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 G ^ y y ( j ω ) {\displaystyle {\hat {G}}_{yy}(j\omega )} at discrete frequencies ω = ω i {\displaystyle \omega =\omega _{i}} .
- Do a singular value decomposition of the power spectral density, i.e. G ^ y y ( j ω i ) = U i S i U i H {\displaystyle {\hat {G}}_{yy}(j\omega _{i})=U_{i}S_{i}U_{i}^{H}} where U i = [ u i 1 , u i 2 , . . . , u i m ] {\displaystyle U_{i}=[u_{i1},u_{i2},...,u_{im}]} is a unitary matrix holding the singular vectors u i j {\displaystyle u_{ij}} , S i {\displaystyle S_{i}} is the diagonal matrix holding the singular values s i j {\displaystyle s_{ij}} .
- For an
n
{\displaystyle n}
degree of freedom system, then pick the
n
{\displaystyle n}
dominating peaks in the power spectral density using whichever technique you wish (or manually). These peaks correspond to the mode shapes.3
- Using the mode shapes, an input-output system realization can be written.
See also
- Eigensystem realization algorithm - an input/output identification technique
References
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. http://vbn.aau.dk/ws/files/203990023/Modal_Identification_of_Output_Only_Systems_using_Frequency_Domain_Decomposition.pdf ↩
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. http://www.svibs.com/solutions/literature/2000_2.pdf ↩
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. http://vbn.aau.dk/ws/files/203990023/Modal_Identification_of_Output_Only_Systems_using_Frequency_Domain_Decomposition.pdf ↩