## 3 Numerical Simulations  
### Numerical Model  
A four-DOF linear system is considered in this paper as shown in Figure 1. The mass, stiffness and damping of each story are 200 kg, 7200 N/m, and 120 N-s/m, respectively. Assuming the structure is excited by the Hanshin earthquake wave, which has the peak ground acceleration (PGA) of 0.5 g as shown in Figure 2, the motion equation of the structure can be written as:  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=-\mathbf{M}\mathbf{I}_{4\times 1} \ddot{\mathbf{g}} \tag{14}\]  
\[\mathbf{M}=\left[\begin{array}{cccc}m_{1}&&&\\ &m_{2}&&\\ &&m_{3}&\\ &&&m_{4}\end{array}\right]\]  
\[\mathbf{K}=\left[\begin{array}{cccc}k_{1}+k_{2}&-k_{2}&&\\ -k_{2}&k_{2}+k_{3}&-k_{3}&\\ &-k_{3}&k_{3}+k_{4}&-k_{4}\\ &&&-k_{4}&k_{4}\end{array}\right]\]  
\[\mathbf{C}=\left[\begin{array}{cccc}c_{1}+c_{2}&-c_{2}&&\\ -c_{2}&c_{2}+c_{3}&-c_{3}&\\ &&-c_{3}&c_{3}+c_{4}&-c_{4}\\ &&&-c_{4}&c_{4}\end{array}\right]\]  
\[\mathbf{x}=\left[\begin{array}{cccc}x_{1}&x_{2}&x_{3}&x_{4}\end{array}\right]^{T}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices, respectively, and vectors \(\mathbf{x}\), \(\dot{\mathbf{x}}\) and \(\ddot{\mathbf{x}}\) denote the displacement, velocity, and acceleration response, respectively; \(\ddot{g}\) and \(\mathbf{I}_{4\times 1}\) are the earthquake acceleration and four-order unit column vector, respectively; \(-\mathbf{M}\ddot{g}\) is the equivalent external force vector caused by the base movement. The dynamic response can be calculated using Newmark-\(\beta\) method (\(\gamma\) is 1/2, \(\beta\) is 1/6, and the time step is 0.02 s).  
Figure 1: Schematic diagram of the four-degree-of-freedom system.  
In this paper, the observation vector is the absolute acceleration of structure which can be expressed as:  
\[\ddot{\mathbf{x}}_{a}=\ddot{\mathbf{x}}+\mathbf{I}_{4\times 1}\ddot{\mathbf{g}} \tag{15}\]  
The stiffness \(k_{i}\) (\(i\) = 1, 2, 3, 4) and damping \(c_{i}\) (\(i\) = 1, 2, 3, 4) are the system parameters to be identified. Therefore, the augmented state vector can be expressed as:  
\[\mathbf{X}=\left[\begin{array}{ccc}\mathbf{x}&\dot{\mathbf{x}}&\Theta\\ \end{array}\right]^{T} \tag{16}\]  
where \(\Theta=\left[\begin{array}{cccccc}k_{1}&\cdots&k_{4}&c_{1}&\cdots&c_{4}\\ \end{array}\right]^{T}\).  
Then, the system equation and the observation equation can be written as:  
\[\mathbf{X}_{k+1}=\mathbf{X}_{k}+\int_{t_{k}}^{t_{k+1}}f\big{(}\mathbf{X},\ddot{\mathbf{g}} \big{)}dt+\mathbf{w}_{k} \tag{17}\]  
\[f\big{(}\mathbf{X},\ddot{\mathbf{g}}\big{)}=\left\{\begin{array}{c}\dot{\mathbf{x}}\\ -I\ddot{\mathbf{g}}-M^{-1}\Big{[}(\mathbf{C})_{c_{i}}\dot{\mathbf{x}}+(\mathbf{K})_{k_{i}}\mathbf{ x}\Big{]}\\ \mathbf{0}_{8\times 1}\end{array}\right\}\]  
\[\ddot{\mathbf{y}}_{a}=\mathbf{h}(\mathbf{X})+\mathbf{v} \tag{18}\]  
\[\mathbf{h}(\mathbf{X})=-\mathbf{M}^{-1}\Big{[}(\mathbf{C})_{c_{i}}\dot{\mathbf{x}}+(\mathbf{K})_{k_{i }}\mathbf{x}\Big{]}\]  
In the actual operation, the system equation has the system noise, and the observation equation also has the observation noise (observation error). Therefore, the covariance matrix of the system noise and the observed noise are assumed as follows: \(\mathbf{Q}=10^{-5}\cdot\mathbf{I}_{16}\), and \(\mathbf{R}=\mathbf{I}_{4}\). Moreover, the initial state vector \(\mathbf{\hat{X}}_{0}=\left[\begin{array}{cccc}0.01\cdot\mathbf{I}_{4\times 1}&0.01 \cdot\mathbf{I}_{4\times 1}&0.7k\cdot\mathbf{I}_{4\times 1}&0.7c\cdot\mathbf{I}_{4 \times 1}\\ \end{array}\right]^{T}\) and the covariance matrix \(\hat{P}_{0}=10^{9}\cdot\mathbf{I}_{16}\) are given. With this, the stiffness and damping parameters of the four-DOF system can be identified by the EKF algorithm.