## 2 Stochastic Subspace Identification with Uncertainty
### Stochastic Subspace Identification (SSI)  
SSI is a method for extracting modal parameters based on the stochastic state-space model. Owing to its high degree of accuracy compared to other methods, SSI is preferred for OMA. Assuming a white noise external excitation, the following discrete time state-space model of a linear stationary system can be formulated as follows:\[\begin{array}{l}x_{k+1}=Ax_{k}+v_{k}\\ y_{k}=Cx_{k}+w_{k}\end{array} \tag{1}\]  
where \(x_{k}\in\mathbb{R}^{n_{x}}\) is the state vector; \(y_{k}\in\mathbb{R}^{n_{y}}\) is the output vector; \(v_{k}\in\mathbb{R}^{n_{x}}\) and \(w_{k}\in\mathbb{R}^{n_{y}}\) are, respectively, the process and output-noise vectors; \(A\in\mathbb{R}^{n_{x}\times n_{x}}\) is the state matrix of the system; \(C\in\mathbb{R}^{n_{y}\times n_{x}}\) is the output matrix; and \(n_{x}\) and \(n_{y}\) are, respectively, the model order and the number of measuring sensors.  
A data-driven or covariance-driven technique can be applied to Equation (1). Here, a reference-based covariance-driven technique [26] is adopted for its relatively cheaper analysis burden. First, the block Hankel matrix \(Y_{1|2i-1}\in\mathbb{R}^{n_{y}(2i-1)\times j}\) is derived from the measured data as follows:  
\[Y_{1|2i-1}=\begin{bmatrix}y_{1}&y_{2}&\cdots&y_{j}\\ y_{2}&y_{3}&\cdots&y_{j+1}\\ \vdots&\vdots&\ddots&\vdots\\ y_{2i-1}&y_{2i}&\cdots&y_{n_{d}-1}\end{bmatrix}Y_{0|0}^{\tau}=\begin{bmatrix} y_{0}^{\tau}&y_{1}^{\tau}&\cdots&y_{j-1}^{\tau}\end{bmatrix} \tag{2}\]  
where \(Y_{0|0}^{\tau}\in\mathbb{R}^{n_{y}^{\tau}\times j}\) is constructed from the data measured by the reference sensors; \(n_{y}^{\tau}\) is the number of reference sensors; and \(n_{d}\) is the number of time steps. If the whole set of data is used, \(j=n_{d}-2i\), where the time lag \(i\) is an integer satisfying the following equation [26]:  
\[i\geq\frac{f_{s}}{2f_{0}} \tag{3}\]  
where \(f_{s}\) and \(f_{0}\) are, respectively, the sampling rate and the lowest frequency of interest.  
Since the exact output covariance matrix cannot be obtained, unbiased estimates \(\Lambda_{1|2i-1}^{r}\in\mathbb{R}^{n_{y}(2i-1)\times n_{y}^{\tau}}\) are derived from Equation (2) by the following equation.  
\[\Lambda_{1|2i-1}^{r}=\frac{1}{j}Y_{1|2i-1}Y_{0|0}^{\tau}=\begin{bmatrix} \Lambda_{1}^{r}\\ \Lambda_{2}^{r}\\ \vdots\\ \Lambda_{2i-1}^{r}\end{bmatrix} \tag{4}\]  
The block Toeplitz matrix \(L_{1|i}^{r}\in\mathbb{R}^{n_{y}j\times n_{y}^{\tau}}\) can be obtained from Equation (4).  
\[L_{1|i}^{r}=\begin{bmatrix}\Lambda_{1}^{r}&\Lambda_{i-1}^{r}&\cdots&\Lambda_{ 1}^{r}\\ \Lambda_{i+1}^{r}&\Lambda_{i}^{r}&\cdots&\Lambda_{2}^{r}\\ \vdots&\vdots&\ddots&\vdots\\ \Lambda_{2i-1}^{r}&\Lambda_{2i-2}^{r}&\cdots&\Lambda_{i}^{r}\end{bmatrix}=U_{ i}\Sigma_{i}V_{i}^{T}=O_{i}C_{i}^{r} \tag{5}\]  
The singular value decomposition of the block Toeplitz matrix in Equation (5) provides the observability matrix \(O_{i}\in\mathbb{R}^{n_{y}i\times n_{x}}\) and the controllability matrix \(C_{i}^{r}\in\mathbb{R}^{n_{x}\times n_{y}^{\tau}}\) according to the model order.  
\[\begin{array}{l}O_{i}=U_{i}\Sigma_{i}^{1/2}\\ C_{i}^{r}=\Sigma_{i}^{1/2}V_{i}^{T}\end{array} \tag{6}\]  
In Equation (1), \(A=O_{i}^{\tau}\overline{O_{i}}\) is the state matrix, and the output matrix Subsequently, the \(k\)-th modal parameters, like the frequency \(f_{k}\in\mathbb{R}\) (Hz), damping ratio \(\zeta_{k}\in\mathbb{R}\) and mode shape \(\phi_{k}\in\mathbb{C}^{ny}\), can be calculated.  
\[\begin{split} f_{k}&=\frac{|f_{k}\ln\lambda_{k}|}{2 \pi}\\ \tilde{\zeta}_{k}&=\frac{\text{Re}(\ln\lambda_{k})}{| \ln\lambda_{k}|}\\ \phi_{k}&=C\phi_{k}\end{split} \tag{8}\]