Decentralized modal identification of structures using parallel factor decomposition and sparse blind source separation

In this paper, a novel decentralized modal identification method is proposed utilizing the concepts of sparse blind source separation (BSS) and parallel factor decomposition. Unlike popular ambient modal identification methods which require large arrays of simultaneous vibration measurements, the decentralized algorithm presented here operates on partial measurements, utilizing a sub-set of sensors at-a-time. Mathematically, this leads to an underdetermined source separation problem, which is addressed using sparsifying wavelet transforms. The proposed method builds on a previously presented concept by the authors, which utilizes the stationary wavelet packet transform (SWPT) to generate an over-complete dictionary of sparse bases. However, the redundant SWPT can be computationally intensive depending on the bandwidth of the signals and the sampling frequency of the vibration measurements. This issue of computational burden is alleviated through a new method proposed here, which is based on a multi-linear algebra tool called PARAllel FACtor (PARAFAC) decomposition. At the core of this method, the wavelet packet decomposition coefficients are used to form a covariance tensor, followed by PARAFAC tensor decomposition to separate the modal responses. The underdetermined source identifiability of PARAFAC enables source separation in wavelet packet coefficients with considerable mode mixing, thereby relaxing the conditions to generate over-complete bases, thus reducing the computational burden. The proposed method is validated using a series of numerical simulations followed by an implementation on recorded ambient vibration measurements obtained from the UCLA factor building.