8 Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations 77 ˆy =argmin y∈R|A| E yT (X) −M(X) 2 +λ y 1 (8.15) where ˆy 1 =#α∈A | yα | is a regularization term to select low rank solutions of the minimization problem. The solution reads: ˆy = ATA −1 ATY (8.16) whereY={M(X1), . . . ,M(XM)} andAij =Ψj(x (i)), i =1, . . . , M, j =0, . . . , P−1 is the experimental matrix containing the values of all the orthogonal polynomials at the experimental design points. InKriging meta-modelling, the system response is approximated with a realization of a Gaussian process: YKrig =βTf (x) +σ 2 Z(x,ω) (8.17) whereβ Tf(x), the trend, corresponds to the mean of the Kriging meta-model with{ βj, j =1, . . . , P} the regression coefficients and {fj, j =1, . . . , P} the regression functions, Z(x, ω) is a zero-mean, unit variance, stationary Gaussian process and σ 2 is the variance of the Gaussian process [21]. Changes in trend types result in different Kriging meta-models. Common trend types are the constant (also called ordinary Kriging), linear, quadratic and polynomial. The trend of the Kriging meta-model used for this study is the quadratic and the polynomial. The latter corresponds to the PCK case and is addressed later on. The stationary process Z(x, ω) is determined by a probability space ωand a correlation function R=R(xi, xj; θ). The latter characterize the correlation between two samples based on the hyperparameters θ of the correlation function. The Kriging meta-model is trained with a set of experimental design points {X, Y} = {{xi, i =1, . . . ,n}, {M(xi), i =1, . . . ,n}} and, based on that, provides predictions of the model response for new points x. The mean and variance of the Kriging predictor read: μˆY (x) =f(x)Tβ +r(x)TR−1 (y −Fβ) σ 2 ˆY (x) =σ 2 1−rT(x)R−1 r (x) +uT (x) FTR−1 F −1 u(x) (8.18) where r(x) = R(x, xi; θ), Fij = fj(xi), β = (FTR−1F)−1FTR−1y is obtained by generalized least-squares estimate and u(x) =FTR−1r(x) −f(x). PCK is a combination of the above two surrogate techniques [22]. The Kriging methodology is used and PCE is employed as its polynomial trend type. The system’s response approximation follows: YPCK = α∈A yα α (X) +σ 2 Z(x,ω) (8.19) For the calculation of the PCE coefficients, the LARS methodology is used again. 8.6 Global Sensitivity Analysis Global sensitivity analysis (GSA) aims at quantifying to what extent each one of the stochastic input variables affects the model response. The outcome of such analysis provides engineers with viable information regarding the system design. If, for instance, if a stochastic input variable has a negligible influence to the output, then this variable could be replaced by a deterministic value, reducing the model complexity and the computational cost of simulating the model. Moreover, if a variable has a significant effect on the system response, then the variability of this variable should be, if possible, decreased in order to achieve higher robustness to uncertainties. In this paper, PCE-based Sobol’ indices, also called ANOVA, are employed in order to conduct GSA. The idea behind Sobol’ indices is to decompose the full computational model into submodels which depend on an increasing number of input variables [23]. By rearranging the PCE model we get:
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