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.^[1]^[2]

Algorithm

Estimate the power spectral density matrix ${\hat {G}}_{{yy}}(j\omega )$ at discrete frequencies $\omega =\omega _{i}$ .
Do a singular value decomposition of the power spectral density, i.e. ${\hat {G}}_{{yy}}(j\omega _{i})=U_{i}S_{i}U_{i}^{H}$ where $U_{i}=[u_{{i1}},u_{{i2}},...,u_{{im}}]$ is a unitary matrix holding the singular values $u_{{ij}}$ , $S_{i}$ is the diagonal matrix holding the singular values $s_{ij}$ .
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]
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". Smart Materials and Structures. 10 (3): 441. doi:10.1088/0964-1726/10/3/303.
↑ 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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