PSI - Issue 64

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

1252

5

3.1. State and model parameter update with ensemble-based KF The first KF used is the Ensemble Kalman filter (EnKF), depicted in Fig. 4 (a).

(a)

(b)

Fig. 4. (a) algorithm of the Ensemble Kalman filter, (b) algorithm of the Ensemble Square Root Filter. In the initialization step, an ensemble of different extended state parameter vectors x at time ( −1) is generated, which are assembled in the matrix X . This matrix is called the augmented state parameter matrix because it contains the state parameter vectors , which consists of the displacement, velocity and acceleration of every investigated degree of freedom as well as the model parameter vectors , which consist of the ensemble variation of the stiffness, mass and damping values at each degree of freedom. Thus, each ensemble member is an independent variation of the model which consists of parameters that need to be updated withing this algorithm. In the next step, each ensemble member will be propagated to the next update step at time by evaluating the forecast function (⋅) with the given state at the previous time step and a given input , as in equation (2). Since the dynamic behavior is to be analyzed in this work, the Newmark integration algorithm for the equations of motion is applied to generate each ensemble model prediction for the next time step. Furthermore, each model forecast is subjected to a gaussian distributed model noise term , which represents the model uncertainties. Thereafter, the ensemble mean of the forecast ̅ f and the ensemble error covariance matrix f will be determined through equations (3) to (5). In the correction step, the augmented state vectors will be updated by calculating the difference between the noisy measurement data and the predicted states ( ⋅ f ) at the sensor positions from equation (7). This difference is then weighted by the Kalman gain from equation (6), which takes into account the variation of the states as well as the measurement noise. The matrix is the noise covariance matrix and the matrix is a mapping matrix, that converts the predicted state parameters into the measurement domain. To prevent filter divergence due to undersampling and overconfidence in the model forecast, an additional noise term is added to the measured data Burgers et al. (1998). At the end of the