# A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models
## 3 Identification Algorithm
### Identification of Orders and Separation of Parameters
The H-GOLS, OMP, basis pursuit denoising (BPDN) [32], and the least absolute shrinkage and selection operator (LASSO) [33] algorithms are employed to identify the system; Figure 3 illustrates the parameter estimation errors for varying data lengths with the noise variance \(\delta^{2}=0.10^{2}\). It is observed that the H-GOLS, OMP, and LASSO algorithms effectively estimate system parameters even with restricted data lengths, with the H-GOLS algorithm exhibiting a higher degree of precision compared to the other two. In cases where  
Figure 1: The BIC curves versus \(k\). Figure 2: Output error criterion \(J(\alpha)\) versus the commensurate differentiation \(\alpha\). the data length is sufficient, all four methods are capable of accurately estimating the system parameters; however, the H-GOLS algorithm maintains the utmost estimation accuracy. With a data length fixed at \(m=500\), the system was characterized using the H-GOLS, BPDN, OMP, and LASSO algorithms. Figure 4 displays the parameter estimation errors across various noise variance levels. At lower noise variances, all four algorithms demonstrate effective parameter estimation capabilities, with the H-GOLS algorithm exhibiting the highest parameter estimation accuracy. Nevertheless, as the noise variance rises, both the BPDN and OMP algorithms exhibit growing parameter estimation errors, indicating that the H-GOLS algorithm not only achieves superior estimation accuracy but also exhibits better robustness against interference. Computational complexity serves as a critical performance metric for algorithm evaluation, quantifying the computational resources required for execution. It is conventionally characterized by the number of floating-point operations (FLOPs), primarily encompassing multiplication and addition operations. Let \(m=1000\); the computational requirements for the H-GOLS, BPDN, OMP, and LASSO algorithms are listed in Table 1, where \(K_{B}=50\) and \(K_{L}=100\) are the given iterations for the BPDN algorithm and the LASSO algorithm, respectively. Since \(K\) is significantly smaller than \(n\) (i.e., \(K\ll n\)), the H-GOLS algorithm apparently achieves the lowest computational complexity among the four methods. Figure 4: The parameter estimation errors \(\delta\) of different algorithms under different noise variances \(\sigma^{2}\). Figure 3: The parameter estimation errors \(\delta\) of different algorithms under different data lengths \(m\). Setting \(n_{a}=2\), \(l=3\), \(p=5\), and \(\varepsilon_{0}=0.05\), the H-GOLS algorithm is utilized to identify the following fractional Hammerstein model:  
\[y(t)= \sum_{i=1}^{2}\frac{B_{i}(s)}{A(s)}\dot{u}_{i}(t)+\frac{1}{A(s)}p( t),\] \[\hat{c}_{i,q}=\frac{1}{\hat{n}_{b_{i}}}\sum_{q=1}^{\hat{n}_{b_{i}}}\frac{\hat{ c}_{i,q}\hat{R}_{i,d}}{\hat{R}_{i,d}},\quad i=1,\cdots,r. \tag{40}\]0 \[\hat{c}_{i,q}=\frac{1}{\hat{n}_{b_{i}}}\sum_{q=1}^{\hat{n}_{b_{i}}}\frac{\hat{ c}_{i,q}\hat{R}_{i,d}}{\hat{R}_{i,d}},\quad i=1,\cdots,r. \tag{40}\]1 \[\hat{c}_{i,q}=\frac{1}{\hat{n}_{b_{i}}}\sum_{q=1}^{\hat{n}_{b_{i}}}\frac{\hat{ c}_{i,q}\hat{R}_{i,d}}{\hat{R}_{i,d}},\quad i=1,\cdots,r. \tag{40}\]2  
where \(\{v(t)\}\) is a noise sequence with zero mean and constant variance of \(\sigma^{2}=0.10^{2}\). By using the stopping criterion (37), the process stops after completing five iterations. Therefore, the sparsity level is \(K=4\). The estimated parameter vector is  
\[\hat{c}_{i,q}=\frac{1}{\hat{n}_{b_{i}}}\sum_{q=1}^{\hat{n}_{b_{i}}}\frac{\hat{ c}_{i,q}\hat{R}_{i,d}}{\hat{R}_{i,d}},\quad i=1,\cdots,r. \tag{40}\]3  
where \(P=n_{a}+lpr=32\). Figure 7 displays the output fittings for both the training and validation data. Figure 8 depicts the curve of the output error criterion \(J(\alpha)\) as a function of \(\alpha\), yielding an estimated fractional order \(\hat{\alpha}\) of \(0.10\).