78 S. Marwitz and V. Zabel Here Gref denotes the next-state output covariance matrix and Rref i the output-covariance matrix at time-lag i. In the case of reference-based stochastic subspace identification these matrices are computed only between the output-channels and selected reference channels. Equation 7.8 is especially important, as the covariances can be estimated directly from the recorded timeseries. Several covariance matrices at ascending time lags i are arranged into a block-Toeplitz-Matrix OTref 1ji D 2 666 64 ORref i ORref i 1 : : : O Rref 1 ORref iC1 O Rref i : : : ORref 2 : : : : : : : : : : : : ORref 2i 1 O Rref 2i 2 : : : O Rref i 3 777 75 (7.9) where ıO denotes “estimate”. The matrix Tref 1ji can also be written in terms of the state variables Tref 1ji D 2 666 4 C CA : : : CAi 1 3 777 5 Ai 1Gref : : : AGref Gref DOi i ; (7.10) Oi and i denote the observability matrix and controllability matrix, respectively. A singular value decomposition OTref 1ji D U1 U2 S1 0 0 S2 V| 1 V| 2 (7.11) and the truncation of singular values at the desired model order is employed to obtain the matrices Oi DU1S0:5 1 ; and i DS0:5 1 V | 1: (7.12) The output matrix Ccan be extracted from the first l rows of Oi, with l being the number of analyzed channels. There exist several alternatives for the computation of the state matrix A. One common method is given by ADO"i O#i (7.13) where ı denotes a pseudoinverse. O"i andO#i are obtained by removing the last and the first block rows of Oi, respectively. An eigendecomposition of the state matrix Aleads to the eigenvalues ƒDdiag. r/ and eigenvectors ‰, which occur in complex conjugate pairs. The natural frequencies fr and damping values r can be computed from the eigenvalues r D ln. r/ t ; fr D j rj 2 ; r D Re. r/ j rj : (7.14) Multiplying the matrix of eigenvectors with the output matrix yields the mode shape matrix ˆDC‰: (7.15) 7.2.3 Multi-setup Merging Strategies In operational modal analysis it is frequently required to carry out measurements on multiple points of a structure in order to obtain an adequate representation of the mode shapes of a structure. Limitations regarding the number of available sensors may apply and consequently, consecutive measurements using different sensor location setups are performed. In order to recombine these afterwards, one or more common output reference sensors, which remain at the same position throughout
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