# A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models
## 3 Identification Algorithm
### Identification of Orders and Separation of Parameters  
Given the block sparse nature of \(\mathbf{\theta}\), it becomes possible to obtain an efficient parameter estimate, denoted as \(\hat{\mathbf{\theta}}\), which preserves the identical sparse structure. Within this sparse structure, \(\hat{\mathbf{\theta}}\) comprises \(\sum_{i=1}^{r}\hat{m}_{i}+1\) zero blocks. Let \(Z_{\varrho}\) represent the number of zeros in each zero block, where \(\varrho\) ranges from \(1\) to \(\sum_{i=1}^{r}\hat{m}_{i}+1\). Subsequently, the size of \(Z_{\varrho}\) can be detected sequentially. Upon detecting that \(Z_{\varrho}>l\), the nonlinear order for each input channel can be estimated as  
\[\hat{m}_{i}=\varrho-\sum_{j=0}^{i-1}\hat{m}_{j}-1,\ \hat{m}_{0}=0,\ i=1,2, \cdots,r. \tag{39}\]  
Based on the structure of \(\hat{\mathbf{\theta}}\), we can deduce that the number of elements within each non-zero block, ranging from the \((\sum_{j=1}^{i}\hat{m}_{j-1}+2)\)-th to the \((\sum_{j=1}^{i}\hat{m}_{j}+1)\)-th non-zero block, is constant. The linear orders \(\hat{n}_{b_{i}}\) can be determined by counting the number of elements in any of these non-zero blocks. According to Equations (13)-(15), it is evident that all non-zero blocks in \(\hat{\mathbf{\theta}}\) contain mixed parameters, with the exception of the first block. We can obtain \(\hat{\mathbf{a}}\) and \(\hat{\mathbf{b}}_{i}\) from the structure of \(\mathbf{\theta}\). Since \(c_{i,1}=1\), the \((\sum_{j=1}^{i}\hat{m}_{j-1}+2)\)-th non-zero block corresponds to \(\hat{\mathbf{R}}_{i}^{T}\). The estimations of \(\hat{c}_{i,q},q=2,\ldots,\hat{m}_{i}\) can be derived using the parameters from the \((\sum_{j=1}^{i}\hat{m}_{j-1}+2)\)-th to the \((\sum_{j=1}^{i}\hat{m}_{j}+1)\)-th non-zero block and \(\hat{\mathbf{R}}_{i}\). Notably, \(\hat{c}_{i,q}\) can be computed using any one of \(\frac{\hat{c}_{i,q}\hat{R}_{i,d}}{\hat{R}_{i,d}}\). Thus, the average of these computed values is taken as the estimation of \(\hat{c}_{i,q}\), i.e.,  
\[\hat{c}_{i,q}=\frac{1}{\hat{n}_{b_{i}}}\sum_{q=1}^{\hat{n}_{b_{i}}}\frac{\hat{ c}_{i,q}\hat{R}_{i,d}}{\hat{R}_{i,d}},\quad i=1,\cdots,r. \tag{40}\]  
In order to obtain the fractional order, we introduce an output error criterion function as [16]  
\[J(\alpha)=10\log\frac{\|\mathbf{e}(t,\hat{\mathbf{\theta}}_{\alpha})\|^{2}}{\|\mathbf{y}( t)\|^{2}}, \tag{41}\]  
where \(\mathbf{e}(t,\hat{\mathbf{\theta}}_{\alpha})=\mathbf{y}(t)-\mathbf{\Phi}\hat{\mathbf{\theta}}_{ \alpha}\), \(\hat{\mathbf{\theta}}_{\alpha}\) is the estimated parameter vector for the fractional order \(\alpha\). By minimizing this criterion function over different fractional orders, we can determine the approximate fractional order \(\hat{\alpha}\). The H-GOLS algorithm is systematically detailed in the following Algorithm 1, offering a comprehensive suite of implementation procedures. \[\hat{\mathbf{\vartheta}}=[-0.4007,0.7998,0.8493,0.6256,1.2661,1.2017,0.5983,0.7680,0.4 984,0.4071,0.6937,0.5963]^{\mathrm{T}}.\]  
Defining the parameter estimation error as \(\delta=\frac{\|\mathbf{\vartheta}-\hat{\mathbf{\vartheta}}\|}{\|\hat{\mathbf{\vartheta}}\| }\times 100\%\), we obtain \(\delta=5.91\%\). Based on Equations (39)-(40), the following estimates are obtained:  
\[\hat{m}_{1}=3,\;\hat{m}_{2}=3,\;\hat{n}_{b_{1}}=3,\;\hat{n}_{b_{2}}=3.\]  
The curve of the output error criterion \(J(\alpha)\) for different values of \(\alpha\) is shown in Figure 2, allowing the estimated fractional order to be determined as \(\hat{\alpha}=0.30\). From the above results, it is clear that the proposed H-GOLS algorithm effectively estimates the system's structure with limited data. Moreover, by implementing stopping rule (37), the algorithm reduces the computational burden while enhancing parameter estimation accuracy.