## 2 Mechanism Model Construction and Identification Data Collection
### Design of Subspace Identification Recursive Algorithm Based on Correction Quantity
, \tag{27}\]  
where \(T_{l}=\text{diag}(\sqrt{\beta^{l-1}}I_{1},\dots,I_{l})\), \(T_{m}=\text{diag}(\sqrt{\beta^{l-1}}I_{m},\dots,I_{m})\). The column vector length of the initial Hankel matrix is \(s\)-\(i\). When calculating the identification algorithm with a column number interval of \([p\,\,\,p+s-i)\) for the Hankel matrix, due to the existence of the initial values of \(YY_{0}^{T}\), \(YU_{0}^{T}\), and \(P_{0}\), the correction within this interval can only be indirectly calculated. Therefore, in the recursive process, \(p+s-i\) is used as the switch, and the previous method is used for recursion, while the subsequent recursive process increases the correction amount \(\Delta\Xi_{N}\). The following conclusions can be drawn from the recursive steps of the new algorithm: (1) When the length of the column vector \(N\) of the recursive data is less than the threshold \(p\), the 4 SID identification algorithm with a fixed forgetting factor mechanism is used for recursion; when \(N\geq p\), increase the computation step of correction calculation \(\Delta\Xi_{N}\) and correct the original data compression matrix \(\Xi_{N}\). (2) The magnitude of the correction effect \(\Delta\Xi_{N}\) is determined by both the forgetting factor \(\beta\) and the threshold \(p\). When choosing a larger forgetting factor to reduce sensitivity to noise and \(\beta\) is fixed, the larger \(p\), the longer the length of the memory data, and the smaller the effect of the correction amount \(\Delta\Xi_{N}\). On the contrary, the shorter the length of the memory data, the greater the effect of the correction amount \(\Delta\Xi_{N}\). After obtaining the data compression matrix \(\Xi_{N}\), the SVD decomposition method is used to obtain the estimated value \(\hat{\Gamma}_{N}\). The estimated values of \(\hat{A}_{N}\) and \(\hat{C}_{N}\) can be obtained according to the following equation  
\[ \begin{array}{l}\hat{C}_{N}=\hat{\Gamma}_{N}(1:l,1:n)/\sqrt{\beta^{l-1}}\\ \hat{A}_{N}=\sqrt{\beta}(\hat{\Gamma}_{N}(1:l(i-1),:))^{\dagger}\hat{\Gamma}_{ N}(l+1:l_{i},:)\end{array} . \tag{28}\]  
After obtaining \(\hat{A}_{N}\) and \(\hat{C}_{N}\), the Toeplitz matrix \(\hat{H}_{N}\) can be estimated using \(\hat{\Gamma}_{N}\), and then the matrices \(\hat{B}_{N}\) and \(\hat{D}_{N}\) can be estimated using the least squares method. A performance evaluation method is proposed in this work based on system invariants to address the theoretical characteristics of parameter identification in state space models. At the level of system modeling, the state space equation has the characteristic of parameter matrix reconstruction under similar transformations, which poses a theoretical challenge to the performance evaluation of identification algorithms due to the non-uniqueness of equivalent system descriptions. It is worth noting that although there are multiple equivalent expressions for parameter matrices, they can all converge to a unified standard form after being transformed by specifications, providing a theoretical basis for the construction of evaluation systems. Starting from the essential characteristics of the system, the eigenvalues, as the core invariants reflecting the dynamic characteristics of the system, can effectively characterize the performance of the identification algorithm based on the convergence characteristics between the estimated trajectory and the true value. The existing evaluation system mainly includes two typical methods: one is to compare the convergence process of the true eigenvalue trajectory of the system matrix \(A\) with its estimated value \(\hat{A}\). The second approach is to conduct matching analysis based on modal parameters derived from eigenvalues. This study adopts the former to construct an evaluation framework, quantifying the dynamic response ability of the identification algorithm to time-varying parameters through the time-domain tracking accuracy of the eigenvalue trajectory. The iterative steps of the subspace identification recursive algorithm based on the forgetting factor correction are shown in Figure 4.