## 2 Mechanism Model Construction and Identification Data Collection
### Design of Subspace Identification Recursive Algorithm Based on Correction Quantity  
Assuming that \(P_{N-1}^{-1}:=U_{N-1}U_{N-1}^{\mathrm{T}},\mathcal{X}_{N-1}:=Y_{N-1}P_{\{\hat{U} _{N-1}^{-}Y_{N-1}^{T}\}}\) and \(Y_{N}-1U_{N-1}^{T}\) from the previous step are known, when obtaining a new set of input-output data \(\{u_{N},y_{N}\}\), the data compression matrix \(\Xi_{N}\) can be updated as follows:  
\[\left\{ \begin{array}{l}\alpha_{N}=\left(1+u_{N}^{\mathrm{T}}P_{N-1}u_{N} \right)^{-1}\\ e_{N}=y_{N}-YU_{N-1}^{\mathrm{T}}P_{N-1}u_{N}\\ P_{N}=P_{N-1}-\alpha_{N}P_{N-1}u_{N}u_{N}^{\mathrm{T}}P_{N-1}\\ YU_{N}^{\mathrm{T}}=YU_{N-1}^{\mathrm{T}}+y_{N}u_{N}^{\mathrm{T}}\\ E_{N}=E_{N-1}+\alpha_{N}e_{N}^{\mathrm{T}}\end{array} \right.. \tag{15}\]  
After obtaining the data compression matrix \(\Xi_{N}\), the SVD method can be used to decompose and obtain the estimation result \(\hat{\Gamma}\), which can then be used to estimate the parameter matrices \(A\), \(B\), \(C\), and \(D\) of the system. If \(A\) and \(B\) are the matrices of \(p\times(m+n)\) and \(q\times(m+n)\) dimensions and satisfy the condition \(A=\left[A_{1}\vdots A_{2}\right]\) and \(B=\left[B_{1}\vdots B_{2}\right]\), then:  
\[AB^{\mathrm{T}}=A_{1}B_{1}^{\mathrm{T}}+A_{2}B_{2}^{\mathrm{T}}. \tag{16}\]  
where \(A_{1},\ A_{2},\ B_{1},\ B_{2}\) are \(p\times m,\ p\times n,\ q\times m,\ q\times n\) dimensional matrices. According to the assumption of \(A=[A_{1}\vdots A_{2}]\) and the law of matrix addition, it can be concluded that  
\[ \begin{array}{l}A=[A_{1}\vdots 0_{p\times n}]+[0_{p\times m}\vdots A_{2}]\\ AB^{\mathrm{T}}=([A_{1}:0_{p\times n}]+[0_{p\times m}:A_{2}])B^{\mathrm{T}}\end{array} , \tag{17}\]  
Then, according to the law of matrix multiplication, we can obtain  
\[AB^{\mathrm{T}}=[A_{1}:0_{p\times n}][B_{1}:B_{2}]^{\mathrm{T}}+[0_{p\times m} :A_{2}][B_{1}:B_{2}]^{\mathrm{T}}=A_{1}B_{1}^{\mathrm{T}}+A_{2}B_{2}^{\mathrm{ T}}. \tag{18}\]  
According to Equation (15), the recursive calculation of the data compression matrix \(\Xi_{N}\) is the key to the recursive N4 SID subspace identification algorithm. If the influence \(\Delta\Xi_{N}\) of the old data to be removed on the data compression matrix \(\Xi_{N}\) in Formula (15) can be separated, it can effectively block the role of the old data and improve the tracking speed of the identification algorithm for time-varying parameters. Based on this idea, this study uses Equations (15) and (16) to implement the calculation steps of the correction amount \(\Delta\Xi_{N}\) for removing old data. Assuming that \(YU_{N-1}^{T}\), \(YY_{N-1}^{T}\), \(P_{N-1}\), and \(\Xi_{N-1}\) from the previous step are known, the correction amount \(\Delta\Xi_{N}\) of the data compression matrix \(\Xi_{N}\) has the following relationship:  
\[\Delta\Xi_{N}=YY_{1}^{T}-YU^{T}(P-P_{2})UY^{T}-Y_{1}U_{1}^{T}P_{2}UY^{T}-(YU^ {T}-Y_{1}U_{1}^{T})P_{2}U_{1}Y_{1}^{T} \tag{19}\]  
where \(Y_{1}\) and \(U_{1}\) are the matrix blocks that need to be removed from the input and output Hankel matrices \(Y\) and \(U\) in the recursion, respectively, and they can be represented as \(P_{2}=\left(P^{-1}-U_{1}U_{1}^{T}\right)^{-1}=P+PU_{1}U_{1}^{T}P\left(I-U_{1}U _{1}^{T}P\right)^{-1}\). To simplify the equation description, the data compression matrix \(\Xi_{N}\) in Equation (15) is abbreviated as \(\Xi\) afterwards. According to the definition of the data compression matrix \(\Xi_{N}\) in Equation (15), the following equation can be obtained:  
\[\Xi_{N}=Y\prod_{U}^{\perp}Y^{T}=YY^{T}-YU^{T}(UU^{T})^{-1}UY^{T}, \tag{20}\]  
According to Equation (16), the input Hankel matrix \(U\) has the following relationship:  
\[UU^{T}=[U_{1}^{T}\vdots U_{2}^{T}][U_{1}^{T}\vdots U_{2}^{T}]^{T}=U_{1}U_{1} ^{T}+U_{2}U_{2}^{T}, \tag{21}\]  
where \(U_{2}\) is the matrix composed of the data pairs that the Hankel matrix \(U\) needs to retain. Similarly, it can be concluded that  
\[