## 2 Recursive Formulation
### Subspace Identification (SI)  
To develop SI, the state-space equations need to be transformed into \"Matrix Input-Output Equations\" [31] as  
\[\mathbf{Y}_{p}=\mathbf{\Gamma}_{i}\mathbf{X}_{p}+\mathbf{H}_{i}\mathbf{U}_{p }+\mathbf{G}_{i}\mathbf{W}_{p}+\mathbf{V}_{p} \tag{3}\]  
\[\mathbf{Y}_{f}=\mathbf{\Gamma}_{i}\mathbf{X}_{f}+\mathbf{H}_{i}\mathbf{U}_{f} +\mathbf{G}_{i}\mathbf{W}_{f}+\mathbf{V}_{f} \tag{4}\]  
\[\mathbf{X}_{f}=\mathbf{A}^{i}\mathbf{X}_{p}+\Delta_{i}\mathbf{U}_{p} \tag{5}\]where \(\mathbf{Y}_{p}\) and \(\mathbf{Y}_{f}\) are past and future output data Hankel matrices with \(i\) rows and \(j\) columns (both in samples), respectively. \[\mathbf{Y}_{p}= \begin{bmatrix}\mathbf{y}_{1\,|\,i}&\mathbf{y}_{2\,|\,i+1}&\cdots &\mathbf{y}_{j\,|\,i+j-1}\end{bmatrix}  \tag{6}\]  
\[\mathbf{Y}_{f}= \begin{bmatrix}\mathbf{y}_{i+1\,|\,2i}&\mathbf{y}_{i+2\,|\,2i+1 }&\cdots&\mathbf{y}_{i+j\,|\,2i+j-1}\end{bmatrix}  \tag{7}\]  
\[\mathbf{y}_{1\,|\,i}= \begin{bmatrix}\mathbf{y}_{1}^{T}&\mathbf{y}_{2}^{T}& \cdots&\mathbf{y}_{i}^{T}\end{bmatrix} ^{T} \tag{8}\]  
Similarly, \(\mathbf{U}_{p}\) and \(\mathbf{U}_{f}\) are past and future input data Hankel matrices. As a result, the dimension of \(\mathbf{Y}_{p}\) and \(\mathbf{Y}_{f}\) is \(im\times j\) and the dimension of \(\mathbf{U}_{p}\) and \(\mathbf{U}_{f}\) is \(il\times j\). \(\mathbf{\Delta}_{i}\) is the reversed extended controllability matrix; \(\mathbf{H}_{i}\) and \(\mathbf{G}_{i}\) are the lower block triangular Toeplitz matrices. \(\mathbf{\Gamma}_{i}\) is extended observability matrix which contains information of system matrices (\(\mathbf{A}\) and \(\mathbf{C}\)) and is the primary outcome of subspace identification. Details about those matrices can be also found in the literature [31]. So, the problem of identifying the system matrices is identifying the extended observability matrix. To extract the key matrix, a geometric tool, called projection in the field of linear algebra, is utilized to obtain the extended observability matrix from the matrix input-output equations. Both orthogonal and oblique projections can be used to extract the extended observability matrix and the \"Multivariable Output-Error State sPace (MOESP)\" is especially efficient for performing the oblique projection [32]. Instead of a series of matrix multiplication, MOESP implements LQ decomposition to the Hankel matrices as  
\[ \begin{bmatrix}\mathbf{U}_{f}\\ \mathbf{\Xi}_{p}\end{bmatrix} = \begin{bmatrix}\mathbf{L}_{11}&0&0\\ \mathbf{L}_{21}&\mathbf{L}_{22}&0\\ \mathbf{L}_{31}&\mathbf{L}_{32}&\mathbf{L}_{33}\end{bmatrix}\begin{bmatrix} \mathbf{Q}_{1}^{T}\\ \mathbf{Q}_{2}^{T}\\ \mathbf{Q}_{3}^{T}\end{bmatrix}  \tag{9}\]  
where \(\mathbf{\Xi}_{p}\) denotes an instrumental variable (IV) matrix consisting of both past input and output data Hankel matrices [33]. \[\mathbf{\Xi}_{p}= \begin{bmatrix}\mathbf{U}_{p}\\ \mathbf{Y}_{p}\end{bmatrix} = \begin{bmatrix}\mathbf{\xi}_{1\,|\,i}&\mathbf{ \xi}_{2\,|\,i+1}&\cdots&\mathbf{\xi}_{j\,|\,i+j-1}\end{bmatrix}  \tag{10}\]  
It is noted that the left-hand side matrix in Equation (9) is suggested to be a square matrix to secure a correct division after the LQ decomposition, meaning that the number of columns, \(j\), has no choice but  
\[j=2i\,(m+l) \tag{11}\]  
where the dimension of \(\mathbf{L}_{32}\) is hence specified, \(il\times i(m+l)\). Consequently, \(\mathbf{L}_{32}\mathbf{Q}_{2}^{T}\) is equal to the oblique projection, \(\mathbf{O}\), and the column space of \(\mathbf{L}_{32}\) is the same with the column space of the extended observability matrix. It is the essence of why the algorithms are named \"subspace\" identification. To retrieve the column space of the extended observability matrix, one can take advantage of singular value decomposition (SVD)  
\[\mathbf{O}=\mathbf{U}\mathbf{S}\mathbf{V}^{\mathrm{T}}= \begin{bmatrix} \mathbf{U}_{1}&\mathbf{U}_{2}\end{bmatrix}\begin{bmatrix}\mathbf{S}_{1}&0\\ 0&\mathbf{S}_{2}\end{bmatrix}\begin{bmatrix}\mathbf{V}_{1}^{T}\\ \mathbf{V}_{2}^{T}\end{bmatrix}