## 2 Basic Theory
### Eigensystem Realization Algorithm  
The input condition of the ERA is the impulse response function. By solving the response of the damped multi-degree of freedom vibration system under arbitrary displacement excitation, it is found that the response solution under the initial displacement excitation can replace the impulse response function as the input of the ERA.  
For a finite, discrete-time linear time-invariant system, the state equation can be expressed in the following form:  
\[\left\{\begin{array}{c}x(k+1)=Ax(k)+Bu(k)\\ y(k)=Cx(k)\end{array}\right. \tag{22}\]  
Among them, \(x\in R^{n},u\in R^{m},y\in R^{p}\) are the state vector, input vector and output vector, respectively, and \(A\), \(B\), \(C\) are the state matrix, input matrix and output matrix of the system, respectively.  
The Hankel matrix is constructed using the response function under arbitrary initial displacement excitation, namely:  
\[H_{rs}(k-1)=\left[\begin{array}{cccc}Y(k)&Y(k+t_{1})&\cdots&Y(k+t_{s-1})\\ Y(j_{1}+k)&Y(j_{1}+k+t_{1})&\cdots&Y(j_{1}+k+t_{s-1})\\ \vdots&\vdots&&\vdots\\ Y(j_{r-1}+k)&Y(j_{r-1}+k+t_{1})&\cdots&Y(j_{r-1}+k+t_{s-1})\end{array}\right] \tag{23}\]  
where \(Y(k)\in R^{l\times p}\) is the response function matrix under arbitrary initial displacement excitation, namely.  
\[Y(k)=\left[\begin{array}{cccc}h_{11}(k)&h_{12}(k)&\cdots&h_{1p}(k)\\ h_{21}(k)&h_{22}(k)&\cdots&h_{2p}(k)\\ \vdots&\vdots&&\vdots\\ h_{L1}(k)&h_{L2}(k)&\cdots&h_{Lp}(k)\end{array}\right]_{(l\times p)} \tag{24}\]  
In the formula, \(h_{ij}(k)\) is the displacement excitation response function between excitation point \(j\) and response point \(i\) at time \(k\). Perform singular value decomposition on \(H_{rs}(0)=UVW^{T}\); the order determined by singular value decomposition obtains the minimum realization of the system:  
\[A=V^{-\frac{1}{2}}U^{T}H_{rs}(1)WV^{-\frac{1}{2}} \tag{25}\]  
\[B=V^{\frac{1}{2}}W^{T}E_{m} \tag{26}\]  
\[C=E_{p}{}^{T}UV^{\frac{1}{2}} \tag{27}\]  
where: \(E_{p}{}^{T}=[I_{p},\,0_{p},\,\ldots,\,0_{p}]\), \(E_{m}{}^{T}=[I_{m},\,0_{m},\,\ldots,\,0_{m}]\).  
Carry out the eigenvalue decomposition of the matrix \(A\) to obtain the eigenvalue matrix \(G\), and then obtain the eigenvector matrix \(\varphi\).  
\[\varphi^{-1}A\varphi=G,\,G=diag(g_{1},g_{2},\cdots,g_{r}) \tag{28}\]  
where: \(g_{r}\) is the eigenvalue of matrix \(A\), and \(r\) is the modal order.  
Determine the modal frequency \(\omega_{r}\) and modal damping \(\zeta_{r}\) according to the relationship between the eigenvalue gr of the matrix \(A\) and the system eigenvalue \(\lambda_{r}\):  
\[\lambda_{\rm i}=\frac{1}{\Delta t}\ln g_{i}=\lambda_{r}{}^{R}+j\lambda_{r}{}^ {I} \tag{29}\]  
\[\omega_{r}=\sqrt{\left(\lambda_{r}{}^{R}\right)^{2}+\left(\lambda_{r}{}^{I} \right)^{2}} \tag{30}\]  
\[\zeta_{r}=-\frac{\lambda_{r}^{R}}{\sqrt{\left(\lambda_{r}^{R}\right)^{2}+ \left(\lambda_{r}^{I}\right)^{2}}} \tag{31}\]  
In the formula: \(\lambda_{r}{}^{R}\) represents the real part of the system eigenvalue \(\lambda_{r},\lambda_{r}{}^{I}\) represents the imaginary part, and \(r\) is the modal order.  
According to the output matrix \(C\) and the eigenvector matrix \(\varphi\), the mode shape matrix \(\phi\) can be determined:  
\[\phi=C\varphi=E_{p}{}^{T}UV^{\frac{1}{2}}\varphi \tag{32}\]  
Use the Modal Assurance Criterion (MAC) to check the independence and consistency between the two modes:  
\[MAC_{uv}=\frac{\left|\phi_{u}{}^{T}\phi_{v}\right|^{2}}{\phi_{u}{}^{T}\phi_{u }\phi_{v}{}^{T}\phi_{v}} \tag{33}\]  
Among them, \(\phi_{u}\) and \(\phi_{v}\) are mode vectors, which are column vectors, and \(u\) and \(v\) are modal orders. In the formula, modal displacement needs to be normalized, and its value ranges from \([0,1]\). The MAC value of \(1\) indicates that the two modes are completely correlated, while a value of \(0\) indicates that the two modes are completely unrelated.  
The modal identification process of ultralow-frequency flexible structure is shown in Figure 3: