The current Bayesian FFT algorithm relies on direct differentiation to obtain the posterior covariance matrix (PCM), which is time-consuming, memory-intensive, and hard to code, especially for the multi-setup operational modal analysis (OMA). Aiming at accelerating the uncertainty quantification in multi-setup OMA, an expectation-maximization (EM)-based algorithm is proposed by reformulating the Hessian matrix of the negative log-likelihood function (NLLF) as a sum of simplified components corresponding to the complete-data NLLF. Matrix calculus is employed to derive these components in a compact manner, resulting in expressions similar to those in the single-setup case. This similarity allows for the reuse of existing Bayesian single-setup OMA codes, simplifying implementation. The singularity caused by mode shape norm constraints is addressed through null space projection, eliminating potential numerical errors from the conventional pseudoinverse operation. A sparse assembly strategy is further adopted, avoiding unnecessary calculations and storage of predominant zero elements in the Hessian matrix. The proposed method is then validated through a comprehensive parametric study and applied to a multi-setup OMA of a high-rise building. Results demonstrate that the proposed method efficiently calculates the PCM within seconds, even for cases with hundreds of parameters. This represents an efficiency improvement of at least one order of magnitude over the state-of-the-art method. Such performance paves the way for a real-time modal identification of large-scale structures, including those with closely-spaced modes.

翻译：当前贝叶斯快速傅里叶变换算法依赖直接微分计算后验协方差矩阵，该方法计算耗时、内存密集且编码复杂，在多测点工作模态分析中尤为突出。为加速多测点工作模态分析中的不确定性量化，本文提出一种基于期望最大化的算法，通过将负对数似然函数的Hessian矩阵重构为完整数据负对数似然函数对应简化分量的和形式。运用矩阵微积分以紧凑形式推导这些分量，所得表达式与单测点情形具有相似结构。这种相似性使得现有贝叶斯单测点工作模态分析代码可直接复用，显著简化了实现过程。针对振型归一化约束导致的奇异性问题，通过零空间投影法进行处理，避免了传统伪逆运算可能引发的数值误差。进一步采用稀疏装配策略，规避了Hessian矩阵中主要零元素的不必要计算与存储。通过系统参数研究验证所提方法，并将其应用于某高层建筑的多测点工作模态分析。结果表明：即使针对数百个参数的情况，所提方法亦能在数秒内高效计算后验协方差矩阵，较现有先进方法的计算效率提升至少一个数量级。该性能为大型结构（包括密集模态结构）的实时模态识别奠定了技术基础。