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Collaborative Research: Operator theoretic methods for identification and verification of dynamical systems

合作研究：动力系统识别和验证的算子理论方法

基本信息

批准号：
2027976
负责人：
Joel Rosenfeld
金额：
$ 22.94万
依托单位：
University of South Florida
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2027976&HistoricalAwards=false
关键词：
Collaborative Research Operator theoretic methods

项目摘要

Widespread use of automation in many sectors of society has yielded a large amount of data regarding historical behaviors for a variety of dynamical systems, such as unmanned aerial, marine, and ground vehicles, biological systems, and weather systems. This project aims to develop novel algorithms to discover governing rules that explain the observed behaviors (i.e., trajectories) of dynamical systems. Discovery of underlying models, while useful for analysis and control, can be computationally challenging. For example, traditional modeling methods rely on derivatives, and can be hampered by even modest amounts of measurement noise that derails numerical differentiation. Such methods treat each measurement of the output of a dynamical system as a separate data point. The data points are then related back to the underlying model using numerical differentiation. Instead, in this project, the entire trajectory is treated as a unit of interest. The sequence of measured data points is treated as a sampled, noisy representation of that trajectory, and is related back to the underlying model using numerical integration. It is hypothesized that treating trajectories of dynamical systems as the fundamental unit of data can yield better data-driven techniques for analysis and control of dynamical systems, and this project aims to develop such data-driven identification and verification techniques. To broaden the impact of the research, the team will also develop week-long workshops for undergraduate students that teach data science and artificial intelligence (AI) concepts through video games. To facilitate early introduction to machine learning, the team will also develop versions of the AI workshops that are suitable to be offered during high school summer camps. The specific aim of this project is to develop a new theoretical framework to process a large amount of time-series data and to apply the framework to yield robust and flexible tools for the study of nonlinear dynamical systems. In the proposed approach, trajectory information is embedded in a reproducing kernel Hilbert space (RKHS) through what are called occupation kernels. The occupation kernels are tied to the original dynamics through a densely defined operator, the Liouville operator. Occupation kernels and Liouville operators result in a nontrivial generalization of contemporary methods that study finite-dimensional nonlinear optimization problems by lifting them into infinite dimensional linear programs over the spaces of measures. The proposed approach facilitates lifting into linear programs over function spaces instead of measure spaces, and as a result, tools from function theory and approximation theory become available for design and analysis of algorithms. The specific aims of the project include: studying fundamental properties of occupation kernels and Liouville operators over RKHSs, applications to nonlinear system identification, study of the pre-inner product space that results from the action of the adjoint of a Liouville operator on an occupation kernel, and applications of the framework to solve motion tomography problems. The developed tools will be validated by solving identification and verification problems for unmanned ground, air, or underwater vehicles.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.

自动化在社会许多领域的广泛应用已经产生了大量关于各种动力系统历史行为的数据，如无人驾驶的空中、海上和地面车辆、生物系统和天气系统。该项目旨在开发新的算法来发现解释观察到的动态系统行为（即轨迹）的控制规则。底层模型的发现虽然对分析和控制有用，但在计算上具有挑战性。例如，传统的建模方法依赖于导数，即使是少量的测量噪声也会阻碍数值微分。这种方法将动力系统输出的每个测量值视为一个单独的数据点。然后使用数值微分将数据点关联回底层模型。相反，在这个项目中，整个轨迹被视为一个兴趣单元。测量数据点序列被视为该轨迹的采样、噪声表示，并使用数值积分与底层模型相关。假设将动力系统的轨迹作为数据的基本单位可以产生更好的数据驱动技术来分析和控制动力系统，本项目旨在开发这种数据驱动的识别和验证技术。为了扩大研究的影响，该团队还将为本科生举办为期一周的研讨会，通过视频游戏教授数据科学和人工智能（AI）概念。为了促进机器学习的早期介绍，该团队还将开发适合在高中夏令营期间提供的人工智能研讨会版本。该项目的具体目标是开发一个新的理论框架来处理大量的时间序列数据，并应用该框架来产生用于非线性动力系统研究的鲁棒和灵活的工具。在提出的方法中，轨迹信息通过所谓的占用核嵌入到再现核希尔伯特空间（RKHS）中。通过一个密集定义的操作符，即Liouville操作符，将占用内核绑定到原始动态。职业核和Liouville算子通过将有限维非线性优化问题提升到测度空间上的无限维线性规划，从而对当代研究有限维非线性优化问题的方法进行了非平凡的推广。所提出的方法有助于在函数空间而不是测量空间上提升到线性规划，因此，函数理论和近似理论的工具可以用于算法的设计和分析。该项目的具体目标包括：研究RKHSs上的职业核和Liouville算子的基本性质，在非线性系统识别中的应用，研究由Liouville算子对职业核的伴随作用产生的预内积空间，以及将该框架应用于解决运动层析问题。开发的工具将通过解决无人地面、空中或水下航行器的识别和验证问题进行验证。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。