet transformation has long been a theoretical tool for such problems, allowing for a change in system representation that effectively shifts the periodic time dependence out of the internal dynamics into the input/output ports. Only recently has it been practical to use this approach for small to medium scale problems. The initial focus of this research will be on the development of scalable, numerically effective algorithms for large-scale Floquet transformations, enabling the high-fidelity reduced models for time-varying periodic flow dynamics. By combining operator splitting approaches with optimal model reduction methods for quadratic systems, new capabilities for input-independent model reduction for nonlinear dynamics will be developed and analyzed. The model reduction framework that is proposed here offers improved numerical efficiency and greater accuracy at modest cost. Fundamental to this approach will be the integration of an accurate representation of dynamics into reduced-order models, better respecting the properties of the underlying optimal control problem. Robust computational tools aiding the simulation and modeling of large-scale oscillatory dynamics will be developed and provided to the science and engineering community.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.