## 4 Numerical Investigations  
### SI under Simulated Damage Scenarios  
To test the combination of the selected dual adaptive filtering techniques and the proposed state-space model based on regularization, a shear building having 6 DOFs in Figure 4 is investigated. The lumped mass of each floor is assumed as 1 kg, and Young's modulus of all inter-story is equal to 1 Pa (i.e., inter-story stiffness of each floor is 24 N/m). Rayleigh damping, which is proportional to mass and stiffness matrices, is utilized by assuming a damping ratio of 5% for the first and second modes. For the linear system, Young's modulus of each component is estimated using an EQ ground motion. The time-history of the 180-degree component of ground acceleration recorded at El Centro station during the earthquake event in 1940 with M\({}_{\rm w}\) = 6.95 is used in this numerical investigation. The length of this ground acceleration history is 42 s, and discretized by 0.01 s. To consider potential damage scenarios, the EQ time-history is first applied to the shear building without damage events. The relative displacements between adjacent stories are calculated to find the peak values at each story and the occurrence-time as shown in Table 1.  
Figure 3: Algorithm of Master-Slave Kalman Filter for noise measurement estimation.  
Based on the simulated sequence of peak displacements, five damage scenarios are created in terms of sudden decrease of [PERSON]'s moduli as summarized in Table 2. Scenarios 1 and 2 depict sequence of sudden decreases of story stiffness. Scenario 3 simulates a sequential failure initiated by the maximum-peak response at 5 th story. Scenarios 4 and 5 are assumed to depict more complicated sequential scenarios. In Scenario 4, the second story experiences the additional sudden decrease 7.42 s after the initial decrease of corresponding story's stiffness. Scenario 5 consists of three sudden decreases of different stories. After simulating the above scenarios, the condition of sparse measurements is defined such that accelerations at the first, second, fourth, and sixth story can be measured. From this assumption, the selection matrix \(\mathbf{S}\) will be set up as \(\mathbf{S}_{13}=\mathbf{S}_{23}=\mathbf{S}_{43}=\mathbf{S}_{63}=1\) and \(\mathbf{S}_{ij}=0\) otherwise. Then, the noise-history, which is simulated from a uniform distribution between 95% and 105% of RMS of measured accelerations, is added to simulate noise in the measurements.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario No.** & **Description of Scenario** \\ \hline Scenario 1 & \(E_{2}\) decreases by 25\% at 4.94 s & \(E_{1}\) decreases by 33\% at 8.58 s \\ Scenario 2 & \(E_{4}\) at 7.50 s (25\%) & \(E_{1}\) at 8.58 s (33\%) \\ Scenario 3 & \(E_{5}\) at 9.92 s (28\%) & \(E_{2}\) at 14.89 s (24\%) \\ Scenario 4 & \(E_{3}\) at 5.23 s (24\%) & \(E_{2}\) at 8.58 s (24\%) & \(E_{2}\) at 16.00 s (41\%) \\ Scenario 5 & \(E_{6}\) at 6.15 s (25\%) & \(E_{4}\) at 7.50 s (22\%) & \(E_{1}\) at 8.58 s (33\%) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Damage scenarios based on simulated peak relative displacements in Table 1.  
Figure 4: Schematic view of a shear building.