Multi-setup Bayesian FFT modal identification with multiple modes leveraging on second order information

Conventional multi-setup Bayesian operational modal analysis (OMA) algorithms may suffer from low efficiency and false convergence when dealing with closely-spaced modes. To address these challenges, this paper proposes a Newton-type Bayesian FFT algorithm to minimize the negative log-likelihood function (NLLF). The gradient vector and Hessian matrix, i.e., the first- and second-order derivatives of NLLF with respect to modal parameters, are obtained semi-analytically by exploring a latent variable structure of NLLF through the expectation–maximization (EM) formulation. The sparsity of the Hessian matrix is explored to enable a fast and robust implementation. In the optimization, an existing EM algorithm is employed in the first round of iterations to improve the positive semidefiniteness of the Hessian matrix, and Newton iterations are then performed to refine the estimate. Since the inverse of Hessian matrix is available at each iteration, the posterior covariance matrix is obtained as a by-product under the Laplace approximation, providing uncertainty quantification without additional computation. A synthetic example of a 30-story shear-type building and a field test of the 292 m SEG Plaza are applied to illustrate the performance of the proposed algorithm. Results show that the proposed algorithm achieves high accuracy and fast convergence, while the existing method may fail in challenging scenarios, e.g., multiple closely-spaced modes.