Bayesian two-stage structural identification with equivalent formulation and EM algorithm - Zhu, Jia-Xin (2024)
For structural model identification using a Bayesian two-stage approach, modal properties (e.g., natural frequencies) are first extracted from measured data in Stage I and then used for determining structural properties (e.g., stiffness) in Stage II. With sufficient data that allow the problem to be globally identifiable, the computational problem often reduces to optimizing a ‘measure-of-fit’ function to yield the ‘most probable value’ (MPV) that informs the ‘best estimate’, and determining the Hessian of the function to yield the posterior covariance matrix that informs the remaining uncertainty. Recent developments deal with the presence of model error, and the increasing complexity calls for proper computational strategy. In this spirit, assuming Gaussian model error, this work develops a hypothetical yet mathematically equivalent formulation for the two-stage problem, which facilitates development of effective algorithms for MPV using Expectation-Maximization techniques. By treating hypothetically the MPV of modal properties in Stage I as ‘data’ and model error as latent variables, the Q-function in the M−step can be expressed as a sum of two terms that can be optimized with respect to structural parameters and model error parameters. The optimization of structural parameters reduces to that of a Stage II problem without model error, for which existing algorithms can be applied. Analytical expressions are also derived for computing the posterior covariance matrix. The proposed methodology is investigated with synthetic data for model errors associated with sensor misalignment and model simplification, lab data for the effect of number of modes and measured degrees of freedom (DoFs), and field data for reality tests.