## 3 Identification Algorithms  
### The AM-SG Algorithm  
According to the over-parameterized linear regression approach [36], define the information vector \(\boldsymbol{\varphi}_{i}(k)\) and the parameter vector \(\boldsymbol{\theta}_{i}\) as:  
\[\boldsymbol{\varphi}_{i}(k) :=\begin{bmatrix}\boldsymbol{\varphi}_{iw}(k)\\ \boldsymbol{\varphi}_{iu}(k)\end{bmatrix}\in\mathbb{R}^{n_{0}},\quad \boldsymbol{\theta}_{i}:=\begin{bmatrix}\boldsymbol{a}_{i}\\ \boldsymbol{\theta}_{iu}\end{bmatrix}\in\mathbb{R}^{n_{0}},\ n_{0}=n+(n+1)n_{c},\] \[\boldsymbol{\varphi}_{iw}(k) :=\begin{bmatrix}-w_{i-1}(k)\\ -w_{i-2}(k)\\ \vdots\\ -w_{i-n}(k)\end{bmatrix}\in\mathbb{R}^{n},\quad\boldsymbol{\phi}_{ij}(k):= \begin{bmatrix}f_{1}[u_{i-j}(k)]\\ f_{2}[u_{i-j}(k)]\\ \vdots\\ f_{n_{c}}[u_{i-j}(k)]\end{bmatrix}\in\mathbb{R}^{n_{c}},\ j=0,\,1,\,2,\cdots, \,n,\] \[\boldsymbol{\varphi}_{iu}(k) :=\begin{bmatrix}\boldsymbol{\phi}_{i0}(k)\\ \boldsymbol{\phi}_{i1}(k)\\ \boldsymbol{\phi}_{i2}(k)\\ \vdots\\ \boldsymbol{\phi}_{in}(k)\end{bmatrix}\in\mathbb{R}^{(n+1)n_{c}},\quad \boldsymbol{\theta}_{iu}:=\begin{bmatrix}b_{i0}\boldsymbol{c}\\ b_{i1}\boldsymbol{c}\\ b_{i2}\boldsymbol{c}\\ \vdots\\ b_{in}\boldsymbol{c}\end{bmatrix}\in\mathbb{R}^{(n+1)n_{c}}.\]  
Using Equation (1), Equation (2) can be written in the following vector form:  
\[w_{i}(k) =-\sum_{j=1}^{n}a_{ij}w_{i-j}(k)+\sum_{j=0}^{n}b_{ij}\left[\sum_{ m=1}^{n_{c}}c_{m}f_{m}[u_{i-j}(k)]\right]\] \[=\boldsymbol{\varphi}_{i}^{\tau}(k)\boldsymbol{\theta}_{i}. \tag{4}\]  
Substituting Equation (4) into Equation (3), we have:  
\[y_{i}(k)=\boldsymbol{\varphi}_{i}^{\tau}(k)\boldsymbol{\theta}_{i}+v_{i}(k). \tag{5}\]  
Equation (5) is the identification model of the periodically non-uniformly sampled-data Hammerstein system, in which the parameter vector \(\boldsymbol{\theta}_{i}\) includes the products of parameters \(b_{ij}\) and \(c_{m}\). To guarantee a unique parametrization, the first coefficient \(c_{1}\) of the nonlinear function is assumed to be one [14; 37]. Furthermore, the information vector \(\boldsymbol{\varphi}_{i}(k)\) contains unmeasurable noise-free outputs \(w_{i-j}(k)\). A solution to this difficulty is to replace \(w_{i-j}(k)\) with their estimates \(\hat{w}_{i-j}(k)\) based on the auxiliary model identification idea [38; 39; 40]. Accordingly, define the estimate of \(\boldsymbol{\varphi}_{i}(k)\) as:  
\[\boldsymbol{\hat{\varphi}}_{i}(k) :=\begin{bmatrix}\boldsymbol{\hat{\varphi}}_{iw}(k)\\ \boldsymbol{\varphi}_{iu}(k)\end{bmatrix},\] \[\boldsymbol{\hat{\varphi}}_{iw}(k) :=[-\hat{w}_{i-1}(k),\ -\hat{w}_{i-2}(k),\ \cdots,\ -\hat{w}_{i-n}(k)]^{\tau}.\]  
Using the estimates \(\boldsymbol{\hat{\varphi}}_{i}(k)\) and \(\boldsymbol{\hat{\theta}}_{i}(k)\) to replace \(\boldsymbol{\varphi}_{i}(k)\) and \(\boldsymbol{\theta}_{i}\) in (4), respectively, yields  
\[\hat{w}_{i}(k)=\boldsymbol{\hat{\varphi}}_{i}^{\tau}(k)\boldsymbol{\hat{ \theta}}_{i}(k). \tag{6}\]  
Applying the stochastic gradient search principle to minimize the following criterion function:  
\[J_{1}(\boldsymbol{\theta}_{i})=\mathrm{E}\left\{[y_{i}(k)-\boldsymbol{\hat{ \varphi}}_{i}^{\tau}(k)\boldsymbol{\theta}_{i}]^{2}\right\},\]an auxiliary model-based stochastic gradient (AM-SG) algorithm can be derived for identifying the parameter vector \(\mathbf{\theta}_{i}\) in (5):  
\[\hat{\mathbf{\theta}}_{i}(k) =\hat{\mathbf{\theta}}_{i}(k-1)+\frac{\hat{\mathbf{\varphi}}_{i}(k)}{r_{i} (k)}[y_{i}(k)-\hat{\mathbf{\varphi}}_{i}^{\mathrm{ T}}(k)\hat{\mathbf{\theta}}_{i}(k-1)], \tag{7}\] \[r_{i}(k) =r_{i}(k-1)+\|\hat{\mathbf{\varphi}}_{i}(k)\|^{2},\ r_{i}(0)=1. \tag{8}\]