Accelerating computations in two-stage Bayesian system identification with Fisher information matrix and eigenvalue sensitivity

Structural system identification aims at identifying the parameters of a theoretical (e.g., finite element) model using measured data of a constructed structure. As a means of extracting relevant information from noisy vibration data, one common strategy is to adopt a ‘two-stage’ approach, where natural frequencies and mode shapes are first estimated from Stage I (modal identification), and then used for further estimating the structural model parameters (e.g., stiffness, mass) in Stage II. In a Bayesian statistical context with sufficient data so that model parameters can be identified uniquely, the identification result is characterized by the ‘most probable value’ (MPV) that gives the best estimate, and a covariance matrix that quantifies the remaining uncertainty. Computationally, determining the MPV involves solving an optimization problem with a measure-of-fit function; the covariance matrix involves taking the Hessian of the function. Exploiting the mathematical structure of the measure-of-fit function in the two-stage problem, this work proposes a Fisher scoring method for determining the MPV, where the Hessian matrix in a Newton's iteration is replaced by the Fisher information matrix to eliminate repeated computations of second order derivatives and to improve convergence robustness. Advanced techniques of eigenvalue sensitivity are applied so that the gradient and Hessian involved in MPV and covariance matrix computations can be obtained accurately and efficiently by solving full-rank matrix equations for the subject modes only. The proposed methodology is investigated with synthetic and laboratory experimental data. Performance in terms of convergence robustness and efficiency is compared with existing methods, including simplex search and Newton's method.