PSI - Issue 64

Philipp Kähler et al. / Procedia Structural Integrity 64 (2024) 1248–1255 Kähler / Petryna / Structural Integrity Procedia 00 (2019) 000 – 000

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correction step, the updated ensemble mean value as well as the updated ensemble error covariance matrix can be calculated to check the convergence of the filter according to equations (8) to (10). The second KF used in this work is the Ensemble Square Root Filter (EnSRF), shown in Fig. 4 (b), which employs a different formulation of the correction step to address the problem of filter divergence. According to Hamill/Whitaker (2002), the correction involves adjusting the mean values of the state parameters and the individual ensemble members using different Kalman gains. A major advantage of the EnSRF is the sequential evaluation of the p sensor observations at each update time step, making the algorithm very fast, since the calculation of the inverse measurement noise covariance matrix from equation (6) is reduced to scalar values in equation (20). To further enhance the KF and make it viable for online monitoring, it can be extended through data reduction algorithms, model reduction algorithms, localization algorithms and parallelization of the forecast step. These two KF described above are combined in a sequential update scheme to maximize the accuracy of the update while keeping the computation time low. Whenever an update step is initiated, the EnSRF is used to update the state and model parameters at the same time. However, if the input force, which is used to calculate the model predictions in the KF, is subject to uncertainties, an additional iterative EnKF is used every tenth update step to solve the inverse problem with iterations to increase the accuracy. In this Ensemble Kalman Inversion (EKI), a pseudo-time is introduced at the update time step to determine the most optimal model parameters which provide the best accuracy for the previously adjusted state parameters Schillings/Stuart (2016). The described scheme of the algorithm is summarized in Fig. 5.

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Fig. 5. KF update scheme with EnSRF and EKI.

3.2. Investigations on a laboratory structure In the following, the KF update scheme is tested on a laboratory structure. The test object is a simple supported stainless-steel beam, depicted in Fig. 6 (a). The beam has a length of beam = 4.4 m and has a rectangular hollow profile with a height ℎ=30 mm , a width = 50 mm and a wall thickness of = 2.6 mm , making its total mass approximately beam ≈ 14.6 kg . Additionally, five plates of plate ≈ 100 g each, are attached to the beam for the mounting of the measurement sensors. Acceleration measurements were taken in the vertical direction at five sensor positions. The system was loaded with an attached additional mass, which was suddenly removed from the system, causing a free damped vibration. Additionally, system changes were induced by additional masses ranging from Δ = 100 g to Δ = 500 g to the system. The location of the applied load mass and the location of the additional masses for system changes were varied according to Fig. 6 (b).