频响函数概率模型驱动的结构系统识别不确定性量化与传播机理研究

As a classical model reflecting the input-output relationship in the frequency domain, Frequency Response Functions (FRFs) play a fundamental role in system identification. Unfortunately, its applicability is widely affected by uncertainty. This project aims to quantify uncertainty, understand uncertainty and reduce uncertainty by accommodating measurement noise and modeling errors. This project deeply investigates three issues including statistical inference for FRFs, uncertainty quantification for system identification and uncertainty propagation law of system identification, thereby building an uncertainty analysis framework for system identification driven by the probabilistic model of FRFs in the complex domain. The main contents are shown as follows: (1) The project is expected to prove probability density transformation of random vector in the complex domain so as to reduce the dimension of integration for the statistical inference problem of FRFs, based on which the probability density function of FRFs can be achieved by Gaussian integration algorithm. (2) The Bayesian system identification approach driven by the probabilistic model of FRFs will be proposed by properly considering the correlation and variation among different prediction errors. Also, efficient algorithms will be developed to obtain the optimal values of the model parameters as well as their uncertainties. (3) The project will analytically derive functional relationship between uncertainties and different driving factors to reveal the uncertainty propagation law of measurement noise. Analytical formulas will also be developed to calculate the evidence of different theoretical models employing asymptotic analysis, which is then utilized to obtain the probability of selecting different model classes. On the basis of uncertainty analysis, the project aims to provide solid foundations for guiding structural dynamic testing and model selection, and it is also expected to present more precise dynamic analysis, which is of theoretical significance and practical values in real applications.

作为描述系统输入-输出关系的经典模型，频响函数是系统识别的重要基础，然而其工程适用性受不确定性的制约。项目以考虑测试和建模随机性为突破口，以量化不确定性、认识不确定性和抑制不确定性为目标，以频响函数概率模型推断、特征参数不确定度量化和不确定性传播机理分析为主线，从基础理论、数值模拟和试验三方面，对频响函数概率模型驱动的系统识别不确定性分析方法展开研究：(1)推导复数域随机向量概率密度转换原理实现频响函数概率模型积分降维，引入高斯积分方法实现快速求解;(2)基于频响函数概率模型构建考虑预测误差相关性和方差变异性的贝叶斯系统识别方法，开发新算法提高不确定性量化运算效率;(3)建立参数不确定性与测试随机性之间的近似显式函数关系，推导表征模型精度的证据表达式量化建模随机性。通过探索随机性起源、传播和累积，项目旨在为动力测试及系统建模提供指导，为提高动力分析精度提供依据，具有较大的理论和实用价值。

频响函数作为描述系统输入-输出关系的经典模型，是系统识别和结构健康监测的重要基础，然而其工程应用受到不确定性的制约。项目充分利用贝叶斯在不确定性度量方面优势，以考虑测试和建模随机性为突破口，对频响函数概率模型推断，系统识别不确定度量化与不确定性传播机理分析三个层次的基础理论、算法和计算机实现展开深入研究，主要工作包括：(i)推导出了复数域随机向量概率密度转换原理，实现多元相关频响函数概率模型积分降维，并应用高斯数值积分和稀疏网格算法快速推断出频响函数概率分布；(ii)基于频响函数概率模型构建考虑预测误差相关性的贝叶斯随机系统识别框架，引入代数算法、向量化算法和并行算法大幅度提高了参数不确定性量化计算效率；(iii)建立了表征动力测试随机源对特征参数识别不确定性贡献的近似显式函数关系，计算表征建模随机性的证据表达式，剖析测试和建模随机性的传播机理；(iv)将贝叶斯随机系统识别不确定性量化算法拓展应用于基于超声导波测试的无损检测，求解结构物理参数和损伤参数的后验不确定性；(v)采用数值模拟和实验模型振动测试验证了频响函数概率模型的鲁棒性及系统识别不确定性分析方法的准确性。以上研究工作通过揭示随机性的起源和传递规律，达到量化不确定性、认识不确定性和抑制不确定性的目标，为改善动力测试条件及其优化数学模型提供理论依据，为提高工程结构动力测试和结构监测的分析精度提供科学基础，具有较大的理论意义和工程实用价值。本项目在国内外著名期刊发表论文17篇，其中国际权威期刊论文13篇。此外，依托本项目授权中国发明专利1项，培养毕业博士研究生2人及硕士研究生5人。