## 2 Mechanism Model Construction and Identification Data Collection
### Problem Description of Longitudinal Tire Force Estimation Model  
The mechanism model used for tire longitudinal force estimation in Equation (9) can be represented as the following discrete subspace model:  
\[\left\{ \begin{array}{l}x_{t+1}=Ax_{t}+Bu_{t}+w_{t}\\ y_{t}=Cx_{t}+Du_{t}+v_{t}\end{array} \right., \tag{10}\]  
where \(u_{t}\in R^{m}\) is the input, \(y_{t}\in R^{l}\) is the output, and \(x_{t}\in R^{n}\) is the state vector of the system, respectively. In addition, \(w_{t}\in R^{n}\) is the process noise, and \(v_{t}\in R^{l}\) is the measurement noise, respectively. To ensure the identifiability and convergence of the system, it is assumed that the dimension of the identified system remains unchanged during the time period of interest and that (\(A\), \(B\)) is controllable, (\(A\), \(C\)) is observable, and the system is the minimum implementation; external input and noise are uncorrelated and meet sufficient excitation conditions. The focus of the subspace identification algorithm is to use the following Hankel matrix relationship to project future data pairs onto past data pairs in order to estimate the generalized observation matrix; that is,  
\[Y_{0,i,N}=\Gamma_{i}X_{0,N}+H_{i}U_{0,i,N}+N_{0,i,N}, \tag{11}\]  
where \(U_{0,i,N}\) is the input matrix, \(Y_{0,i,N}\) is the output matrix, and \(N_{0,i,N}\) is the measurement noise Hankel matrix, which can be described as \(Y_{0,i,N}= \begin{bmatrix}y_{0}&\dots&y_{N-i+1}\\ \vdots&\dots&\vdots\\ y_{i-1}&\dots&y_{N}\end{bmatrix} \). When \(i\) is determined, \(Y_{0,i,N}\) can be simplified and labeled as \(Y_{N}\), \(X_{N}=[x_{0}\cdots x_{N-i}]\) is the state matrix, and the generalized observability matrix \(X_{N}=[x_{0}\cdots x_{N-i}]\) and Markov coefficient matrix \(H\) are \(\Gamma= \begin{bmatrix}C\\ CA\\ \vdots\\ CA^{i-1}\end{bmatrix} \) and \(H= \begin{bmatrix}D&0&\dots&0\\ CB&D&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ CA^{i-2}B&CA^{i-3}B&\dots&D\end{bmatrix} \), respectively. The system matrix parameters \(A\), \(B\), \(C\), and \(D\) can be calculated from the estimated value \(\hat{\Gamma}\) of the generalized observability matrix. The following two projection matrices are defined as  
\[ \begin{array}{l}\Pi_{U_{N}}=U_{N}^{\mathrm{T}}{(U_{N}U_{N}^{\mathrm{T}})}^{ -1}U_{N}\\ \Pi_{U_{N}}^{\perp}=I-\Pi_{U_{N}}\end{array} . \tag{12}\]  
The above projection matrix has the following properties:  
\[ \begin{array}{l}\Pi_{U_{N}}^{\mathrm{T}}=\Pi_{U_{N}}\cdot\ {(\Pi_{U_{N}}^{ \mathrm{T}})}^{\mathrm{T}}=\Pi_{U_{N}}^{\perp}\\ U_{N}\Pi_{U_{N}}^{\perp}=0\\ \Pi_{U_{N}}^{\perp}\times\Pi_{U_{N}}^{\perp}=\Pi_{U_{N}}^{\perp}\end{array} . \tag{13}\]  
Equation (11) can be right-hand multiplied by the matrix \(\Pi_{U_{N}}^{\perp}\) on both sides to obtain the following equation:  
\[Y_{N}\Pi_{U_{N}}^{\perp}=\Gamma X_{N}\Pi_{U_{N}}^{\perp}. \tag{14}\]  
By using the singular value decomposition method, the estimated value \(\hat{\Gamma}\) of the generalized observable matrix can be obtained. Aiming at the optimization requirements of the existing recursive N4 SID algorithm, an improvement plan is proposed while maintaining computational efficiency. In terms of reducing computational complexity, existing research often uses matrix operation frameworks to construct recursive N4 SID algorithms. However, as the data scale expands, the time-varying characteristics of system parameters can lead to dynamic tracking performance degradation of traditional identification algorithms, mainly due to the continuous accumulation effect of historical data. Although the forgetting factor mechanism can moderately improve the ability to identify time-varying parameters through exponential decay weighting, it still cannot completely eliminate the residual influence of old data, in essence. Compared to other strategies, although the sliding window strategy can achieve data truncation, it faces the dual limitations of high sensitivity to window parameters and a heavy real-time computational burden.