Multifidelity Nonsmooth Optimization and Data-Driven Model Reduction for Robust Stabilization of Large-Scale Linear Dynamical Systems

从汽车到飞机再到医疗设备，自主系统在工程应用和整个社会中发挥着越来越重要的作用。真正的自主系统必须能够在不确定的情况下采取行动并做出决策。决定自治系统采取什么行动的关键组件是系统的控制器，它保证系统始终保持稳定和安全的状态。因此，设计控制器来稳定系统是包括在环境中作用的虚拟或物理系统的广泛应用中的重要问题。该项目将开发的计算方法旨在仅从数据中可靠地稳定大规模系统，即使只有很少的数据和受噪音污染的数据。这些算法有可能对自治系统的效率、安全性和可靠性等关键问题产生重大影响。该项目将促进从机器学习到控制理论到数值分析再到科学计算的跨学科合作，并将通过创建新颖的课程和推广活动来支持教育和多样性，这些课程和活动将上述学科中代表性不足的群体整合在一起。鲁棒稳定通常需要解决非光滑、非凸优化问题，这些问题在计算和数学上都具有挑战性。此外，在许多情况下，感兴趣的系统的模型不可用。相反，数据是从系统中采样的，必须通过从这些数据中学习来实现稳定。该项目开发和集成了通过梯度采样和数据驱动（非侵入性）模型简化通过Loewner框架的非光滑优化的新方法。第一个贡献将是一个多保真度版本的梯度采样算法的非光滑优化，利用低成本，低保真度的梯度近似的计算昂贵的目标，以加速梯度的估计。如果成功的话，这种多保真度近似有可能使大规模优化问题的易于处理的梯度采样，同时保持梯度采样已知的严格收敛保证。第二个贡献是利用鲁棒控制器的稳定半径来减少学习稳定系统的简化模型所需的数据点（样本）的数量。为此，提出了一种从数据中学习简化模型的新方法，该方法允许学习的模型尽可能多地偏离真实的系统动态特性，并通过鲁棒性（稳定半径）进行补偿，以有利于减少数据点的数量。如果该项目成功，所开发的方法将能够有效和严格地稳定大规模系统，并且从中获得很少的数据点和/或高噪声数据。该奖项反映了NSF的法定使命，并被认为值得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来支持。

Autonomous systems play an increasingly important role in engineering applications and in society as a whole, from cars to airplanes to medical devices. Truly autonomous systems will have to be able to act and make decisions under uncertainty. The key component that decides what action an autonomous system takes is the controller of the system, which guarantees that the system always remains in stable and safe states. Thus, designing controllers to stabilize systems is an important problem in a wide range of applications that include virtual or physical systems acting in an environment. The computational methodologies that will be developed in this project aim towards a reliable stabilization of large-scale systems from data alone, even when only little data and data polluted with noise are available. These algorithms have the potential to have significant impact on critical issues such as efficiency, safety, and reliability of autonomous systems. The project will promote cross-disciplinary collaborations from machine learning to control theory to numerical analysis to scientific computing and will support education and diversity by creating novel courses and outreach activities that integrate underrepresented groups in the above disciplines.Robust stabilization typically requires solving nonsmooth, nonconvex optimization problems that are computationally and mathematically challenging. Furthermore, in many situations, models of the systems of interest are unavailable. Rather, data are sampled from the systems and stabilization has to be achieved via learning from these data. This project develops and integrates new methods for nonsmooth optimization via gradient sampling and data-driven (nonintrusive) model reduction via the Loewner framework. The first contribution will be a multifidelity version of the gradient sampling algorithm for nonsmooth optimization that exploits low-cost, low-fidelity gradient approximations of a computationally expensive objective to accelerate the estimation of gradients. If successful, this multifidelity approximation has the potential to make tractable gradient sampling for large-scale optimization problems and at the same time maintain the rigorous convergence guarantees that gradient sampling is known for. The second contribution is to exploit the stability radius of robust controllers to reduce the number of data points (samples) that are required to learn reduced models for stabilizing systems. To that end, a new approach for learning reduced models from data is proposed that allows the learned models to divert from the real system dynamics by as much as can be compensated for by the robustness (stability radius) in favor of reducing the number of data points. If the project is successful, the developed methodologies will enable efficiently and rigorously stabilizing systems that are large-scale and from which few data points and/or high-noise data are available.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.