566 F. Abid et al. 12345678910 100 Singular value number σ i 0 1 2 3 4 5 −1000 −500 0 Time (s) ci (K) 0 1 2 3 4 5 −200 0 200 400 600 Time (s) Temperature (K) Fig. 55.3 SVD computed on the reference FE model;(top left) the singular values (circle); (top right) modal contributions of Proper Orthogonal Mode (POM): POM1 (solid curve), POM2 (dashed curve) and POM3 (dashed-dotted curve); (bottom) Temperature response at DOF 1: FE response (solid curve), SVD with2POM(dashed curve) and SVD with 3 POM (dashed-dotted curve) 55.4.1 Filtering Step Now we apply EKF and UKF to the FE reference model. Our goal is to identify the discrete ROM by using known input forcing vector u D imp.t/, a square signal (see Fig. 55.2 (top)), and available temperature data collected at DOF 1 and 10 of the FE reference model (see Fig. 55.2 (bottom)). For simplicity, the state and observation noise covariance matrices are set as QD 2 wIne ne andRD 2 v Im m,where 2 w and 2 v are the state and observation noise variances. The notations Ine ne and Im m denote the ne ne and m midentity matrices, and ne D10; mD2 stand for the dimension of the extended state vector and observation vector. The initial state estimate covariance is set as P0 Dp0 Diag.vect/, where p0 is the initial state error variance andvect DŒ10; 10; 1; 1; 0:01; 0:01; 1; 1; 1; 1 a vector of dimensionne. 55.4.2 Sensitivity Analysis In this study, we show how: (1) the initial state estimate covariance P0 representing the confidence in the initial state estimate, (2) state model covariance Q representing the confidence in the Kalman model and (3) the observation covariance R representing the confidence in the measurements, affect the performance of both EKF and UKF. The performance of the EKF and UKF can be measured by: (1) comparison of the true observed and estimated temperature and the corresponding terms in P (not shown here); (2) evolution of the identified parameters and the corresponding terms in P; and finally (3) the evolution of the residual term, which is the difference between the predicted and true observed temperatures at DOF of observation (1 and 10 in the reference FE model). 55.4.2.1 Sensitivity to the State Model Covariance Figures 55.4–55.7, illustrate the sensitivity of EKF and UKF to the state covariance by comparing values from w D2:10 5 to w D 2:10 9 and from w D 10 2 to w D 10 11, respectively. Increasing the state model covariance increases the convergence speed (Fig. 55.4) and sensitivity to measurement (a significant decrease of the residual term when w goes from 10 8 (dashed-dotted curve; error up to 6%) to 10 5 (dashed curve; error up to 1:6%) in Fig. 55.7 (left)). Increasing the state model covariance too far results in parameter identification failure (dashed and solid curves in Fig. 55.6 (left)) and
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