Convergent Data-Based Algorithms for Model Reduction and Feedback Control of PDEs

在复杂的应用中，如节能建筑物的设计和反馈控制，以及波浪或风能阵列，现有的代码通常可用于仿真，但这些代码并不总是适合于模型简化或反馈控制设计等目的。本研究的成果是一套计算效率和可证明收敛的算法，在PDE模型降阶和反馈控制的大挑战问题。 所提出的工作集中在两个基本问题类：（1）计算用于反馈控制设计的无限维李雅普诺夫和黎卡提方程的近似解，和（2）PDE平衡模型降阶算法。这项工作利用具体的解决方案数据从PDE系统产生可证明收敛的逼近PDE模型的减少和反馈控制问题。由于算法是基于数据的，因此该方法可以使用具有专门的离散化方案和工具（诸如自适应求解器、并行算法和多重网格方法）的现有模拟代码来提高计算效率和准确性。由于算法可以利用这种先进的模拟技术，算法可以扩展到处理高度复杂的大挑战问题。 偏微分方程系统模型降阶问题的精确解对于快速仿真、易处理的优化、反馈控制设计和真实的控制实现等许多应用都具有重要意义。偏微分方程的反馈控制在许多领域都有潜在的应用。 例如，商业建筑中的气流反馈控制可以显著降低能耗。在另一个领域，波能转换器中流体-结构相互作用的反馈控制可以导致有效的海洋波能提取。所提出的工作主要考虑线性PDE系统的模型降阶和反馈控制算法的开发和分析，这些降阶模型和反馈律的扩展可用于处理涉及非线性PDE的问题。该算法可以利用先进的现有的仿真代码，因此，所提出的工作的完成将打开大门，在计算反馈控制具有挑战性的非线性PDE系统的大的进步。

In complex applications such as design and feedback control of energy efficient buildings, and wave or wind energy arrays, existing codes are often available for simulation but these codes are not always suitable for purposes such as model reduction or feedback control design. The outcome of this research is a suite of computationally efficient and provably convergent algorithms for grand challenge problems in PDE model reduction and feedback control. The proposed work focuses on two fundamental problem classes: (1) computing approximate solutions of infinite dimensional Lyapunov and Riccati equations used in feedback control design, and (2) algorithms for PDE balanced model reduction. This work utilizes specific solution data from PDE systems to produce provably convergent approximations for PDE model reduction and feedback control problems. Since the algorithms are data-based, the methods may use existing simulation codes with specialized discretization schemes and tools such as adaptive solvers, parallel algorithms, and multigrid methods to increase computational efficiency and accuracy. Since the algorithms can exploit such advanced simulation techniques, the algorithms can scale to treat highly complex grand challenge problems. The accurate solution of model reduction problems for partial differential equation systems is important for many applications including fast simulations, tractable optimization, feedback control design, and real time control implementation. Feedback control of PDEs has potential applications in many areas. For example, feedback control of air flow in commercial buildings could lead to dramatic reduction in energy consumption. In another area, feedback control of fluid-structure interactions in wave energy converters could lead to effective ocean wave energy extraction. The proposed work primarily considers the development and analysis of model reduction and feedback control algorithms for linear PDE systems, as extensions of these reduced models and feedback laws can be used to treat problems involving nonlinear PDEs. The algorithms can exploit advanced existing simulation codes, and therefore the completion of the proposed work will open the door for large advances in computational feedback control to challenging nonlinear PDE systems.