Improved modal parameter estimation for lowly damped systems using non-parametric exponential windowing techniques

Estimation of nonparametric noise and FRF models for multivariable systems—Part II: Extensions, applications

“…with ZðkÞ ¼ YðkÞ The FRF estimate (16) is also known as the instrumental variables estimator [14,15]. The drawback of this approach is that the part of the true system input u 1 ðtÞ (see Fig.…”

“…(5) and (6) applied to model (14)). The estimateŷ SML has the following asymptotic ðF ¼ OðNÞ-1Þ properties: (i) if M in (7) satisfies MZn y þ 3 then, in the absence of modelling errors,ŷ SML is consistent (converges to the true value); and (ii) if MZn y þ 9 thenŷ SML is asymptotically normally distributed (for single output systems, n y ¼ 1, a less conservative bound MZ7 can be derived [7]) and (11) remains valid (proof: see Appendix F).…”

“…and their uncertainty (see, for example [17]). If the noise leakage is the dominant noise error in (14), then, following the lines of Section 3.2, it is removed in the sample meanẐ ðkÞ and sample covariance matrixĈẐ ðkÞ. This is done by applying (12) to the errors-in-variables case (14).…”

“…If the noise leakage is the dominant noise error in (14), then, following the lines of Section 3.2, it is removed in the sample meanẐ ðkÞ and sample covariance matrixĈẐ ðkÞ. This is done by applying (12) to the errors-in-variables case (14). In (18) and (19)Ẑ ðkÞ,ĈẐ ðkÞ, and T G ðO k ; yÞ are then replaced by, respectively,Ẑ T ðkÞ,ĈẐ T ðkÞ, and 0.…”

“…The bottom plot of Fig. 9 illustrates the influence of the nonparametric leakage removal (12) on the estimates in case the system and noise leakage errors can be neglected in (14). It can be seen that the increase is smaller than 1 dB and that it decreases with increasing values of m. Hence, the nonparametric leakage removal (12) is recommended as the standard choice.…”