108 K.A. Kvåle et al. correlation between all the measurement channels l, with varying time shifts, as follows: ŒHi D 2 66 64 ŒR1 ŒR2 : : : ŒRi ŒR2 ŒR3 : : : ŒRiC1 : : : : : : : : : : : : ŒRi ŒRiC1 : : : ŒR2iC1 3 77 75 (14.1) Here, 2i corresponds to the maximum number of time lags, or equally valid, i corresponds to the number of block rows. The correlation matrices are formally defined as: ŒRk DE fynCkgfyng T (14.2) where the vector fyng corresponds to the monitored quantities (accelerations) from all channels, for sample n. The sampleshift k is related to the time lag through t D kf 1 s , where fs is the sampling rate and t represents the time lag. The cross-correlation matrices may be efficiently computed based on FFT and IFFT, as is how the xcorr function built-in to MATLAB is functioning. The block-Hankel matrix can be decomposed into the corresponding observability and controllability matrices as follows: ŒHi DŒOi ŒCi (14.3) which further are defined as follows: ŒOi D 2 66 66 64 ŒC ŒC ŒA ŒC ŒA 2 : : : ŒC ŒA i 1 3 77 77 75 ; ŒCi D ŒG ŒA ŒG : : : ŒA i 1ŒG (14.4) The matrices ŒA and ŒC refer to the discrete state matrix and discrete output matrix, respectively, from the stochastic state space model describing the problem: fznC1gDŒA fzngCfwng (14.5) fyngDŒC fzngCfvng (14.6) where fzng, fyng, fwng, and fvng correspond to the state vector, output vector, process noise and measurement noise, respectively. The matrix ŒG is formally defined as follows: ŒG DE fznC1gfyng T (14.7) After pre-multiplication with ŒW1 and post-multiplication with ŒW2 T, the block-Hankel matrix is decomposed using singular value decomposition (SVD), and thereafter truncated, as follows: ŒW1 ŒHi ŒW2 T D ŒU1 ŒU2 Œ†1 Œ0 Œ0 Œ†2 ŒV1 T ŒV2 T (14.8) ŒU1 ŒU2 Œ†1 Œ0 Œ0 Œ0 ŒV1 T ŒV2 T (14.9) DŒU1 Œ†1 ŒV1 T (14.10) The truncation above represents an approximation due to noise and imperfections in the system; a system of finite order will have non-zero values for Œ†2 in practice, and a manual specification of the order has to be made. The optimal order is not known a priori, and a stabilization plot is used to distinguish physical poles from spurious ones, using multiple truncation
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