## 4 Simulation Example  
Assume that the nonlinear block in Figure 1 is described by:  
\[\overline{u}(t)=u(t)+0.5u^{2}(t)+0.25u^{3}(t),\]  
and the linear continuous process \(P\) is described by:  
\[P(s)=\frac{1}{4s^{2}+2s+1}.\]  
Over a frame period of \(T=2\sqrt{2}+1\,\mathrm{s}\), the input-output data are non-uniformly-sampled twice (i.e., \(q=2\)) at \(t_{0}=0\,\mathrm{s}\) and \(t_{1}=\sqrt{2}\,\mathrm{s}\). According to Theorem 1 in [35], the following \(\delta^{-1}\)-based transfer function model is derived:  
\[w_{0}(k) =\frac{1-0.61623\delta^{-1}+0.50931\delta^{-2}}{1-0.8086\delta^{- 1}+0.25514\delta^{-2}}\overline{u}_{0}(k),\] \[w_{1}(k) =\frac{1-0.49522\delta^{-1}+0.75553\delta^{-2}}{1-0.94779\delta^{ -1}+0.57794\delta^{-2}}\overline{u}_{1}(k).\]  
Therefore, the parameters of this periodically non-uniformly sampled-data Hammerstein system are:  
\[\mathbf{a}_{0} =[-0.8086,\ 0.25514]^{\mathrm{ T}},\quad\mathbf{b}_{0}=[1,\ -0.61623,\ 0.50931]^{\mathrm{ T}},\] \[\mathbf{a}_{1} =[-0.94779,\ 0.57794]^{\mathrm{ T}},\quad\mathbf{b}_{1}=[1,\ -0.49522,\ 0.75553]^{\mathrm{ T}},\] \[\mathbf{c} =[1,\ 0.5,\ 0.25]^{\mathrm{ T}}.\]  
In the simulation, take the non-uniform inputs \(\{u_{0}(k)\}\) and \(\{u_{1}(k)\}\) as two uncorrelated random sequences with zero mean and unit variance, \(\{v_{0}(k)\}\) and \(\{v_{1}(k)\}\) as two white noise sequences with zero mean and variance \(\sigma^{2}=0.10^{2}\). Based on 5000 non-uniform input-output dataset, applying the AM-MISG algorithm in (14)-(25) with \(\lambda_{0}=\lambda_{1}=0.95\) to estimate the system parameters, the results for \(p=1\), \(p=5\) and \(p=12\) are shown in Tables 1-3, respectively, where \(\varepsilon\) is the estimation error defined as:  
\[\varepsilon=\sqrt{\frac{\|\hat{\mathbf{a}}_{0}-\mathbf{a}_{0}\|^{2}+\|\hat{\mathbf{b}}_{0}- \mathbf{b}_{0}\|^{2}+\|\hat{\mathbf{a}}_{1}-\mathbf{a}_{1}\|^{2}+\|\hat{\mathbf{b}}_{1}-\mathbf{b} _{1}\|^{2}+\|\hat{\mathbf{c}}-\mathbf{c}\|^{2}}{\|\mathbf{a}_{0}\|^{2}+\|\mathbf{b}_{0}\|^{2} +\|\mathbf{a}_{1}\|^{2}+\|\mathbf{b}_{1}\|^{2}+\|\mathbf{c}\|^{2}}}\times 100\%.\]  
Meanwhile, the estimation errors \(\varepsilon\) versus \(k\) are shown in Figure 3. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(k\) & **100** & **200** & **500** & **1000** & **2000** & **3000** & **4000** & **5000** & **True Values** \\ \hline \(a_{01}\) & \(-\)0.21097 & \(-\)0.29207 & \(-\)0.48337 & \(-\)0.62718 & \(-\)0.78913 & \(-\)0.82242 & \(-\)0.82167 & \(-\)0.81394 & \(-\)0.80860 \\ \(a_{02}\) & \(-\)0.09616 & \(-\)0.00999 & 0.04310 & 0.14720 & 0.23542 & 0.25631 & 0.25755 & 0.25552 & 0.25514 \\ \(b_{00}\) & 0.46831 & 0.60206 & 0.82476 & 0.92527 & 0.99463 & 0.99562 & 0.99966 & 1.00088 & 1.00000 \\ \(b_{01}\) & 0.05982 & 0.04225 & \(-\)0.06860 & \(-\)0.27395 & \(-\)0.53707 & \(-\)0.61472 & \(-\)0.62645 & \(-\)0.62093 & \(-\)0.61623 \\ \(b_{02}\) & 0.27987 & 0.32848 & 0.34589 & 0.38704 & 0.42283 & 0.47864 & 0.49652 & 0.50931 \\ \(a_{11}\) & \(-\)0.42997 & \(-\)0.51235 & \(-\)0.60352 & \(-\)0.72537 & \(-\)0.84006 & \(-\)0.90571 & \(-\)0.93254 & \(-\)0.94364 & \(-\)0.94779 \\ \(a_{12}\) & \(-\)0.00011 & 0.04270 & 0.21788 & 0.32975 & 0.47520 & 0.53405 & 0.56383 & 0.57217 & 0.57794 \\ \(b_{10}\) & 0.44256 & 0.61696 & 0.76526 & 0.92836 & 1.00028 & 0.99858 & 0.99669 & 1.00004 & 1.00000 \\ \(b_{11}\) & 0.01971 & 0.05871 & 0.01162 & \(-\)0.14602 & \(-\)0.33453 & \(-\)0.42806 & \(-\)0.46968 & \(-\)0.48801 & \(-\)0.49522 \\ \(b_{12}\) & 0.11625 & 0.16567 & 0.24897 & 0.36557 & 0.55820 & 0.68047 & 0.73284 & 0.75125 & 0.75553 \\ \(c_{2}\) & 0.88043 & 0.76552 & 0.64129 & 0.52932 & 0.49418 & 0.49855 & 0.50087 & 0.49915 & 0.50000 \\ \(c_{3}\) & 1.04485 & 0.79819 & 0.45534 & 0.30318 & 0.25542 & 0.25328 & 0.25163 & 0.25057 & 0.25000 \\ \(\varepsilon\) (\%) & 73.56646 & 62.33004 & 45.43354 & 29.28325 & 12.29280 & 4.72652 & 1.75023 & 0.55900 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The auxiliary model-based multi-innovation stochastic gradient (AM-MISG) parameter estimates and errors (\(p=1\)).