to deal with the cases where there are less number of sensors available than the number modes to be estimated. The suggested approach therefore is a step towards expanding the applicability of BSS based approaches to Operational Modal Analysis applications. 1. Introduction Several recent works have shown how second order blind source separation (SO-BSS) techniques, such as Second Order Blind Identification (SOBI) [1, 2], can be utilized for the purpose of output-only modal analysis [3-7]. However, in spite of encouraging results shown by these algorithms, applicability of BSS based algorithms for OMA purposes has been quite limited. This can be attributed to certain limitations associated with BSS based algorithms for OMA. Mathematical formulation of SO-BSS based OMA algorithms suggest that they are more suitable for lightly damped systems having real normal modal vectors. The fact that SO-BSS algorithms only estimate real modal vectors is a serious issue as this is seldom a case in real life. A methodology based on Hilbert transform is suggested in [8] to estimate complex mode shapes. However, the robustness of the method is yet to be ascertained. Yet another issue which restricts applicability of these algorithms is that they can only estimate as many modes as the number of output responses being measured. Despite these limitations SO-BSS techniques present an interesting outlook with regards to operational modal analysis, as they differ from traditional OMA algorithms in terms of estimating the modal parameters of a system. In [9], the authors showed how SO-BSS algorithms, such as SOBI, are related to well known Stochastic Subspace Identification (SSI) algorithm [10, 11]. In this work it was shown that whereas SSI estimate the modes of a system by putting a constraint on the poles of the system, SO-BSS based algorithms use a joint diagonalization procedure to obtain the modal parameters by estimating orthogonal vectors that diagonalize the correlation matrices of observed responses. These orthogonal vectors are estimates of the modal vectors which are then used to obtain modal frequencies and damping by means of modal expansion theorem [12]. Since the two algorithms, SSI and SO-BSS, that share similar mathematical foundations, estimate modal parameters using different approaches, it is intriguing to pursue an algorithm that can combine the advantages of both the algorithms. In this paper, an Alternative Least Squares (ALS) [13] based algorithm is proposed that combines the advantages of SO-BSS and SSI algorithms in order to overcome the limitations SO-BSS algorithms suffer from in terms of their application for OMA purposes. This algorithm can be explained within the framework of Parallel Factor (PARAFAC) theory [14]. Mathematical development of this algorithm is presented in the next section and preliminary results of this algorithm are shown in Section 3 by means of studies conducted on an analytical system. Finally, conclusions are made, in light of the results obtained and performance of the algorithm in general, with regards to its further development and suitability for OMA applications. 2. A PARAFAC based BSS Algorithm for OMA 2.1 General Background This section recalls the main results and notations used in [8] as these results serve as the background and main motivation for the ALS based PARAFAC algorithm suggested in this paper. Consider the following n degree of freedom (DOF) discrete-time state-space system, + + = + = + [ ] [ ] [ ] [ ] [ ] [ ] [ 1] k k k k k k k d d d d ν y C x D f A x B f x (1) where x[k] is the 2 1×n state vector, y[k] the 1×m measurement vector, f[k] the unknown force vector, ν[k] the measurement noise vector, and Ad the n n 2 2× state matrix. Let 1− = ΣΨ Ψ A d d (2) be the eigenvalue decomposition of the state-matrix where the n n 2 2× diagonal matrix, where { } { } Σ = × × s n n n n s d T T * exp exp Λ 0 0 Λ (3) (Ts = sampling period and *= conjugate operator) contains the modal parameters of interest, ( ) n diag λ λ ,..., 1 =Λ with λi the i-th pole of the system, and 180
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