## 2 Mechanism Model Construction and Identification Data Collection
### Design of Subspace Identification Recursive Algorithm Based on Correction Quantity
\begin{array}{l}YU^{T}=[Y_{1}\vdots Y_{2}][U_{1}\vdots U_{2}]^{T}=Y_{1}U_{1 }^{T}+Y_{2}U_{2}^{T}\\ YY^{T}=[Y_{1}\vdots Y_{2}][Y_{1}\vdots Y_{2}]^{T}=Y_{1}Y_{1}^{T}+Y_{2}Y_{2}^{T }\end{array} . \tag{22}\]  
Defining that \(P_{2}{}^{-1}=U_{2}U_{2}{}^{T}\), and then according to the matrix inverse lemma, the following equation can be calculated:  
\[P_{2}=(P^{-1}-U_{1}U_{1}{}^{T})^{-1}=P+PU_{1}U_{1}{}^{T}\times P(I-U_{1}U_{1} {}^{T}P)^{-1}, \tag{23}\]  
The matrix \(U_{1}U_{1}{}^{T}\) in Equation (23) can be obtained by inverting the previously recursive matrix \(P\). From Equations (20) to (23), it can be concluded that  
\[ \begin{array}{l}\Xi_{N}=(Y_{1}Y_{1}^{T}+Y_{2}Y_{2}^{T})-(Y_{1}U_{1}^{T}+Y_{ 2}U_{2}^{T})P\times(U_{1}Y_{1}^{T}+U_{2}Y_{2}^{T})\\ =(Y_{2}Y_{2}^{T}-Y_{2}U_{2}^{T}P_{2}U_{2}Y_{2}^{T})+(Y_{1}Y_{1}^{T}-YU^{T}(P-P _{2})UY^{T}-\\ YU^{T}P_{2}UY^{T}-(YU^{T}-YU^{T})P_{2}UY^{T})=\Xi_{2}+\Delta\Xi\end{array} . \tag{24}\]  
Thus, it can be proven that Equation (19) is obtained. From the above proof process, it can be seen that the recursive step of implementing the data compression matrix correction \(\Delta\Xi\) requires first calculating five matrices, namely: \(Y_{1}U_{1}^{T}\), \(UY^{T}\), \(U_{1}Y_{1}^{T}\), \(Y_{1}Y_{1}^{T}\) and \(P_{2}\). If the threshold size is determined, the matrix \(Y_{1}U_{1}^{T}\) can be directly obtained from the previously recursive matrix \(YU^{T}\), while \(UY^{T}\) and \(U_{1}Y_{1}^{T}\) are transposes of \(YU^{T}\) and \(Y_{1}U_{1}^{T}\). Therefore, calculating the correction \(\Delta\Xi\) only requires adding the calculation steps of matrix \(YY^{T}\) and matrix \(P_{2}\). Design of Subspace Identification Recursive Algorithm Based on Correction Quantity with Forgetting Factor  
The size of the forgetting factor is a key factor determining the performance of subspace identification algorithms under the forgetting factor mechanism. Generally, identification algorithms choose smaller forgetting factors, which have fast tracking speed but are sensitive to noise; choosing a larger forgetting factor makes the identification algorithm less susceptible to noise interference but slower in tracking speed. On the basis of the recursive algorithm in Equation (19), a new recursive N4 SID subspace identification algorithm with a fixed forgetting factor mechanism is designed using a correction amount. This algorithm not only selects a larger forgetting factor and reduces sensitivity to noise, but also improves the tracking speed of the algorithm for time-varying parameters by using the correction amount and can adjust the threshold size of the correction amount to change the tracking speed of the algorithm for time-varying parameters. Assuming the correction threshold is \(p\) and the \(YU_{N-1}^{T}\), \(YY_{N-1}^{T}\), \(P_{N-1}\), and \(\Xi_{N-1}\) of the previous step are known, the algorithm implementation steps are as follows:  
\[ \begin{array}{l}\phi_{yN}=T_{l}y_{N},\,\phi_{uN}=T_{m}u_{N}\\ YU_{N}^{T}=\beta YU_{N-1}^{T}+\phi_{yN}\phi_{uN}^{T}\\ YY_{N}^{T}=\beta YY_{N-1}^{T}+\phi_{yN}\phi_{yN}^{T}\\ \alpha_{N}=(\beta+\phi_{uN}^{T}P_{N-1}\phi_{yN})^{-1}\\ P_{N}=(P_{N-1}-\alpha_{N}P_{N-1}\phi_{yN}\phi_{yN}^{T}P_{N-1})/\beta\\ e_{N}=\phi_{yN}-YU_{N-1}^{T}P_{N-1}\phi_{uN}\end{array} , \tag{25}\]  
When \(N<p\), the following equation can be obtained by recursively solving the subspace matrix:  
\[\Xi_{N}=\Xi_{N-1}+\alpha_{N}e_{N}^{T}, \tag{26}\]  
When \(N\geq p\), the matrices \(P_{N}^{2}\) and \(\Delta\Xi_{N}\) are calculated, and the data compression matrix \(\Xi_{N}\) is corrected, resulting in the following equation:  
\[ \begin{array}{l}P_{N}^{2}=P_{N}+\beta^{p}P_{N}P_{N-1}^{-1}P_{N}(I-\beta^{p} P_{N-1}^{-1}P_{N})^{-1}\\ \Delta\Xi_{N}=\beta^{p}YY_{N}^{T}-YU_{N}^{T}P_{N}(YU_{N}^{T})^{T}+YU_{N}^{T}P_{ N}^{2}(YU_{N}^{T})^{T}-\\ \beta^{p}YU_{N-p}^{T}P_{N}^{2}(YU_{N-p}^{T})^{T}-\beta^{p}(YU_{N}^{T}-\beta^{p }YU_{N-p}^{T})P_{N}^{2}(YU_{N-p}^{T})^{T}\\ \Xi_{N}=\Xi_{N-1}+\alpha_{N}e_{N}e_{N}^{T}-\Delta\Xi_{N}\end{array}