Compressive Sampling for Uncertainty Modeling and Quantification of Dynamical Systems Subject to Highly Limited/Incomplete Data

本项目将改进模型识别过程中对不确定性的处理。用数学方法表示作用于真实系统的外部影响和系统对这些输入的响应通常是有用的。然后，系统本身可以被建模为一个数学对象，将输入转换为响应。模型识别是基于对输入和输出的观察，确定模型的形式，并为模型可能需要的任何参数赋值的过程。受随机变化影响的系统称为随机系统。当输入应用于随机系统时，很难区分响应的可重复部分和由于随机波动引起的响应部分。这一困难不可避免地导致了最终系统模型的不确定性。这个项目的结果将使这些不确定因素能够尽可能地在目前无法处理的几个重要案件中得到处理。这些是随时间变化的系统，缺少输入/响应数据的系统，以及具有某些类型的复杂动态行为的系统。这些方法的重要性对于系统损坏/故障检测程序的进一步发展至关重要。这种诊断方法可以集成到一个更广泛的框架中，用于实时监测系统的动态行为以及评估其可靠性。该项目将为结构动力学、概率方法、数据采集和信号处理以及结构安全性和可靠性等多个研究领域做出贡献。例如，为了恢复和改善城市基础设施，必须查明和量化与老化机制和环境刺激有关的不确定因素。此外，结合该项目的研究成果，将在网上部署电子学习互动软件工具。该项目将推进不确定性建模和量化领域的知识，重点是随机激励建模和结构系统识别，这些领域的数据非常有限。与不确定性建模和基于输入-输出（激励-响应）数据知识的结构系统特性识别相关的具体挑战包括：(1)测量/可用数据通常具有进化特征，即它们表现出时变行为；(2)大多数情况下存在有限的，不完整的和/或缺失的数据；(3)从数学角度来看，系统控制方程非常复杂，包括非线性和分数阶导数建模。目前，不可能以一致、有效的方式同时解决情况(1)、(2)和(3)。本项目的研究目标是创建一种基于压缩采样的方法，用于在数据高度有限的随机工程动力学领域进行有效的不确定性建模和量化。具体目标包括发展频谱分析和随机过程统计估计技术，以不完整的数据，以及识别具有分数阶导数元素的非线性系统的参数。初步工作表明，高达80%的缺失数据的准确性令人满意。

This project will improve the treatment of uncertainty in the process of model identification. It is often useful to mathematically represent both the external influences acting on a real system, and the response of the system to those inputs. The system itself can then be modeled as a mathematical object that converts inputs into responses. Model identification is the process of determining the form of the model -- and of assigning values to any parameters that it may require -- based on observation of the inputs and outputs. Systems that are subject to random variations are called stochastic. When inputs are applied to a stochastic system, it can be difficult to distinguish between the repeatable part of the response and the part of the response due to random fluctuations. This difficulty inevitably causes uncertainty in the resulting system model. The results of this project will enable these uncertainties to be handled as well as possible in several important cases that cannot currently be treated. These are systems that change over time, systems with missing input/response data, and systems with certain types of complex dynamic behavior. The importance of such approaches is paramount for further development of system damage/fault detection procedures. Such diagnostic methods can be integrated in a broader framework for real-time monitoring of the dynamic behavior of the system as well as assessing its reliability. The project will contribute to diverse research fields such as structural dynamics, probabilistic methods, data acquisition and signal processing, as well as structural safety and reliability. For instance, to restore and improve urban infrastructure, uncertainties related to ageing mechanisms and environmental excitations need to be identified and quantified. In addition, e-learning interactive software tools will be deployed online, incorporating research results from this project. This project will advance knowledge in the fields of uncertainty modeling and quantification, with emphasis on stochastic excitation modeling and structural system identification subject to highly limited data. Specific challenges related to modeling of the uncertainties, and the identification of the structural system properties based on knowledge of input-output (excitation-response) data include: (1) measured/available data most often possess evolutionary features, i.e. they exhibit a time-varying behavior, (2) most often there are limited, incomplete and/or missing data, and (3) the system governing equations are highly complex from a mathematics perspective, including nonlinearities and fractional derivatives modeling. Currently, it is not possible to address cases (1), (2), and (3) simultaneously in a consistent, efficient manner. The research objective of this project is to create a compressive sampling based methodology for efficient uncertainty modeling and quantification in the field of stochastic engineering dynamics subject to highly limited data. Specific goals include the development of techniques for spectral analysis and stochastic process statistics estimation subject to incomplete data, as well as for identifying the parameters of nonlinear systems endowed with fractional derivative elements. Preliminary work suggests satisfactory accuracy for up to 80% missing data.