Strain predictions at unmeasured locations of a substructure using sparse response-only vibration measurements

Strain Virtual Sensing Applied to Industrial Presses for Fatigue Monitoring

“…In the fourth substep (16), the prediction of the states is updated (x(t) updated ) using the error between the measurements (z(t)-Hx(t)) and the calculated gain (K(t)).…”

Information theoretic-based optimal sensor placement for virtual sensing using augmented Kalman filtering

“…with mean ẑ(δ, ϕ) given by ( 11) that depends on the data, and covariance matrix Σ(δ, ϕ) given by (15) for the time update step and by (16) for the measurement update step that does not depend on the data. In particular, the variance of the i-th element z i (t; δ, ϕ) of the response vector z(t; δ, ϕ) is given by the i-th diagonal element of the covariance matrix, which simplify to…”

“…Modal expansion [1][2][3][4][5][6] and filtering techniques [7][8][9][10][11][12][13][14][15] are conventional virtual sensing tools. Although the former technique can reconstruct unobserved history responses, it cannot calculate external loadings applied to the structure.…”

Sequential Bayesian Inference for Uncertain Nonlinear Dynamic Systems: A Tutorial.

“…A particular class of identification schemes, relies on adoption of an observer setup, often in the form of Bayesian filtering methods, such as the Observer/Kalman filter Identification (OKID) [7,8], which aim at identifying the system dynamics by recursively, or even in batch mode [9], assimilating data into the underlying model structure of the system at hand. These filtering formulations are particularly fitting within the context of virtual sensing, i.e., the task of inferring response quantities at unmeasured locations [10,11], or even unknown system properties. Virtual sensing is essential for tasks such as digital twinning, diagnostics of condition in critical yet unreachable locations and control.…”

“…10 • I n p . The measurement noise covariance is equal to R k = diag 10 −11 , 10 −9 , 10 −3 , 10 −10 and the state process covariance matrix is tuned to the following values Q k = diag 10 −10 , 10 −9 , 10 −9 , 10 −8 , 10 −8 , 10 −8 , 10 −25 .…”