## 3 Identification Algorithms
### The AM-MISG Algorithm
Substituting Equation (13) into Equation (12), the auxiliary model-based multi-innovation stochastic gradient (AM-MISG) algorithm can be derived:  
\[\mathbf{V}_{i}(p,k) =[v_{i}(k),\ v_{i}(k-1),\ \cdots,\ v_{i}(k-p+1)]^{\mathrm{ T}}\in\mathbb{R}^{p},\]3 \[\mathbf{V}_{i}(p,k) =[v_{i}(k),\ v_{i}(k-1),\ \cdots,\ v_{i}(k-p+1)]^{\mathrm{ T}}\in\mathbb{R}^{p},\]4 (15) \[\mathbf{V}_{i}(p,k) =[v_{i}(k),\ v_{i}(k-1),\ \cdots,\ v_{i}(k-p+1)]^{\mathrm{ T}}\in\mathbb{R}^{p},\]5 (16) \[\mathbf{V}_{i}(p,k) =[v_{i}(k),\ v_{i}(k-1),\ \cdots,\ v_{i}(k-p+1)]^{\mathrm{ T}}\in\mathbb{R}^{p},\]6 (17) \[\mathbf{\hat{\varphi}}_{i}(k) = \begin{bmatrix}\mathbf{\hat{\varphi}}_{iw}(k)\\ \mathbf{\varphi}_{iu}(k)\end{bmatrix} ,\] (18) \[\mathbf{\hat{\varphi}}_{iw}(k) =[-\hat{w}_{i-1}(k),\ -\hat{w}_{i-2}(k),\ \cdots,\ -\hat{w}_{i-n}(k)]^{ \mathrm{T}},\] (19) \[\mathbf{\varphi}_{iu}(k) =[\mathbf{\varphi}_{10}^{\mathrm{T}}(k),\ \mathbf{\hat{\varphi}}_{11}^{ \mathrm{T}}(k),\ \mathbf{\varphi}_{12}^{\mathrm{T}}(k),\ \cdots,\ \mathbf{\hat{\varphi}}_{in}^{ \mathrm{T}}(k)]^{\mathrm{T}},\] (20) \[\mathbf{\hat{\varphi}}_{ij}(k) =[f_{1}[u_{i-j}(k)],\ f_{2}[u_{i-j}(k)],\ \cdots,\ f_{n_{c}}[u_{i-j}(k)]]^{ \mathrm{T}},\ j=0,\ 1,\ 2,\cdots,\ n,\] (21) \[\hat{w}_{i}(k) =\mathbf{\hat{\varphi}}_{i}^{\mathrm{T}}(k)\mathbf{\hat{\theta}}_{i}(k). \tag{22}\]  
Since \(c_{1}=1\), the estimates of \(a_{ij}\) and \(b_{ij}\) can be directly read from \(\hat{\mathbf{\theta}}_{i}(k)\),  
\[\hat{a}_{ij}(k) =\hat{\mathbf{\theta}}_{i,j}(k),\ j=1,\ 2,\ \cdots,\ n, \tag{23}\] \[\hat{b}_{ij}(k) =\hat{\mathbf{\theta}}_{i,n+n_{c}j+1}(k),\ j=0,\ 1,\ 2,\cdots,\ n, \tag{24}\]  
where \(\hat{\mathbf{\theta}}_{i,j}(k)\) represents the \(j\)-th element of \(\hat{\mathbf{\theta}}_{i}(k)\). Note that \(c_{m}\ (m=2,\ 3,\ \cdots,\ n_{c})\) has been estimated \(n+1\) times at each non-uniform sampling instant \(kT+t_{i}\ (i=0,\ 1,\ 2,\cdots,\ q-1)\). Therefore, we can simply take their average like in [36] as the estimate of \(c_{m}\) over the \(k\)-th frame period, i.e.,  
\[\hat{c}_{m}(k)=\frac{1}{q(n+1)}\sum_{i=0}^{q-1}\sum_{j=0}^{n}\frac{\hat{\mathbf{ \theta}}_{i,n+n_{c}j+m}(k)}{\hat{b}_{ij}(k)},\ m=2,\ 3,\ \cdots,\ n_{c}. \tag{25}\]  
Furthermore, a more numerically-sound SVD-based approach proposed by [PERSON] [41] can be applied to obtain the estimates of \(b_{ij}\) and \(c_{m}\). The flowchart of the AM-MISG algorithm in (14)-(25) for computing the parameter estimates of periodically non-uniformly sampled-data Hammerstein systems can be illustrated in Figure 2, and the detailed implementation steps are summarized as follows:  
1. Initialization: Choose the data length \(L\), the innovation length \(p\) and the forgetting factor \(\lambda_{i}\); give the nonlinear basis functions \(\{f_{m}(\cdot),\ m=1,\ 2,\ \cdots,\ n_{c}\}\); set \(u_{i}(k)=0,y_{i}(k)=0,\ \hat{w}_{i}(k)=0\) for \(k\leqslant 0\) and \(i=0,\ 1,\ 2,\ \cdots,\ q-1\); take the initial values to be \(\hat{\mathbf{\theta}}_{i}(0)=\mathbf{1}_{n_{0}}/p_{0}\), where \(p_{0}=10^{6}\) and \(\mathbf{1}_{n_{0}}\) is a column vector of ones; let \(k=1\) and \(i=0\). 2. Collect the non-uniformly sampled input-output data \(u_{i}(k)\) and \(y_{i}(k)\). Calculate \(f_{m}[u_{i}(k)]\) based on \(u_{i}(k)\); form \(\hat{\mathbf{\varphi}}_{iu}(k)\), \(\mathbf{\varphi}_{iu}(k)\), \(\mathbf{\phi}_{ij}(k)\) by (19)-(21); and construct \(\hat{\mathbf{\varphi}}_{i}(k)\) by (18). 4. Form the stacked output vector \(\mathbf{Y}_{i}(p,k)\) and the stacked information matrix \(\widehat{\mathbf{\Psi}}_{i}(p,k)\) by (16) and (17), respectively. 5. Compute the step size \(r_{i}(k)\) by (15) and update the parameter estimate \(\hat{\mathbf{\theta}}_{i}(k)\) by (14); calculate \(\hat{w}_{i}(k)\) by (22); obtain \(\hat{a}_{ij}(k)\) and \(\hat{b}_{ij}(k)\) based on (23) and (24), respectively. 6. If \(i<q-1\), then increase \(i\) by one, and go to Step 2; otherwise, compute \(\hat{\mathbf{c}}_{m}(k),\ m=2,\ 3,\ \cdots,\ n_{c}\) by (25); let \(i=0\) and go to the next step. 7.