A Bayesian Expectation-Maximization (BEM) methodology for joint input-state estimation and virtual sensing of structures

The joint input-state estimation and virtual sensing of structures are reformulated on a Bayesian probabilistic foundation, focusing on data-driven uncertainty quantification and propagation. The variation of input forces is described via a first-order random walk model, which helps to construct an augmented state vector encompassing both input and state vectors. Then, system detectability is analyzed based on the transfer matrix of the coupled process and observation models, considering different sensor configurations. As a result, input pseudo-observations are included to overcome singularity problems encountered when having acceleration-only responses. Subsequently, the joint posterior distribution of the latent states and the noise parameters is characterized, and a Bayesian Expectation-Maximization (BEM) strategy is established to search for the most probable values iteratively. The E-Step of this algorithm coincides with the backward-forward Kalman smoother, and the M-Step leads to explicit formulations for updating the process and observation noise characteristics. Still, the EM algorithm might require reasonable choices of the noise parameters in the beginning. This issue is tackled using steady-state solutions of the estimator and smoother, prescribed as an initializer. Since the stationary solutions do not require a recursive calculation of the gain and covariance matrices, the associated computational cost is assessed to be lower than the main EM algorithm. Finally, the proposed methodology is tested using numerical and experimental examples. It is demonstrated that this new probabilistic perspective can provide a reliable tool for uncertainty quantification and propagation in this type of problem

Automated summary

This study proposes a joint input-state estimation and virtual sensing method based on Bayesian probability theory, focusing on data-driven uncertainty quantification and propagation. By introducing a random walk model for input forces and including input pseudo-observations, singularity problems are overcome and a Bayesian Expectation-Maximization (BEM) strategy is established for parameter estimation.