14 Covariance-Driven Stochastic Subspace Identification of an End-Supported Pontoon Bridge: : : 109 levels (orders). A pole is deemed physical and stable, as opposed to spurious and unstable, if certain criteria for deviance of modal quantities are fulfilled. One possible implementation of this is discussed in the sub-section below. By combining Eqs. (14.3) and (14.8), the following estimate of the observability matrix is established: ŒOi DŒW1 1ŒU1 Œ†1 1=2 (14.11) Then, the state matrix can be computed as follows: ŒA DŒOdown ŒOup (14.12) where the matrices ŒOdown and ŒOup both are subsets fromŒOi , without the first or last l rows, respectively, and denotes the pseudo-inverse. The output matrix ŒC is required to establish the physical mode shapes, and is retrieved from the first l rows of the observability matrix, as follows: ŒC DŒOi 1Wl (14.13) By performing an eigenvalue decomposition of the discrete state matrix ŒA , the discrete eigenvalues O r and the system eigenvectors Œ‰ are established, which thereafter are transformed to continuous eigenvalues and to eigenvectors with coordinates referring to the sensor coordinates as follows: r Dln O r f 1 s ; Œˆ DŒC Œ‰ (14.14) where the modal transformation matrix Œˆ has columns that refer to the identified mode shapes f rg, where r is themode index. 14.2.1 Selection of Weighting Matrices There are two traditional options for the weighting matrices ŒW1 andŒW2 for Cov-SSI, namely the canonical variate analysis (CVA) weighting and the balanced realization (BR) weighting. The CVA weighting is commonly interpreted as the weights ensuring balanced energy levels for all the system modes (see e.g. [8]). The following two Toeplitz-structured matrices are used as a starting point: ŒRC D 2 66 64 ŒR0 ŒR1 T : : : ŒRi 1 T ŒR1 ŒR0 : : : ŒRi 2 T : : : : : : : : : : : : ŒRi 1 ŒRi 2 : : : ŒR0 3 77 75 ; ŒR D 2 66 64 ŒR0 ŒR1 : : : ŒRi 1 ŒR1 T ŒR 0 : : : ŒRi 2 : : : : : : : : : : : : ŒRi 1 T ŒRi 2 T : : : ŒR 0 T 3 77 75 (14.15) The weights are thereafter defined as follows: ŒW1 DŒLC 1; ŒW2 DŒL 1 (14.16) where ŒLC and ŒL are established from the Cholesky decomposition of ŒRC and ŒR , respectively. This weighting is reported to yield a better identification of less excited modes. The Cholesky decomposition is not straightforward to perform due to the poor conditioning of the input matrices. The authors observed that the block-Cholesky decomposition algorithm described in [9], modified to use Gaxpy-rich Cholesky factorization, was rather robust. BR weighting is analogous to using unit matrices for ŒW1 and ŒW2 .
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