## 4 Result Verification  
In order to demonstrate the effectiveness of the CFF-SIR algorithm proposed in this study in estimating longitudinal tire forces in vehicles, a comparative verification was first conducted in the joint simulation environment of CarSim and Simulink. At the same time, in order to reflect the application effect and applicability of the proposed estimation method under various working conditions, simulation verification was carried out under three working conditions: no noise and no interference, with noise influence, and with  
Figure 4: Iterative steps of the subspace identification recursive algorithm based on the forgetting factor correction. interference influence. Firstly, simulation tests were conducted under J-turn conditions, with the vehicle speed set at 15m/s and the steering wheel angle shown in Figure 5.  
The estimated longitudinal tire force under the J-turn condition is shown in Figure 6. From the figure, it can be observed that, without considering the noise and interference encountered during the model identification process, the curves of the estimated longitudinal tire force values obtained by the CFF-SIR and C-SIR algorithms and the actual longitudinal tire force values can generally remain consistent, indicating that both models have high accuracy in identification. In addition, compared to the C-SIR algorithm, the estimation accuracy under the CFF-SIR algorithm is significantly higher, while the C-SIR algorithm has relatively obvious bias and lag in the estimation result curve, resulting in relatively poorer estimation performance. In order to further evaluate the performance of the proposed estimation model identification method based on the CFF-SIR algorithm in practical application scenarios, a noisy identification data model validation experiment was conducted. At this point, an additional 5% zero mean white noise was added to the identification data, and the joint simulation verification data also added the influence of 5% noise. As shown in the figure, when the identification model is affected by noise interference, the longitudinal tire force estimation values exhibit more obvious oscillation phenomena. Although the estimation result curves under the CFF-SIR and C-SIR algorithms remain roughly unchanged in overall trend, the longitudinal tire force estimation based on C-SIR is more significantly affected by noise, resulting in larger amplitude oscillations and glitches in the state curve. In comparison, the CFF-SIR algorithm still demonstrates higher estimation accuracy in this situation. On this basis, further research was conducted on the situation where driving data contains interference. Similarly, 5% of irregular interference signals were added to the identification data and joint simulation validation data of the estimation model. Compared with the undisturbed situation, the estimation accuracy under the C-SIR algorithm has decreased overall, and the longitudinal tire force curve has shown a more significant deviation. However, the estimated values under the CFF-SIR algorithm were not significantly affected by interference, and the estimation accuracy was also relatively close to the estimation results under no interference, demonstrating strong anti-interference ability. Based on the above comparison results, the CFF-SIR algorithm exhibits better estimation accuracy and stability in the absence of noise and interference, with noise influence, and with interference influence, and has significant advantages compared to the C-SIR algorithm. This indicates that the CFF-SIR algorithm can more accurately identify the es  
Figure 5: Steering wheel angle of vehicle under J-turn condition. sential parameters of the longitudinal tire force estimation model in practical applications, thereby obtaining more accurate longitudinal force calculation results.