Dynamic shape reconstruction of a notched beam by proportional observer and multiresolution analysis

A novel virtual sensing technique aimed at reconstructing the displacement field throughout the struc- ture from pointwise measurements is here exploited for dynamic shape reconstruction of a notched beam. This estimator is a proportional observer that exploits a linear, frequency independent, relation between the estimated state-space vector and the measurements. To improve its accuracy, this concept is augmented to the definition of a sequence of proportional observers, each one acting on a signal decom- position provided by wavelet multi-resolution analysis, or a Multi-Resolution Proportional Observer (MR-PO). The considered experimental test case is a straight, uniform beam with an unmodeled stiff- ness and mass reduction provided by a notch. The strain data are provided by strain gauges positioned 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 observer. The results are compared with the so called Modal Filter (MF) by assessing the prediction capability of the observer on the points that are not exploited for the virtual measurement. A sensitivity analysis is carried out on the number of strain gauges used to predict the state response showing that the use of wavelet multi-resolution analysis is effective in maintaining the prediction capabilities high even when few measurements are taken into consideration for state estimation.