PSI - Issue 62

S. Anastasia et al. / Procedia Structural Integrity 62 (2024) 1061–1068 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 x k+1 = A x k + ω k , y k = C x k + v k , (1) where vectors x and y represent the state and observation vectors, matrices ∈ℝ 2 2 ×2 2 and ∈ℝ ×2 2 indicate the system's state and output matrices, respectively, and ∈ℕ is a generic time step (i.e., ( ) = ∆ = ⁄ with = ∆ −1 the sampling frequency). For additional theoretical details, see reference (Juang, 1994). Vectors w and v represent stochastic white Gaussian noise processes, respectively, that account for the unknown excitation and measurement noise. The eigenvalues ǡ and eigenvectors ǡ of matrix can be used to extract the structure's natural frequencies , damping ratios and complex mode shape as shown in (Filipe Magalhães, 2011): = ln Δ ⇔ =− + √1− 2 , , = , (2) with =√−1 being the imaginary unit. The accuracy, noise-robustness, and computing efficiency of the covariance-driven SSI (Cov-SSI) method are employed in this work to identify the matrices of the system. With =E{ + } , E denoting the expected value operator, the Cov-SSI technique exploits the covariances between the output measurements with positive time delays varying from ∆ to (2 −1)∆ , represented by to 2 −1 Next, the covariances are arranged as follows in an - by- block Toeplitz matrix: 1| = [ −1 ⋯ 1 2 +1 ⋯ 2 ⋯ ⋯ ⋱ ⋯ 2 −1 2 −2 ⋯ ] (3) Cov-SSI makes use of a crucial feature of stochastic state-space models that connects the output covariances and the state vectors in order to determine the system's matrices from 1| (P Van Overschee, 1996): = − , (4) Then, the Toeplitz matrix is decomposed using Singular Value Decomposition (SVD) as follows: 1| = = [ ] [ ] [ ]= . (5) The rank of the system is given by the number of non-zero SVs (model order) concentrated in (assuming ൏ ). From the SVD outputs, the following estimate of the observability and controllability matrices can be calculated using the comparison of Eqs. (5) and (6): = / , = / , (6) which allows determining the state-space model matrices and . Using the recursive property in Eq. (4), the matrix can be extracted from the observability matrix's first l lines [see Eq. (8)]. Conversely, the Balanced Realization (BR) method, which takes advantage of the observability matrix's shift structure as (C Rainieri, 2014): is a popular way to compute : [ ⋯ − ] = [ ⋯ − ] ⇔ [ ⋯ − ] † [ ⋯ − ] = † , (7) where and contain the fist and the last ( −1) lines of O respectively and symbol † stands for Moore Penrose pseudo-inverse. 2.2. Principal Component Analysis (PCA) After the feature extraction process, the time series of the selected damage-sensitive features (in this case, the normalized resonant frequencies) are organized in an observation matrix = [ 1 ,… , ] containing observations. Data normalization is a key process involving the subtraction of reversible variability in selected characteristics 1063 3