## 3 Identification Algorithm
The parameters are estimated by the BPDN criterion and the unmeasurable terms are replaced by the outputs of the auxiliary model in each iteration. Let \(k=1,2,\cdots\) be the iterative number. To get accurate reconstruction from Equation (17), the information matrix should satisfy some conditions, such as the RIP [29] or the exact recovery condition (ERC) [9]. The ERC guarantee and the consistency properties for the identification of the controlled autoregressive models have been investigated in [12]. To meet the ERC, we normalize the information matrix \(\hat{\mathbf{\Phi}}_{k}\) defined in Equations (11)-(13) by dividing the elements in each column by the \(l_{2}\) norm of that column [30; 31]. Denote the element of the \(i\)th row and \(j\)th column in \(\hat{\mathbf{\Phi}}_{k}\) by \(\hat{\mathbf{\Phi}}_{k,ij}\), and the normalized information matrix \(\hat{\mathbf{\Phi}}_{k,n}\) is constructed by  
\[\mathbf{\Phi}_{k}:=\mathbf{\Phi}_{k,n}\mathbf{\Phi}_{l_{2}},\quad\mathbf{\Phi}_{l_{2}}:=\left[ \begin{matrix}\hat{\Phi}_{k}(1)&0&\cdots&0\\ 0&\hat{\[PERSON]}_{k}(2)&&\vdots\\ \vdots&&\ddots&0\\ 0&\cdots&0&\hat{\[PERSON]}_{k}(n)\end{matrix} \right]\in\mathbb{R}^{n\times n}, \tag{18}\]  
where \(\Phi_{k}(j):=\sqrt{\sum_{i=1}^{m}(\Phi_{k,ij})^{2}}\). Similarly, the normalized parameter vector \(\mathbf{\theta}_{k,n}\) can be defined as  
\[\mathbf{\theta}_{n}:=\hat{\mathbf{\Phi}}_{l_{2}}\mathbf{\theta}. \tag{19}\]  
Note that the location of non-zeros in \(\mathbf{\theta}_{n}\) is identical to that in \(\mathbf{\theta}\). Accordingly, the constrained optimization problem in Equation (17) equals  
\[\hat{\mathbf{\theta}}_{n}=\arg\min\|\mathbf{\theta}_{n}\|_{1},\ \ \text{s.t.}\ \ \|Y-\hat{\mathbf{\Phi}}_{k,n}\mathbf{\theta}_{n}\|_{2}^{2}\leq\varepsilon. \tag{20}\]  
The problem in Equation (20) is closely related to the following unconstrained convex optimization problem  
\[\hat{\mathbf{\theta}}_{n}=\min_{\mathbf{\theta}_{n}}\frac{1}{2}\|Y-\hat{\mathbf{\Phi}}_{k,n}\mathbf{\theta}_{n}\|_{2}^{2}+\lambda\|\mathbf{\theta}_{n}\|_{1}, \tag{21}\]  
where \(\lambda\) is a nonnegative parameter. Since the information matrix \(\hat{\mathbf{\Phi}}_{k}\) is normalized, we can set \(\lambda\) to the value \(\lambda=\sigma\sqrt{2\log(n)}\)[16]. The key step of the BPDN approach is to express Equation (21) as a quadratic program (QP) problem [32]. To begin with, two nonnegative vectors \(\mathbf{u}_{n}\) and \(\mathbf{v}_{n}\) are introduced to express \(\mathbf{\theta}_{n}\). Let \(\theta_{nj}\), \(u_{nj}\) and \(v_{nj}\) be the \(j\)th element of the vectors \(\mathbf{\theta}_{n}\), \(\mathbf{u}_{n}\) and \(\mathbf{v}_{n}\), respectively, where \(u_{nj}\geq 0\) and \(v_{nj}\geq 0\) for all \(j=1,2,\cdots,n\). Define  
\[u_{nj}:=(\theta_{nj})_{+},\ \ v_{nj}:=(-\theta_{nj})_{+},\ \ (*)_{+}:=\max\{0,*\}.\]  
Then, \(\mathbf{\theta}_{n}\) can be rewritten as  
\[\mathbf{\theta}_{n}=\mathbf{u}_{n}-\mathbf{v}_{n}.\]  
Accordingly, the \(l_{1}\) regularization term \(\|\mathbf{\theta}_{n}\|_{1}\) can be expressed as  
\[\|\mathbf{\theta}_{n}\|_{1}=\mathbf{1}_{n}^{T}\mathbf{u}_{n}+\mathbf{1}_{n}^{T}\mathbf{v}_{n}=\bm {1}_{2n}^{T}\mathbf{z}_{n}, \tag{22}\]  
where \(\mathbf{1}_{n}^{T}:=[1,\cdots,1]\in\mathbb{R}^{n}\), \(\mathbf{1}_{2n}^{T}:=[1,\cdots,1]\in\mathbb{R}^{2n}\), and  
\[\mathbf{z}_{n}:=[\mathbf{u}_{n}^{T},\mathbf{v}_{n}^{T}]^{T}\in\mathbb{R}^{2n}. \tag{23}\]  
Note that all elements in \(\mathbf{z}_{n}\) are nonnegative. Similarly, the quadratic error term can be written as  
\[ \begin{split}&\|Y-\hat{\mathbf{\Phi}}_{k,n}\mathbf{\theta}_{n}\|_{2}^{2}\\ &=\|Y-[\hat{\mathbf{\Phi}}_{k,n}-\hat{\mathbf{\Phi}}_{k,n}|\mathbf{z}_{n}\|_{ 2}^{2}\\ &=\mathbf{Y}^{T}\mathbf{Y}-\mathbf{Y}^{T}[\hat{\mathbf{\Phi}}_{k,n}-\hat{\mathbf{\Phi }}_{k,n}|\mathbf{z}_{n}-\mathbf{z}_{n}^{T}[\hat{\mathbf{\Phi}}_{k,n}-\hat{\mathbf{\Phi}}_{k,n }]^{T}\mathbf{Y}+\mathbf{z}_{n}^{T}[\hat{\mathbf{\Phi}}_{k,n}-\hat{\mathbf{\Phi}}_{k,n}]^{T}[ \hat{\mathbf{\Phi}}_{k,n}-\hat{\mathbf{\Phi}}_{k,n}]\mathbf{z}_{n}.\end{split}