Mechanical systems virtual sensing by proportional observer and multi-resolution analysis

A technique to reconstruct the displacement field throughout the structure from pointwise measurements under different noise sources is proposed. The developed estimator is a Proportional Observer (PO) that exploits a linear, frequency independent, relation between the estimated state-space vector and the measurements. To improve its accuracy, the PO concept is then extended to the definition of a sequence of proportional observers, each one acting on a signal decomposition provided by wavelet multi-resolution analysis, or a Multi-Resolution Proportional Observer (MR-PO). The considered numerical test case is a straight, uniform beam with an unmodeled stiffness reduction provided by a notch, which allows for characterizing analytically this model uncertainty. The input data is given by virtual strain measurements collected on the top face of the beam, whereas the estimated state variables are the time dependent coordinates of the modal expansion of the vertical displacement along the beam elastic axis. An optimization solver, which minimizes the estimation error, is employed to get the gain matrix of the proposed observers. The effect of different noise sources, like process and measurement noise and unknown excitation, on the estimation accuracy is taken into account with a sensitivity analysis. The obtained results assess the effectiveness of the combination between the PO concept and wavelet multi-resolution analysis as a tool for developing digital twin models based on experimental data.