隐变量模型及其在贝叶斯运营模态分析的应用

Latent variable model and its application to Bayesian operational modal analysis

贝叶斯FFT算法是运营模态分析的最新一代算法,以其准确性高、计算速度快、可有效进行不确定性度量等优点受到广泛关注。然而,现有贝叶斯FFT算法针对不同情况(稀疏模态、密集模态、多步测试等)需采用不同优化算法,且编程实现极为复杂。为此,本文旨在提出针对不同情况的贝叶斯FFT算法的统一框架,并实现模态参数的高效求解;视结构模态响应为隐变量,建立贝叶斯模态识别单步测试和多步测试的隐变量模型框架;针对提出的隐变量模型运用期望最大化算法实现各种情况下模态参数的统一贝叶斯推断,利用隐变量解耦模态参数优化过程,采用Louis等式间接求取似然函数的Hessian矩阵。通过两个实际工程测试案例,并与现有方法对比,验证所提方法的准确性和高效性。分析结果表明,本文所提算法与现有方法结果相同,但其推导简单、易编程,尤其对于密集模态识别问题具有明显的计算优势。本文为贝叶斯模态识别建立起统一的隐变量模型框架,在很大程度上简化原本繁琐且冗长的推导过程,提高计算效率,同时也为应用变分贝叶斯、吉布斯采样等算法求解贝叶斯模态识别问题提供了可能。

As a method for operational modal analysis(OMA),the Bayesian FFT algorithm has garnerd significant attention for its high accuracy and efficiency,as well as its ability of uncertainty quantification.However,different cases of OMA(e.g.well-sep⁃arated mode,closely-spaced modes,and multi-setup OMA)require different optimization strategy,and it is tedious in computer coding.A new framework is proposed in this paper to unify the above-mentioned cases of OMA,and the implement is simplified as a consequence.Regarding the structural modal response as a latent variable,the single-setup and multi-setup Bayesian OMA is cast as latent variable models,which have been deeply investigated in statistics.An expectation-maximization(EM)algorithm is developed for both single-setup and multi-setup OMA.The introduction of latent variables decouples the parameter optimization in EM,and Louis identity is employed to calculate the Hessian matrix.Two field tests are applied to verify the performance of the proposed approach,with a comparison to the existing algorithm.Consistent results are obtained,and a great advantage in efficiency is observed in the case of closely-spaced modes.The proposed latent variable model unifies the cases of Bayesian OMA,with the advantage of simplified implementation and fast computation.It also paves a way for a further improvement of Bayesian OMA,e.g.with the approach of variational Bayes or Gibbs sampling.