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System Identification Methods for (Operational) Modal Analysis: Review and Comparison

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Abstract

Operational modal analysis deals with the estimation of modal parameters from vibration data obtained in operational rather than laboratory conditions. This paper extensively reviews operational modal analysis approaches and related system identification methods. First, the mathematical models employed in identification are related to the equations of motion, and their modal structure is revealed. Then, strategies that are common to the vast majority of identification algorithms are discussed before detailing some powerful algorithms. The extraction and validation of modal parameter estimates and their uncertainties from the identified system models is discussed as well. Finally, different modal analysis approaches and algorithms are compared in an extensive Monte Carlo simulation study.

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Notes

  1. The term realization originates from analog computer realizations for the integration of differential equations [41]. Since the integrator outputs can be defined as the states of a state-space model [42, p. 63], constructing a state-space model is usually called realization in the wide sense. System realization in the strict sense is the identification of a state-space description from an impulse response sequence [43]. In this text, both denotations are frequently used.

  2. The linearity properties can be defined well in the behavioral approach to system theory [44], while the standard definition of linearity, viz., applying a linear combination of inputs yields a response that is the same linear combination of the responses to the individual inputs, can yield problems [42, 45]. However, the standard definition will do for our purposes; we refer to [42, 45] for a detailed discussion.

  3. Mathematically because δ(t)δ(t) is undefined [46, p. 171].

  4. This is possible since both sides of (41) can be left-multiplied with \(\boldsymbol {A}_{n_{a}}^{-1}\).

  5. This is possible by a change of basis: \(\boldsymbol {x}' = \boldsymbol {A}_{n_{a}}^{-1}\boldsymbol {x}'_{new}\).

  6. Note that \(\boldsymbol {y}_{k}^{s}\) is straightforwardly obtained by subtracting the deterministic part \(\boldsymbol {y}_{k}^{d}\) from y k .

  7. Note that, when taking the double-sided z transform, no initial conditions are taken into account. Therefore, the Kalman filter is time-invariant, i.e., independent of z, as discussed in Sect. 2.4.3.

  8. Sometimes a block Toeplitz structure is proposed instead of a block Hankel structure, see for instance [122]. This is just a matter of notation: the resulting system estimate remains the same.

  9. (u k ) is said to be persistently exciting of order k 2 when \(\boldsymbol {U}_{k_{1}|k_{1}+k_{2}-1}\) has full row rank.

  10. This can be relaxed to ℓ≥n when the direct transmission is nonzero, i.e., when the outputs are accelerations.

  11. If y d represents the deterministic part of a channel and y s the stochastic part, the SNR is calculated in dB as \(20 \log( \frac{\mathrm{RMS}(y^{d})}{\mathrm{RMS}(y^{s})})\).

  12. LMS PolyMAX is a registered trademark of LMS International.

  13. If ȷ≠ȷ B n B , the last block can contain the additional ȷ−ȷ B n B columns. The derivation still holds when this block is scaled with ȷ B +(ȷ−ȷ B n B ) instead of ȷ B .

  14. Since the stochastic outputs are assumed to have a zero mean value, their correlation matrices equal their covariance matrices, hence the name SSI-cov.

  15. Most subspace algorithms use the estimator for \(\hat{\boldsymbol {X}}_{\imath+1|\imath+1}^{s}\), where \(\underline{\hat{\boldsymbol {\mathcal{O}}}_{\imath}}\) was defined in (191). However, with this choice, the estimates of \(\hat{\boldsymbol {X}}_{\imath|\imath}^{s}\) and \(\hat{\boldsymbol {X}}_{\imath+1|\imath+1}^{s}\) are not guaranteed to lie in the same state-space basis, and this introduces an additional estimation error for finite ȷ [162].

  16. The persistency of excitation condition on the input can be relaxed to be of order 2ı+n instead of 2(ı+ℓ)+n.

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Acknowledgements

I would like to thank G. De Roeck, G. Degrande, W. Heylen and B. De Moor from K.U.Leuven, R. Pintelon and P. Guillaume from Vrije Universiteit Brussel, Á. Cunha and F. Magalhães from University of Porto, and B. Peeters from LMS International for helpful discussions.

This research was supported by the Research Foundation—Flanders (FWO), Belgium, under Research Project G.0343.04 and a Postdoctoral Research Fellowship.

The present paper quotes text passages, figures and tables from E. Reynders, System identification and modal analysis in structural mechanics, PhD thesis, Faculty of Engineering, K.U.Leuven, 2009. This material is reproduced here with permission.

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    Edwin Reynders

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Reynders, E. System Identification Methods for (Operational) Modal Analysis: Review and Comparison. Arch Computat Methods Eng 19, 51–124 (2012). https://doi.org/10.1007/s11831-012-9069-x

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