## 6 Experiment  
Consider the following Hammerstein system:  
\[A(z)y(k) = B(z)x(k)+v(k),\] \[x(k) = \left\{ \begin{array}{ll}m_{1}u(k)&u(k)\geq 0\\ m_{2}u(k)&u(k)<0\end{array} \right.,\] \[A(z) = 1+a_{1}z^{-1}+a_{2}z^{-2}\] \[= 1+0.20z^{-1}-0.35z^{-2},\] \[B(z) = 1+b_{1}z^{-1}+b_{2}z^{-2}\] \[= 1+0.80z^{-1}+0.60z^{-2},\] \[m_{1} = 1.50,\ m_{2}=-1.00.\]  
Suppose \(q=2\); then, we have:  
\[\varepsilon(z)=1-0.20z^{-1}-0.35z^{-2}.\]  
The corresponding dual-rate model can be expressed as:  
\[\alpha(z)y(k) = \beta(z)x(k),\] \[\alpha(z) = 1+\alpha_{1}z^{-2}+\alpha_{2}z^{-4}\] \[= 1-0.74z^{-2}+0.1225z^{-4},\] \[\beta(z) = 1+\upbeta_{1}z^{-1}+\upbeta_{2}z^{-2}+\upbeta_{3}z^{-3}+\upbeta_ {4}z^{-4}\] \[= 1+0.20z^{-1}-0.35z^{-2},\]0 \[= 1+0.20z^{-1}-0.35z^{-2},\]1 \[\mathbf{\theta} = \left[ \begin{array}{c}\mathbf{a}\\ \mathbf{b}\end{array} \right],\] \[\mathbf{\alpha} = [-0.74,\ 0.1225]^{\rm{T}},\ \mathbf{\beta}=[0.60,\ 0.09,\ -0.40,\ -0.21]^{\rm{T}},\] \[\mathbf{\Theta} = [m_{2},m_{1}-m_{2},\mathbf{\beta}^{\rm{T}},\mathbf{\alpha}^{\rm{T}}]^{ \rm{T}}.\]In the simulation experiment, the input \(\{u(t)\}\) is taken as an uncorrelated persistently-excited normal distribution signal sequence with zero mean and unit variance, and \(\{v(t)\}\) is taken as a white noise sequence with zero mean and variance \(\sigma^{2}=0.50^{2}\) and \(\sigma^{2}=1.00^{2}\). Apply the proposed algorithm to estimate the parameters of this system. The parameter estimation errors of the dual-rate model are:  
\[\delta:=\sqrt{\frac{\|\hat{m}_{1}(kq)-m_{1}\|^{2}+\|\hat{m}_{2}(kq)-m_{2}\|^{2} +\|\hat{\mathbf{\beta}}(kq)-\mathbf{\beta}\|^{2}+\|\hat{\mathbf{\alpha}}(kq)-\mathbf{\alpha}\|^ {2}}{m_{1}^{2}+m_{2}^{2}+\|\mathbf{\beta}\|^{2}+\|\mathbf{\alpha}\|^{2}}}\times 100\%,\]  
the parameter estimation errors of the single-rate model are:  
\[\delta:=\sqrt{\frac{\|\hat{m}_{1}(kq)-m_{1}\|^{2}+\|\hat{m}_{2}(kq)-m_{2}\|^{2 }+\|\hat{\mathbf{\alpha}}(kq)-\mathbf{\alpha}\|^{2}+\|\hat{\mathbf{b}}(kq)-\mathbf{b}\|^{2}}{ m_{1}^{2}+m_{2}^{2}+\|\mathbf{a}\|^{2}+\|\mathbf{b}\|^{2}}}\times 100\%.\]  
The parameter estimates and their errors are shown in Tables 1 and 2. Obviously, the parameter estimation errors become (generally) smaller and smaller with the data length \(k\) increasing. This shows that the proposed algorithm is effective. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{\sigma^{2}}\) & \multicolumn{3}{c}{\(\mathbf{0.50^{2}}\)} & \multicolumn{3}{c}{\(\mathbf{1.00^{2}}\)} \\ \hline \(t\) & 100 & 1000 & 2000 & 3000 & 100 & 1000 & 2000 & 3000 \\ \hline \(m_{1}\)= 1.5000 & 1.64510 & 1.49306 & 1.48069 & 1.48674 & 1.64332 & 1.47369 & 1.45489 & 1.46773 \\ \(m_{2}\)=-1.0000 & \(-\)1.10137 & \(-\)1.00671 & \(-\)0.98836 & \(-\)0.99494 & \(-\)1.07085 & \(-\)0.98956 & -0.96557 & -0.98115 \\ \(\beta_{1}\)= 0.6000 & 0.44924 & 0.56619 & 0.58761 & 0.60112 & 0.46040 & 0.54639 & 0.58138 & 0.60706 \\ \(\beta_{2}\)= 0.0900 & \(-\)0.02088 & 0.04436 & 0.07612 & 0.08695 & \(-\)0.07241 & 0.04529 & 0.09448 & 0.10813 \\ \(\beta_{3}\)=\(-\)0.4000 & \(-\)0.10421 & \(-\)0.33348 & \(-\)0.35710 & \(-\)0.36657 & \(-\)0.13676 & \(-\)0.33817 & \(-\)0.35584 & \(-\)0.36336 \\ \(\beta_{4}\)=\(-\)0.2100 & \(-\)0.16787 & \(-\)0.18590 & \(-\)0.20263 & \(-\)0.20855 & \(-\)0.14399 & \(-\)0.16044 & -0.19223 & -0.20651 \\ \(\alpha_{1}\)=\(-\)0.7400 & \(-\)0.70650 & \(-\)0.75683 & \(-\)0.74221 & \(-\)0.73669 & \(-\)0.77498 & \(-\)0.75709 & \(-\)0.72961 & \(-\)0.72063 \\ \(\alpha_{2}\)= 0.1225 & 0.15543 & 0.14357 & 0.12683 & 0.12660 & 0.18953 & 0.13963 & 0.11305 & 0.11506 \\ \hline \(\delta\) (\(\%\))=0.0000 & 18.97213 & 4.54356 & 2.51473 & 1.76071 & 18.54456 & 5.34749 & 3.71454 & 2.84976 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The dual-rate parameter estimates and errors.