## 3 Numerical Case Study
### Noise Robustness  
Noise robustness of the CVA, UPC, and PC subspace algorithms are evaluated by plotting the extracted pole values. The obtained poles of a damped system are in the form of complex values. Though \"complex poles\" is a common term in analytical and experimental modal analysis, there is not any unified theory to normalize the obtained complex results [44; 45]. Complex poles in a linear system contain real and imaginary parts. In an undamped system the real component of the complex modes are zero and the poles fall on imaginary axis. In the numerical model, the robustness of the subspace algorithms to noise uncertainty is evaluated by oscillation pattern of pole values in the noise-induced and reference states. Figure 5 shows the first four orders of the system poles in the noise-free setting. The poles of the system are in the form of complex conjugates for each of the four orders. Hence, \(\lambda_{1},\ \lambda_{2},\ \lambda_{3}\), and \(\lambda_{4}\) are the first orders of the numerical model and \(\lambda_{1}^{*},\ \lambda_{2}^{*},\ \lambda_{3}^{*}\), and \(\lambda_{4}^{*}\) are their conjugates. In a separate experiment incorporating noise ratios ranging from zero to 30% it was shown that he noise ratio of 5% provided a clear image transition of the patterns from the aggregated into the oscillated pattern. Hence the noise ratio of 5% was selected for plotting the pole analysis in the numerical model.  
The identified results of the first four-orders using different SSI algorithms are plotted in complex plane as shown in Figure 7 (The orders 1 and 2 and orders 3 and 4 are shown in separate figures). In order to provide the required data for this subsection, the response of the simulation model with the  
Figure 6: The variance accounted for (VAF) values for (**a**) CVA, (**b**) UPC, and (**c**) PC algorithms.  
sampling frequency of 500 Hz was used for pole analysis. One hundred sets of different noise patterns were induced to the extracted response signal for evaluation of noise robustness.  
Figure 8 shows the oscillation patterns of complex poles identified using the PC, UPC, and CVA. The conjugate pairs of the obtained pole values for the numerical model are symmetrically distributed about the real axes.  
As it can be seen in the figure, aggregated complex pole values of the reference state can be traced from those of spattered noise-induced poles. The scattering of pole values from the reference state is  
Figure 8: The estimated poles of the numerical model for: (**a**) 1 st and 2 nd and (**b**) 3 rd and 4 th orders in PC algorithm. (**c**) 1 st and 2 nd and (**d**) 3 rd and 4 th orders in UPC algorithm. (**e**) 1 st and 2 nd and (**f**) 3 rd and 4 th orders in CVA algorithm.  
Figure 7: The poles of noise-free simulation model in complex plane.  
similar in UPC (Figure 8a,b) and PC (Figure 8c,d) algorithms whereas pole values obtained by UPC are slightly less affected by the noise variation). The CVA (Figure 8e,f) algorithm has the highest scattering among all subspace algorithms. The obtained result of the subspace algorithms show that the PC has the highest robustness against noise and for UPC algorithm is more affected by noise inclusion, while the CVA has the lowest robustness against noise in the conducted study.