# A Greedy Pursuit Hierarchical Iteration Algorithm
## 2 Systems Model  
Consider the following systems shown in Figure 1.  
The systems can be depicted as  
\[\mathcal{A}(t)y(d)=\sum_{i=1}^{r}t^{-\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac {\mathcal{F}(t)}{\mathcal{C}(t)}e(d), \tag{1}\]  
where \(u_{i}(d)\)and \(y(d)\) are the available input and output sampled data, and \(e(d)\) is the noise satisfying \(e(d)\sim N(0,\sigma^{2})\), respectively. \(\mathcal{A}(t),\mathcal{B}_{i}(t),\mathcal{C}(t),\mathcal{F}(t)\) are  
\[\mathcal{A}(t):=1+\sum_{j=1}^{n_{a}}a_{j}t^{-j},\mathcal{B}_{i}(t):=\sum_{j=1} ^{n_{h}}b_{ij}t^{-j},\mathcal{C}(t):=1+\sum_{j=1}^{n_{c}}c_{j}t^{-j},\mathcal{ F}(t):=1+\sum_{j=1}^{n_{f}}f_{j}t^{-j}.\]  
where \(n_{a},n_{bi},n_{c}\), and \(n_{f}\) are the system known orders and where the input time delay \(\tau_{i}\) of each channel is unknown. Introduce the internal noise item \(w(d)\)  
\[w(d):=\frac{\mathcal{F}(t)}{\mathcal{C}(t)}e(d)=-\sum_{j=1}^{n_{c}}c_{j}w(d-j )+\sum_{j=1}^{n_{f}}f_{j}e(d-j)+e(d)\]  
Figure 1: The multi-input systems with input time delays and colored noise. \[=\mathbf{\varphi}_{n}^{\mathrm{r}}(d)\mathbf{\theta}_{n}+e(d), \tag{2}\]  
where  
\[\mathbf{\varphi}_{n}(d):=[-w(d-1),-w(d-2),\cdots,-w(d-n_{c}),e(d-1),e(d-2), \cdots,e(d-n_{f})]^{\mathrm{r}}\in\mathbb{R}^{n_{c}+n_{f}}, \tag{3}\]  
\[\mathbf{\theta}_{n}:=[c_{1},c_{2},\cdots,c_{n_{c},f_{1}},f_{2},\cdots,f_{n_{f}}]^{ \mathrm{r}}\in\mathbb{R}^{n_{c}+n_{f}}. \tag{4}\]  
The subscript \(n\) denotes noise. Then, system model (1) can be rewritten as  
\[y(d)=[1-A(t)]y(d)+\sum_{i=1}^{r}t^{-\mathrm{r}}\mathcal{B}_{i}(t)u_{i}(d)+w(d). \tag{5}\]  
Since the time delay of each input channel is unknown, define the data regression length \(l\), which is large enough to satisfy \(l\geqslant\max(\tau_{i}+n_{li})\). Define the over-parameterized vector \(\mathbf{\varphi}_{s}(t)\) and \(\mathbf{\theta}_{s}\) as  
\[\mathbf{\varphi}_{s}(d):=[\mathbf{\varphi}_{n}^{\mathrm{r}}(d),\mathbf{\varphi}_{b}^{ \mathrm{r}}(d)]^{\mathrm{r}}\in\mathbb{R}^{n_{a}+lr}, \tag{6}\]  
\[\mathcal{A}(t)y(d)=\sum_{i=1}^{r}t^{-\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac {\mathcal{F}(t)}{\mathcal{C}(t)}e(d), \tag{1}\]0  
where  
\[\mathcal{A}(t)y(d)=\sum_{i=1}^{r}t^{-\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac {\mathcal{F}(t)}{\mathcal{C}(t)}e(d), \tag{1}\]1 \[\mathcal{A}(t)y(d)=\sum_{i=1}^{r}t^{-\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac {\mathcal{F}(t)}{\mathcal{C}(t)}e(d), \tag{1}\]2 \[\mathcal{A}(t)y(d)=\sum_{i=1}^{r}t^{-\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac {\mathcal{F}(t)}{\mathcal{C}(t)}e(d), \tag{1}\]3 \[\mathcal{A}(t)y(d)=\sum_{i=1}^{r}t^{-\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac {\mathcal{F}(t)}{\mathcal{C}(t)}e(d), \tag{1}\]4 \[\mathcal{A}(t)y(d)=\sum_{i=1}^{r}t^{-\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac {\mathcal{F}(t)}{\mathcal{C}(t)}e(d), \tag{1}\]5  
The subscript \(s\) denotes the system, and \(\mathbf{0}_{x}\) denotes the zero block where the subscript means the number of zero elements. Then, the following over-parameterized model can be indicated from Equation (5)  
\[ \begin{array}{l}y(d)=\mathbf{\varphi}_{s}^{\mathrm{r}}(d)\mathbf{\theta}_{s}+w(d)= \mathbf{\varphi}_{s}^{\mathrm{r}}(d)\mathbf{\theta}_{s}+\mathbf{\varphi}_{n}^{\mathrm{r}} (d)\mathbf{\theta}_{n}+e(d)\\ =\mathbf{\varphi}^{\mathrm{r}}(d)\mathbf{\theta}+e(d),\end{array}  \tag{8}\]  
where  
\[ \begin{array}{l}\mathbf{\varphi}(d):=[\mathbf{\varphi}_{s}^{\mathrm{r}}(d),\mathbf{ \varphi}_{n}^{\mathrm{r}}(d)]^{\mathrm{r}}\in\mathbb{R}^{N},N:=n_{a}+lr+n_{c} +n_{f},\\ \mathbf{\theta}:=[\mathbf{\theta}_{s}^{\mathrm{r}},\mathbf{\theta}_{n}^{\mathrm{r}}]^{ \mathrm{r}}\in\mathbb{R}^{N}.\end{array} \]  
Consider the sampled data from \(d=1\) to \(d=L\) and define \(\mathbf{Y},\mathbf{\Phi},\mathbf{E}\) as  
\[ \begin{array}{l}\mathbf{Y}:=[y(1),y(2),\cdots,y(L)]^{\mathrm{r}}\in\mathbb{R}^{L },\\ \mathbf{\Phi}:=[\mathbf{\varphi}(1),\mathbf{\varphi}(2),\cdots,\mathbf{\varphi}(L)]^{\mathrm{r} }\in\mathbb{R}^{L\times N},\\ \mathbf{E}:=[e(1),e(2),\cdots,e(L)]^{\mathrm{r}}\in\mathbb{R}^{L}.\end{array}