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

本项目将改进模型辨识过程中不确定性的处理。用数学方法表示作用在真实的系统上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将输入转换为响应的数学对象。模型识别是根据对输入和输出的观察确定模型形式并为其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。这种困难不可避免地会导致系统模型的不确定性。该项目的结果将使这些不确定性能够在目前无法处理的几个重要案件中得到尽可能好的处理。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。 这种方法的重要性是至关重要的系统损坏/故障检测程序的进一步发展。这种诊断方法可以集成在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。该项目将有助于不同的研究领域，如结构动力学，概率方法，数据采集和信号处理，以及结构安全性和可靠性。例如，为了恢复和改善城市基础设施，需要查明和量化与老化机制和环境激励因素有关的不确定性。此外，将在网上部署电子学习互动软件工具，纳入该项目的研究成果。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受到高度有限的数据。与不确定性建模相关的具体挑战，以及基于输入-输出知识的结构系统特性识别（激发-响应）数据包括：（1）测量的/可用的数据通常具有演化特征，即它们表现出随时间变化的行为，（2）通常存在有限的、不完整的和/或缺失的数据，以及（3）从数学角度来看，系统控制方程是高度复杂的，包括非线性和分数阶导数建模。目前，不可能以一致、有效的方式同时处理情况（1）、（2）和（3）。该项目的研究目标是创建一种基于压缩抽样的方法，用于在数据非常有限的随机工程动力学领域进行高效的不确定性建模和量化。具体目标包括发展技术的频谱分析和随机过程统计估计不完整的数据，以及用于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，令人满意的准确性高达80%的缺失数据。

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.