An expectation-maximization algorithm for Bayesian operational modal analysis with multiple (possibly close) modes

The Bayesian FFT method has gained attention in operational modal analysis of civil engineering structures. Not only the most probable value (MPV) of modal parameters can be computed efficiently, but also the identification uncertainty can be rigorously quantified in terms of posterior covariance matrix. A recently developed fast algorithm for general multiple (possibly close) modes was found to work well in most cases, but convergence could be slow or even fail in challenging situations. The algorithm is also tedious to computer-code. Aiming at resolving these issues, an expectation-maximization (EM) algorithm is developed by viewing the modal response as a latent variable. The parameter-expansion EM and the parabolic-extrapolation EM are further adopted, allowing mode shape norm constraints to be incorporated and accelerating convergence, respectively. A robust implementation is provided based on the QR and Cholesky decompositions, so that the computation can be done efficiently and reliably. Empirical studies verify the performance of the proposed EM algorithm. It offers a more efficient and robust (in terms of convergence) alternative that can be especially useful when the existing algorithm has difficulty to converge. In addition, it opens a way to compute the MPV in the Bayesian FFT method for other unexplored cases, e.g., multi-mode multi-setup problem.