# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## 3 Simulated Data Analysis
### Lattice Tower Example
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**DOF/Mode** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline
1 & 0.4468 & \(-\)0.6874 & 0.3127 & 0.3392 & 0.6939 & 0.1024 \\
2 & \(-\)0.6972 & \(-\)0.4527 & \(-\)0.1825 & \(-\)0.6996 & 0.3094 & \(-\)0.0583 \\
3 & \(-\)0.1535 & 0.0972 & 0.6919 & 0.1836 & 0.1287 & \(-\)0.0468 \\
4 & 0.4489 & \(-\)0.6965 & 0.0014 & 0.3186 & 0.6843 & 0.0022 \\
5 & \(-\)0.7059 & \(-\)0.4370 & 0.3573 & \(-\)0.7115 & 0.3235 & 0.1151 \\
6 & 0.1269 & 0.1106 & 0.6946 & \(-\)0.1849 & 0.0980 & \(-\)0.0492 \\ \hline \end{tabular}
\end{table} 
Table 6: Real modes of the lattice tower structure used in simulation examples. \begin{table}
\begin{tabular}{c c c c c} \hline
**Mode** & **Type** & \(f_{m_{i}}\) [Hz] & \(\xi_{m_{i}}\) [\%] & \(m_{i}\) [Kg] \\ \hline
1 & 1 st bending mode in Y direction (BY1) & 1.2869 & 1.0 & 2608.8271 \\ \hline
2 & 1 st bending mode in X direction (BX1) & 1.2937 & 1.0 & 2592.8286 \\ \hline
3 & 1 st torsional mode (T1) & 2.2251 & 1.0 & 281.0965 \\ \hline
4 & 2 nd bending mode in Y direction (BY2) & 3.8713 & 1.0 & 1431.1279 \\ \hline
5 & 2 nd bending mode in X direction (BX2) & 3.8932 & 1.0 & 1410.8619 \\ \hline
6 & 2 nd torsional mode (T2) & 6.1745 & 1.0 & 54.2551 \\ \hline \end{tabular}
\end{table} 
Table 7: Modal configurations of the FE model of the lattice structure. Figure 13: Sensors and measured directions. the ML estimator needs to iterate more times to reach convergence. Similarly to the previous application example, the optimization with the MLE-PMM was carried out using again the full frequency band, i.e., with no out-of-band modes. The variation of the natural frequency and damping ratio estimates for modes 2 and 5 obtained both with the LSCF and MLE-PMM over the Monte Carlo simulations are shown in Figures 16 and 17 and the results obtained in terms of relative bias error for all the identified modes are synthesized in Table 8.  
Figure 16: Monte Carlo simulation results for the 2 nd identified natural frequency (**a**) and damping ratio (**b**): exact quantity (solid black line), LSCF estimates (blue dots) and MLE-PMM estimates after 20 iterations (red dots). Figure 14: Element(1,1) of the FRF matrix contaminated with 5% noise: exact and noisy FRF, noise standard deviation and exact natural frequencies (vertical lines) (**a**); and details of the close spaced modes around 1.29 Hz (**a**) and 3.88 Hz (**b**). Figure 15: Typical stabilization diagram constructed with the LSCF for model orders ranging from 30 to 60. By comparing the results obtained before and after the optimization with MLE-PMM shown in this table, it is verified that no significant reduction in terms of the bias error was obtained after 20 iterations of the MLE-PMM with regard to the LSCF estimates. The mean of standard deviation estimates for the natural frequencies and damping ratios of modes 2 and 5 predicted by the proposed MLE-PMM after 20 iterations are compared to their respective sample standard deviations in Figures 18 and 19. By inspecting these figures, one verifies that, even in case of closely spaced modes, the MLE-PMM provides very good estimates for the sample standard deviations. These results are also verified in Figure 20 and Table 9, where the sample standard deviations of all the identified natural frequencies and damping ratios are compared to those predicted by the MLE-PMM. In Figures 18 and 19 and in Table 9, the estimates for the standard deviations provided by the MLE-PMM are also compared to the estimates obtained with MLE-MM proposed by Reference [14] after 20 iterations. From this comparison it is concluded that, even though the MLE-PMM provides very accurate estimates for the uncertainty bounds of its optimized modal parameter estimates, these uncertainty intervals are significantly higher than those provided by the MLE-MM. This leads to the conclusion that, in case of closely spaced modes, the optimisation with the MLE-PMM is not as efficient as in case of reasonably or well separated modes.