Frequency domain decomposition

The frequency domain decomposition (FDD) is an output-only system identification technique popular in civil engineering, in particular in structural health monitoring. As an output-only algorithm, it is useful when the input data is unknown. FDD is a modal analysis technique which generates a system realization using the frequency response given (multi-)output data.

Algorithm

Estimate the power spectral density matrix ${\hat{G}}_{y y} (j ω)$ at discrete frequencies $ω = ω_{i}$ .
Do a singular value decomposition of the power spectral density, i.e. ${\hat{G}}_{y y} (j ω_{i}) = U_{i} S_{i} U_{i}^{H}$ where $U_{i} = [u_{i 1}, u_{i 2}, ..., u_{i m}]$ is a unitary matrix holding the singular vectors $u_{i j}$ , $S_{i}$ is the diagonal matrix holding the singular values $s_{i j}$ .
For an n degree of freedom system, then pick the n dominating peaks in the power spectral density using whichever technique you wish (or manually). These peaks correspond to the mode shapes.
1. Using the mode shapes, an input-output system realization can be written.

1 2 Brincker, R.; Zhang, L.; Andersen, P. (2001). "Modal identification of output-only systems using frequency domain decomposition" (PDF). Smart Materials and Structures. 10 (3): 441. Bibcode:2001SMaS...10..441B. doi:10.1088/0964-1726/10/3/303. S2CID 250917814.
↑ Brincker, R.; Zhang, L.; Andersen, P. (February 7–10, 2000). "Modal Identification from Ambient Response Using Frequency Domain Decomposition" (PDF). Proc. of the 18th International Modal Analysis Conference. San Antonio, TX. Retrieved March 11, 2012.

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