260 R. Feldmann et al. paper are assumed calibrated, so that there are no calibration parameters θ. The discrepancy function δ(x) is modelled by a Gaussian process δ(X) ∼N(μ,K(X,X)), where the mean scale μand the covariance matrix Kare assumed constant and squared exponential, respectively. Additionally, a noise level parameter is considered to account for the measurement error εn in (28.6). The behaviour of the Gaussian process is governed by its hyperparameters that are inherent to covariance and mean function. Next, a training data set is needed to fit the Gaussian process representing the discrepancy functions δp,zr , δp,Fef , δp,Fsd for the three outputs and P =4 model candidates. The input training dataset is given by the input matrix X=(x1, . . . , xN) defined in Sect. 28.2.3. The output training datasets are given by the difference vectors zp,zr =yzr −ηp,zr =(δp,zr (x1) +εp,zr,1, . . . ,δp,zr (xN) +εp,zr,N) (28.7a) zp,Fef =yFef −ηp,Fef =(δp,Fef (x1) +εp,Fef,1, . . . ,δp,Fef (xN) +εp,Fef,N) (28.7b) zp,Fsd =yFsd −ηp,Fsd =(δp,Fsd(x1) +εp,Fsd,1, . . . ,δp,Fsd(xN) +εp,Fsd,N) , (28.7c) where the entries of the difference vectors (hereafter denoted by z for simplicity) are assumed realizations of the Gaussian process to be fitted. In order for the Gaussian process to represent the training data set (X, z) the best, its hyperparameters are obtained through a hyperparameter optimization. Using the optimal set of hyperparameters, the 95% confidence intervals for the discrepancy functions can be readily specified analytically. The confidence intervals for discrepancy functions of different model candidates subsequently yield a measure to assess model form uncertainty: As a perfect model would have a zero discrepancy function, it is natural to assume a better model if its confidence interval lies closer to zero. This way a model selection can be conducted. For a more detailed overview about the methodology, the reader is referred to [7]. 28.3.2 Definition of a Case Study The methodology outlined in Sect. 28.3.1 has been applied to the four model candidates of the 2DOF model of the MAFDS in a case study, in which two different sets of initial conditions (28.2) have been adopted. In a first case, an inadequate initial condition model is assumed, while in the second case, measurements are used as initial conditions, that represent true values for the initial conditions except for measurement noise. In the latter, no model form uncertainty is inherent to the initial conditions used for simulation. Model form uncertainty will subsequently be assessed for both cases and the comparison of the results will reveal how the model form uncertainty of the initial condition model propagates to the system model. Case 1: Analytically Derived Initial Condition Model In the first case, the analytically derived initial condition model zu(t0) =0 ˙zu(t0) = 2gh (28.8a) zl(t0) =0 ˙zl(t0) = 2gh, (28.8b) such as those in [7], is regarded. In previous research[7], it was concluded that the functional relationship assumed with this initial condition model is inadequate. In fact, damping or friction effects, that might occur during fall between the frame and the guidance rails 6 and 7 in Fig. 28.1a are completely neglected. Consequently, the resulting velocities ˙zu(t0) and ˙zl(t0) of the upper and lower truss at the moment of impact t0 =0 will be smaller in reality than determined by (28.8). This inadequacy inherent to the initial conditions (28.8) indicates model form uncertainty, whose propagation to the system model shall be investigated in this contribution. Case 2: Measured Initial Conditions The second case is laid-out as reference to the first case: The simulation model is now excited by measured initial conditions in order to exclude effects of inadequacies of the initial condition model. Measurements provide values of the real system
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