## 4 Numerical Investigations
### Tuning Process Using Particle Swarm Optimization
In this case, the total number of particles is assumed as 81 for 2-dimentional problem. There may be more efficient alternatives than using these recommendations. However, it is essential to perform PSO iteratively to find a near-optimal solution because of intrinsic randomness in PSO. Table 3 presents results of PSO using 81 particles about the damage scenarios with the selected boundary conditions given as [lower bound, upper bound]. In Scenarios 3 and 4, PSO was re-performed because the results were located at one of the boundaries. With the tuning parameters obtained by PSO, the MS-KF\({}_{\text{ME}}\) is conducted for the scenarios. For normal-UKF (n-UKF) and adaptive tracking technique-based UKF (AUKF) in Appendix A, PSO using Equation (15) (with the variable \(x_{2}\) excluded) is performed to determine \(x_{1}\) as 3.23 and 3.88, respectively. MS-KF\({}_{\text{PE}}\) (in Appendix B) was not performed due to the difficulty in selecting the size of time-window. Figure 5a-c shows the estimation results by the proposed method, n-UKF and AUKF, respectively, under Scenario 1. From the results, it is confirmed that, while the results of n-UKF and AUKF show biased estimations, the MS-KF\({}_{\text{ME}}\) derives accurate estimations about all parameters. Although divergence in n-UKF and AUKF was avoided by adopting the regularization term into state-space model, those filters still converged into wrong values. These results show that filter methods, which control artificial process noise, cannot adapt to measurements, whose statistics change due to damage. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{
\begin{tabular}{c} **Results of PSO** \\ (\(x_{1}\),\(x_{2}\)) \\ \end{tabular} } & \multicolumn{2}{c}{**Initial Boundary**} & \multicolumn{2}{c}{**Changed Boundary**} \\ \cline{3-6}  & & **for \(x_{1}\)** & **for \(x_{2}\)** & **for \(x_{1}\)** & **for \(x_{2}\)** \\ \hline Scenario 1 & (3.51, 2.33) & [2, 4] & [1, 5] & - & - \\ Scenario 2 & (3.81, 2.19) & [2, 4] & [1, 5] & - & - \\ Scenario 3 & (3.98, \(\lx@overaccentset{{\circ}}{2}\)) (3.88, 6.99) & [2, 4] & [1, 5] & - & [3, 7] \\ Scenario 4 & (\(\lx@overaccentset{{\circ}}{4}\), 1.98) (4.21, 1.73) & [2, \(\lx@overaccentset{{\circ}}{4}\)] & [1, 5] & [3, 5] & - \\ Scenario 5 & (3.95, 2.68) & [2, \(\lx@overaccentset{{\circ}}{4}\)] & [1, 5] & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Tuning parameters identified by particle swarm optimization. Figure 5: Estimation by MS-KFME (proposed), n-UKF, AUKF: Scenario 1 with RMS-noise 5%. Next, more complicated sequential damage scenarios are considered to test stable tracking-capability of the proposed method. The PSO's result for Scenario 4 (the second story experiences the additional sudden damage after the initial damage) was derived at one of the boundaries. Thus, PSO was re-performed to find new tuning parameters reported in Table 3. The estimations with the newly derived tuning parameters are presented in Figure 8. In this figure, the first story's parameter is not assumed to be damaged. By the way, the range from around 10 to 25 s shows slightly biased estimation of the first story, and this bias is being severe before the estimation approaches to the exact value at around 35 s. This is probably because the corresponding acceleration for the damaged third story is not measured, and the damage occurred at adjacent stories. It may be a burden for slave filter to infer accurate information based on only innovations from master filter without structural knowledge. However, as mentioned in Section 3.2, the MS-KFME aims to obtain stable estimations from sparse measurements by searching solution space gradually rather than rapid tracking. The results confirm that the estimations about adjacent stories are distinguishable and converged to accurate values respectively after the transient period. Figure 8: Estimation by MS-KFME: Scenario 4 with RMS-noise 5%. Figure 7: Estimation by MS-KFME: Scenario 3 with RMS-noise 5%.