Bott residue formula

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

1. Estimate the power spectral density matrix ${\hat{G}}_{yy}(j\omega )$ at discrete frequencies $\omega ={\omega }_i$.
2. 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 $s_{ij}$.
3. 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.

See also

• Eigensystem realization algorithm - an input/output identification technique

References

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