# A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models
## 5 Conclusions  
For MISO fractional Hammerstein systems, the H-GOLS algorithm is proposed for jointly identifying the model orders and parameters. This algorithm leverages the Householder transformation to triangularize the sub-information matrix, effectively addressing issues related to ill-conditioned matrices and significantly reducing computational complexity. Subsequently, the sparse parameter vector is obtained using a back-substitution approach. Compared to conventional sparse identification algorithms such as BPDN, OMP, and LASSO, the proposed H-GOLS algorithm demonstrates significant advantages in parameter estimation accuracy, noise robustness, and computational efficiency. Notably, even under limited data availability, H-GOLS maintains superior estimation precision. Furthermore, in comparison to integer-order models, fractional models offer a more accurate description of systems with complex nonlinear characteristics, such as the CD-player mechanical arm presented in this paper. Both numerical simulations and the modeling of a CD-player arm system validate the effectiveness of the proposed H-GOLS algorithm.  
In this study, the fractional-order \(\alpha\) is estimated by first obtaining parameter estimates and then computing the system's output error across various \(\alpha\) values. This approach is computationally intensive. In future studies, we aim to investigate methods for simultaneously estimating the parameters and the fractional-order \(\alpha\) during the identification process.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{_RMSE_} & \multicolumn{2}{c}{_MAE_} & \multicolumn{2}{c}{_R_2} \\  & **Training** & **Validation** & **Training** & **Validation** & **Training** & **Validation** \\ \hline H-GOLS-Fractional & 0.1505 & 0.1494 & 0.1193 & 0.1169 & 0.9383 & 0.9542 \\ BPDN & 0.1701 & 0.1798 & 0.1343 & 0.1411 & 0.9202 & 0.9317 \\ OMP & 0.2539 & 0.2432 & 0.1873 & 0.1998 & 0.8489 & 0.8074 \\ LASSO & 0.1625 & 0.1600 & 0.1271 & 0.1286 & 0.9272 & 0.9459 \\ H-GOLS-Integral & 0.1560 & 0.1614 & 0.1255 & 0.1289 & 0.9234 & 0.9477 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the accuracy of the models established by different algorithms.  
Figure 8: Output error criterion \(J(\alpha)\) versus the commensurate differentiation order \(\alpha\) for the CD-player arm system.  
**Author Contributions:** Conceptualization, X.Y. and Y.L.; methodology, X.Y. and Y.L.; software, X.Y.; validation, X.Y. and Y.L.; formal analysis, X.Y.; investigation, Y.L.; resources, X.Y. and Y.L.; data curation, X.Y. and Y.L.; writing--original draft preparation, X.Y.; writing--review and editing, Y.L.; visualization, X.Y. and Y.L.; supervision, X.Y. and Y.L.; project administration, X.Y. and Y.L.; funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by the National Natural Science Foundation of China (Grant No. 61973137), the Natural Science Foundation of Jiangsu Province (Grant No. BK20201339), the China Postdoctoral Science Foundation (Grant No. 2022M711361), and the 111 project (B23008).  
**Data Availability Statement:** The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.  
**Conflicts of Interest:** The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.