The modal characteristics (μ,ψμ) are given by the eigenstructure (λ,Φλ) of F: eδμ =λ ψμ =φλ Δ =HΦλ In the sequel the dimension of the observed output Y is much smaller than the dimension of the state X. 2.2 Identification Procedure Knowing the output data Yk at the time instants k =1,...,N, the eigenstructure (λ,φλ) of system (1) is identified with Stochastic Subspace Identification algorithms. In this work, the covariance driven approach [2, 7] and data driven approach with the Unweighted Principal Component algorithm [6, 7] are used. For both approaches the parameters p and q are chosen as variables with (p+ 1)r ≥qr ≥n with the desired model order n. Usually, p+1=q is set [1]. The data matrices Y+ p+1 def = ⎛ ⎜⎜⎜ ⎜⎜⎜ ⎝ Yq+1 Yq+2 .. . YN −p Yq+2 Yq+3 .. . YN −p+1 .. . .. . .. . .. . Yq+p+1 Yq+p+2 .. . YN ⎞ ⎟⎟⎟ ⎟⎟⎟ ⎠ , and Y−q def = ⎛ ⎜⎜⎜ ⎜⎜⎜ ⎝ Yq Yq+1 .. . YN −p−1 Yq −1 Yq .. . YN −p−2 .. . .. . .. . .. . Y1 Y2 .. . YN −p−q ⎞ ⎟⎟⎟ ⎟⎟⎟ ⎠ (2) are built, and, according to the method, a subspace matrix as follows: • For the covariance drivenapproach, the subspace matrix H cov p+1,q def =Y+ p+1Y−q T is built. It has the factorization property H cov p+1,q =Op+1 F Cq with the matrix of observability Op+1 =⎛ ⎜⎜⎜ ⎝ H HF.. . HFp ⎞ ⎟⎟⎟ ⎠ and the matrix of controllability Cq. Confidence Intervals of Modal Parameters during Progressive Damage Test 239
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