19 Subspace-Based Damage Detection on Steel Frame Structure Under Changing Excitation 169 into the matrices of observability and controllability OpC1 D 2 6 6 6 4 C CA : : : CAp 3 7 7 7 5 ; Cq D GAG : : : Aq 1G : (19.5) From the observability matrix OpC1, the matrices C and A could be recovered [10, 13] and subsequently the modal parameters. However, the fact is used that damages lead to changes in A and C and subsequently in HpC1;q through properties (19.4) and (19.5), which will be directly checked in a statistical test, instead of doing system identification. 19.2.3 The Damage Detection Test In the following, the non-parametric damage detection test [1] based on [2,3] is described. Using measured data .yk/kD1;:::;N, a consistent estimate OHpC1;q of the Hankel matrix is obtained from the empirical output covariances ORi D 1 N N X kD1 yky T k i ; OHpC1;q DHank ORi : (19.6) Let OHref pC1;q be the Hankel matrix in the reference state. Compute its left null space S from the singular value decomposition (SVD) OHref pC1;q D OU1 OU0 O 1 0 0 O 0 OVT 1 OVT 0 (19.7) as S D OU0, where O 1 is of size n nandwhere O 0 0. The characteristic property of the reference state then writes as ST OHpC1;q 0; (19.8) while the product deviates from 0 in the damaged state. To decide whether measured data corresponds to the reference state or not, the residual vector with D pNvec ST OHpC1;q (19.9) is defined [2, 3]. It is tested if this residual is significantly different from zero or not, using the 2 test 2 D T† 1 ; (19.10) where the empirical residual covariance † Dcov. / is computed on several datasets from the reference state. To decide if damage occurred or not, the test value 2 is compared to a threshold, which can also obtained from 2 test values on several datasets from the reference state. 19.3 Robust Damage Detection Test Under Changing Excitation A change in the covariance Qof the unmeasured ambient excitation vk of system (19.2) provokes a change in the crosscovariance G between the states and the outputs and thus in the Hankel matrix estimate OHpC1;q (see (19.3)), even if no structural change occurs. Hence, the residual and the corresponding test value 2 are influenced by changes in the ambient excitation, which may lead to false alarms.
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