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

1. Estimate the power spectral density matrix [math]\displaystyle{ \hat{G}_{yy}(j\omega) }[/math] at discrete frequencies [math]\displaystyle{ \omega = \omega_i }[/math].
2. Do a singular value decomposition of the power spectral density, i.e. [math]\displaystyle{ \hat{G}_{yy}(j \omega_i) = U_i S_i U_i^H }[/math] where [math]\displaystyle{ U_i = [u_{i1},u_{i2},...,u_{im}] }[/math] is a unitary matrix holding the singular values [math]\displaystyle{ u_{ij} }[/math], [math]\displaystyle{ S_i }[/math] is the diagonal matrix holding the singular values [math]\displaystyle{ s_{ij} }[/math].
3. For an [math]\displaystyle{ n }[/math] degree of freedom system, then pick the [math]\displaystyle{ n }[/math] 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.

See also

• Eigensystem realization algorithm - an input/output identification technique

References

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