  Structural identification and damage detection can be generalized as the
simultaneous estimation of input forces, physical parameters, and dynamical
states. Although Kalman-type filters are efficient tools to address this
problem, the calibration of noise covariance matrices is cumbersome. For
instance, calibration of input noise covariance matrix in augmented or dual
Kalman filters is a critical task since a slight variation in its value can
adversely affect estimations. The present study develops a Bayesian
Expectation-Maximization (BEM) methodology for the uncertainty quantification
and propagation in coupled input-state-parameter-noise identification problems.
It also proposes the incorporation of input dummy observations for stabilizing
low-frequency components of the latent states and mitigating potential drifts.
In this respect, the covariance matrix of the dummy observations is also
calibrated based on the measured data. Additionally, an explicit formulation is
provided to study the theoretical observability of the Bayesian estimators,
which helps characterize the minimum sensor requirements. Ultimately, the BEM
is tested and verified through numerical and experimental examples, wherein
sensor configurations, multiple input forces, and abrupt stiffness changes are
investigated. It is confirmed that the BEM provides accurate estimations of
states, input, and parameters while characterizing the degree of belief in
these estimations based on the posterior uncertainties driven by applying a
Bayesian perspective.
