## 5 Conclusions  
This study proposes a joint estimation method that combines data-driven and modified recursive subspace identification algorithms for the longitudinal force estimation problem of tires in four-wheel independent drive electric vehicles. By constructing an electric drive wheel mechanism model and combining an improved proportional integral observer with a dynamically modified recursive subspace identification algorithm, the problem of insufficient estimation accuracy and robustness of traditional methods under noise, interference, and dynamic conditions has been effectively solved. Simulation and experimental verification show that the CFF-SIR algorithm significantly reduces the mean and mean square error of longitudinal force estimation in complex scenarios such as J-turn and variable speed sinusoidal steering compared to the traditional C-SIR algorithm, and maintains high stability in noisy and disturbed environments; the actual test data of the chassis dynamometer further confirms that its comprehensive accuracy has improved by 8.37%. The innovation of this method lies in the collaborative optimization of mechanism model constraints and data-driven compensation, which achieves a high-precision dynamic decoupling estimation of longitudinal forces and provides reliable technical support for active safety control and intelligent driving systems of vehicles. Future work can explore strategies for adaptively adjusting forgetting factors to further enhance the algorithm's ability to track time-varying characteristics and noise interference. The size of the forgetting factor can be automatically adjusted based on the statistical characteristics of real-time data or the dynamic change rate of the system, thereby enhancing the response speed to rapidly changing system characteristics while maintaining robustness to noise. This research direction helps to improve the adaptability and accuracy of longitudinal force estimation algorithms in complex and changing practical driving environments. Meanwhile, more real-time deployment experiments can be conducted in future research to verify the performance and reliability of the proposed algorithm in actual vehicle systems. Field testing can be considered on different types of vehicles, covering various driving and environmental conditions such as different road surface types, weather conditions, and driving modes. Through these experiments, more real vehicle data can be collected to further optimize algorithm parameters and verify their feasibility and effectiveness in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Working Condition** & **Method** & **Mean Error** & **Mean Square Error** \\ \hline \multirow{2}{*}{Without noise or interference} & CFF-SIR & 2.3745 & 1.5968 \\  & C-SIR & 3.9118 & 2.3467 \\  & CFF-SIR & 2.6787 & 1.8487 \\  & C-SIR & 4.6788 & 3.9512 \\  & CFF-SIR & 2.6854 & 1.6738 \\ With interference & C-SIR & 4.7903 & 2.9106 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean error and mean square error of estimation results in chassis dynamometer test. practical applications. In addition, real-time deployment experiments can also help identify and solve hardware compatibility, computing latency, and resource limitations that may arise during the actual operation of algorithms, laying a solid foundation for the final commercial application of algorithms. Conceptualization, X.W. and T.C. ; methodology, X.W., T.C. and J.L. ; software, X.W. and T.C. ; validation, X.W. and T.C. ; writing--original draft preparation, X.W. and T.C. ; writing--review and editing, X.W. and T.C. ; visualization, J.L. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China (No. 52202472), Suzhou Key Industry Technology Innovation-Prospective Application Research Project (No. SYG202128), Jiangsu Province 3C Product Intelligent Manufacturing Engineering Technology Research and Development Center Funding Project (No.