# A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models
## 3 Identification Algorithm
### Orthogonal Least Squares Algorithm Based on Householder Transformation  
We introduce a permutation matrix \(\boldsymbol{P}_{\Xi_{k}}\), which is constructed by performing \(k\) column swaps on an \(n\times n\) identity matrix \(\boldsymbol{I}_{n}\). The first \(k\) columns of \(\boldsymbol{P}_{\Xi_{k}}\) can be expressed as follows [26]:  
\[\boldsymbol{P}_{\Xi_{k}}(\cdot,1:k)=[\boldsymbol{\eta}_{5_{1}},\boldsymbol{ \eta}_{5_{2}},\cdots,\boldsymbol{\eta}_{5_{k}}]\in\mathbb{R}^{n\times k}, \tag{17}\]  
where \(\Xi_{k}=\{\xi_{1},\xi_{2},\cdots,\xi_{k}\}\), \(\xi_{k}\) denote the index of the chosen column in the \(k\)-th iteration, \(\boldsymbol{\eta}_{5_{i}}=[\boldsymbol{0}_{5_{i}-1},1,\boldsymbol{0}_{n-\xi_{ i}}]^{\mathrm{T}}\). Using the permutation matrix \(\boldsymbol{P}_{\Xi_{k}}\), the least squares criterion can be formulated as  
\[J(\boldsymbol{\theta})=\|\boldsymbol{Y}-(\boldsymbol{\Phi}\boldsymbol{P}_{\Xi _{k}})(\boldsymbol{P}_{\Xi_{k}}^{\mathrm{T}}\boldsymbol{\theta})\|^{2}=\| \boldsymbol{Y}-\boldsymbol{\Phi}_{\Xi_{k}}\boldsymbol{\theta}_{\Xi_{k}}\|^{2}, \tag{18}\]  
where \(\boldsymbol{\Phi}_{\Xi_{k}}\in\mathbb{R}^{m\times k}\) comprises the columns of \(\boldsymbol{\Phi}\) that correspond to \(k\) non-zero parameters, and \(\boldsymbol{\theta}_{k}\in\mathbb{R}^{\Xi_{k}}\) is a vector containing the \(k\) non-zero elements of \(\boldsymbol{\theta}\). Since the sub-information matrix \(\boldsymbol{\Phi}_{\Xi_{k}}\in\mathbb{R}^{m\times k}\) corresponding to \(\tilde{\xi}_{K}\) is a column full-rank matrix, there must exist an orthogonal matrix \(\boldsymbol{L}_{k}\in\mathbb{R}^{m\times m}\) and an upper triangular matrix \(\boldsymbol{T}_{k}\in\mathbb{R}^{k\times k}\) such that  
\[\boldsymbol{\Phi}_{\Xi_{k}}\Xi_{k}=\boldsymbol{L}_{k}\Bigg{[} \begin{array}{c} \boldsymbol{T}_{\Xi_{k}}\\ \boldsymbol{0}\end{array} \Bigg{]}, \tag{19}\]  
where \(\boldsymbol{0}\) is a zero matrix. Define  
\[\boldsymbol{Y}=\boldsymbol{L}_{k}\Bigg{[} \begin{array}{c}\boldsymbol{g}_{k} \\ \boldsymbol{h}_{k}\end{array} \Bigg{]}, \tag{20}\]with \(g_{k}\in\mathbb{R}^{k}\) and \(\mathbf{h}_{k}\in\mathbb{R}^{m-k}\); then, the criterion function (18) can be rewritten as  
\[J(\theta_{\Xi_{k}})=\left\|\mathbf{L}_{k}\left(\left[ \begin{array}{c}g_{k}\\ \mathbf{h}_{k}\end{array} \right]-\left[ \begin{array}{c}\mathbf{T}_{k}\\ \mathbf{0}\end{array} \right]\mathbf{\theta}_{k}\right)\right\|^{2}=\|\mathbf{g}_{k}-\mathbf{T} _{k}\theta_{k}\|^{2}+\|\mathbf{h}_{k}\|^{2}, \tag{21}\]  
By minimizing (21), we obtain \(\mathbf{T}_{k}\hat{\theta}_{k}=\mathbf{g}_{k}\) and \(J(\hat{\theta}_{k})=\|\mathbf{h}_{k}\|^{2}\). The solution \(\hat{\theta}_{k}\) can subsequently be derived using the following back-substitution method:  
\[ \begin{cases}\hat{\theta}_{i}=\frac{1}{T_{ii}}(g_{i}-\sum\limits_{k=1}^{k-i}T _{i+k}\hat{\theta}_{i_{k}}),&i=1,\cdots,k-1,\\ \hat{\theta}_{k}=\frac{g_{k}}{T_{ik}},&i=k,\end{cases}  \tag{22}\]  
where \(\hat{\theta}_{i}\) denotes the \(i\)-th element of \(\hat{\theta}_{k}\), and \(T_{ii}\) and \(g_{i}\) denote the \(i\)-th diagonal element of \(\mathbf{T}_{k}\) and the \(i\)-th element of \(g_{k}\), respectively. Using the permutation matrix \(P_{\Xi_{k}}\), we can recover the parameter estimation by  
\[\hat{\mathbf{\theta}}=\mathbf{P}_{\Xi_{k}}^{\mathrm{T}}\left[ \begin{array}{c}\hat{ \theta}_{k}\\ \mathbf{0}\end{array} \right]. \tag{23}\]  
The Householder transformation [27; 28] is adopted for QR decomposition in Equation (19). Let \(\chi_{\xi_{j}}\) denote the column of \(\mathbf{\Phi}\) indexed by \(\xi_{j}\), and \(\mathbf{I}_{j}\) represent the identity matrix of dimension \(j\). Define the Householder matrix as  
\[\mathbf{H}_{j}=\left[ \begin{array}{cc}\mathbf{I}_{j-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{\tilde{H}}_{j}\end{array}