## 2 Methodology
### MEMD-Based Identification Methods
#### 2.2.1 MEMD-Based SSI (MSSI) Method  
Numerous variants of the SSI method are developed, which are mainly based on two visions (i.e., the data-driven SSI (SSI-DATA) method and the covariance-driven SSI method). In this research, the latter is used to identify the modal parameters. For a linear system, after modeling under white noise excitation, the dynamic response (i.e., acceleration response) with the discrete time can express in the following state-space equation [26; 27; 28]:  
\[\mathbf{X}_{\mathbf{k}+\mathbf{I}}=\mathbf{A}\mathbf{X}_{\mathbf{k}}+\mathbf{w}_{\mathbf{k}},\,\mathbf{y}_{\mathbf{k}}= \mathbf{C}\mathbf{X}_{\mathbf{k}}+\mathbf{v}_{\mathbf{k}} \tag{2}\]  
in which \(\mathbf{A}\) and \(\mathbf{C}\) are the state and output matrix, respectively; \(\mathbf{X}_{\mathbf{k}}\in R^{n\times 1}\) and \(\mathbf{y}_{\mathbf{k}}\in R^{l\times 1}\) are the input and output vectors, respectively, \(\mathbf{w}_{\mathbf{k}}\) and \(\mathbf{v}_{\mathbf{k}}\) are assumed to be the zero-mean white noises. In the covariance-driven SSI, the modal parameters are identified from the state and output matrix (i.e., \(\mathbf{A}\) and \(\mathbf{C}\)) of the stochastic state-space model. To this end, the first step is to construct the Hankel data matrix \(\mathbf{H}\) with the measured data \(\mathbf{X}_{\mathbf{k}}\) and then calculate the block Toeplitz matrix \(\mathbf{T}\) with the block output covariance, as respectively shown in Equations (3) and (4). \[\mathbf{H}=\frac{1}{\sqrt{j}}\left[ \begin{array}{cccc}y_{0}&y_{1}&\cdots&y_{j-1} \\ y_{1}&y_{2}&\cdots&y_{j}\\ \vdots&\vdots&\ddots&\vdots\\ y_{i-1}&y_{i}&\cdots&y_{i+j-2}\\ \hline y_{i}&y_{i+1}&\cdots&y_{i+j-1}\\ y_{i+1}&y_{i+2}&\cdots&y_{i+j}\\ \vdots&\vdots&\ddots&\vdots\\ y_{2i-1}&y_{2i}&\cdots&y_{2i+j-2}\\ \end{array} \right]=\left[\frac{Y_{0|i-1}}{Y_{i|2i-1}}\right]=\left[\frac{Y_{P}} {Y_{f}}\right] \tag{3}\]  
\[\mathbf{T}_{1|i}= \begin{bmatrix}\Lambda_{i}&\Lambda_{i-1}&\cdots&\Lambda_{1}\\ \Lambda_{i+1}&\Lambda_{i}&\cdots&\Lambda_{2}\\ \vdots&\vdots&\ddots&\vdots\\ \Lambda_{2i-1}&\Lambda_{2i-2}&\cdots&\Lambda_{i}\\ \end{bmatrix} =Y_{f}(Y_{P})^{T} \tag{4}\]  
where \(Y_{p}\) and \(Y_{f}\) are the past and future measured matrix; \(\Lambda_{i}\) is the block covariance between all outputs \(\mathbf{y}_{\mathbf{k}}\) as \(\Lambda_{i}=E\big{[}y_{k}y_{k-i}^{T}\big{]}\in R^{l\times l}\). The SVD is applied to the Toeplitz matrix \(\mathbf{T}_{1|i}\), decomposed as  
\[\mathbf{T}_{1|i}=\mathbf{U}\mathbf{S}\mathbf{V}^{T}=(\mathbf{U}_{1}\,\,\mathbf{U}_{2}) \begin{pmatrix} \mathbf{S}_{1}&0\\ 0&0\\ \end{pmatrix}\begin{pmatrix}\mathbf{V}_{1}^{T}\\ \mathbf{V}_{2}^{T}\\ \end{pmatrix} =\mathbf{U}_{1}\mathbf{S}_{1}\mathbf{V}_{1}^{T} \tag{5}\]  
where \(\mathbf{U}\) and \(\mathbf{V}\) are the orthonormal matrices; \(\mathbf{S}\) is the diagonal matrix containing the singular values in descending order. By factorizing the Toeplitz matrix into an extend observation matrix and reversed extend control matrix, the system matrix \(\mathbf{A}\) in Equation (2) can be computed as  
\[\mathbf{A}=\mathbf{S}_{i}^{-1/2}\mathbf{U}_{1}^{T}\mathbf{T}_{2|i+1}\mathbf{V}_{1}\mathbf{S}_{1}^{1/2} \tag{6}\]  
Meanwhile, the output matrix \(\mathbf{C}\) equals the first \(l\) rows of \(\mathbf{U}_{1}\mathbf{S}_{1}^{1/2}\). A more detailed process of reasoning can be found in the reference [26]. Then, the estimation of the modal parameters starts with the eigenvalue decomposition of \(\mathbf{A}=\mathbf{\Psi}\mathbf{\Omega}\mathbf{\Psi}^{-1}\), where \(\mathbf{\Omega}=\text{diag}(\lambda_{i})\) is a diagonal matrix containing the discrete time complex eigenvalues and \(\mathbf{\Psi}\) denotes the eigenvectors as columns. The natural frequencies \(\omega_{i}\) and damping ratio \(\xi_{i}\) can be obtained from the continuous time conjugated eigenvalues \(\lambda_{i}^{c}=\frac{ln\lambda_{i}}{\Delta t}\left(\lambda_{i}^{c}\lambda_{i }^{c}-\xi_{i}\omega_{i}\pm\lambda_{i}^{c}\right)\).