## 5 Examples  
**Example 1**: Consider the following multi-input OEAR system:  
\[x_{1}(t) + a_{11}x_{1}(t-1)+a_{12}x_{1}(t-2)=b_{11}u_{1}(t-1)+b_{12}u_{1}(t -2), \tag{70}\] \[x_{2}(t) + a_{21}x_{2}(t-1)+a_{22}x_{2}(t-2)=b_{21}u_{2}(t-1)+b_{22}u_{2}(t -2),\] (71) \[w(t) + c_{1}w(t-1)=v(t),\] (72) \[y(t) = x_{1}(t)+x_{2}(t)+w(t). \tag{73}\]  
The parameter vector to be estimated is  
\[\mathbf{\theta} = [a_{11},a_{12},b_{11},b_{12},a_{21},a_{22},b_{21},b_{22},c_{1}]^{ \mathrm{ T}}\] \[= [0.15,0.25,0.99,-0.78,-0.10,0.35,-0.50,-0.80,0.20]^{\mathrm{ T}}.\]  
The inputs \(\{u_{1}(t),u_{2}(t)\}\) are taken as two persistent excitation signal sequences with zero mean and unit variance, and \(\{v(t)\}\) as a white noise sequence with zero mean and variance \(\sigma^{2}=0.20^{2}\) and \(\sigma^{2}=0.60^{2}\). Applying the F-RGLS algorithm to estimate the parameters of this example system, the parameter estimates and their estimation errors \(\delta:=\|\hat{\mathbf{\theta}}(t)-\mathbf{\theta}\|/\|\mathbf{\theta}\|\) are shown in Table 1. Applying the F-LSI algorithm to estimate the parameters of this example system, when the data length \(L=2000\) the parameter estimates and their estimation errors \(\delta:=\|\hat{\mathbf{\theta}}_{k}-\mathbf{\theta}\|/\|\mathbf{\theta}\|\) are shown in Table 2 with different noise variances. When the data length \(L=4000\), the parameter estimates and their errors \(\delta\) are shown in Table 3 with different noise variances. Under different noise variances and different data lengths, the parameter estimates and their errors \(\delta\) are shown in Table 4 when the iteration \(k=10\). Under different noise variances, the parameter estimation errors \(\delta\) versus \(k\) are shown in Figure 1. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\boldsymbol{\sigma^{2}}\) & \(\boldsymbol{t=L}\) & \(\boldsymbol{a_{11}}\) & \(\boldsymbol{a_{12}}\) & \(\boldsymbol{b_{11}}\) & \(\boldsymbol{b_{12}}\) & \(\boldsymbol{a_{21}}\) & \(\boldsymbol{a_{22}}\) & \(\boldsymbol{b_{21}}\) & \(\boldsymbol{b_{22}}\) & \(\boldsymbol{c_{1}}\) & \(\delta\) (\%) \\ \hline \(0.20^{2}\) & 100 & 0.14240 & 0.23494 & 0.96900 & \(-0.75613\) & \(-0.11071\) & 0.35200 & \(-0.51875\) & \(-0.80734\) & 0.00302 & 12.18486 \\  & 200 & 0.14579 & 0.25005 & 0.96382 & \(-0.78341\) & \(-0.10189\) & 0.35013 & \(-0.49750\) & \(-0.78942\) & 0.11045 & 5.68913 \\  & 500 & 0.15222 & 0.26364 & 0.97647 & \(-0.77308\) & \(-0.09060\) & 0.35005 & \(-0.48724\) & \(-0.78857\) & 0.13262 & 4.41957 \\  & 1000 & 0.15892 & 0.25970 & 0.98546 & \(-0.76410\) & \(-0.10005\) & 0.34897 & \(-0.49297\) & \(-0.78559\) & 0.15257 & 3.28600 \\  & 2000 & 0.15443 & 0.25820 & 0.98663 & \(-0.77005\) & \(-0.01214\) & 0.34954 & \(-0.49407\) & \(-0.79532\) & 0.15569 & 2.85038 \\  & 3000 & 0.15318 & 0.25979 & 0.99026 & \(-0.77528\) & \(-0.10025\) & 0.35272 & \(-0.49559\) & \(-0.79963\) & 0.15830 & 2.63155 \\ \hline \(0.60^{2}\) & 100 & 0.16661 & 0.22088 & 0.97687 & \(-0.76637\) & \(-0.04796\) & 0.28133 & \(-0.58256\) & \(-0.92613\) & \(-0.01077\) & 16.67281 \\  & 200 & 0.14230 & 0.24704 & 0.95213 & \(-0.83607\) & \(-0.06133\) & 0.30376 & \(-0.51120\) & \(-0.82406\) & 0.10977 & 7.91161 \\  & 500 & 0.15181 & 0.28154 & 0.96971 & \(-0.78348\) & \(-0.06034\) & 0.33148 & \(-0.46893\) & \(-0.78559\) & 0.14472 & 5.25893 \\  & 1000 & 0.17142 & 0.27161 & 0.98984 & \(-0.74692\) & \(-0.09093\) & 0.33498 & \(-0.48227\) & \(-0.76910\) & 0.15722 & 4.45410 \\  & 2000 & 0.16154 & 0.27069 & 0.98704 & \(-0.75643\) & \(-0.10049\) & 0.34152 & \(-0.48438\) & \(-0.79306\) & 0.16109 & 3.31329 \\  & 3000 & 0.15847 & 0.27633 & 0.99597 & \(-0.76992\) & \(-0.09664\) & 0.35335 & \(-0.48844\) & \(-0.80372\) & 0.16376 & 2.95257 \\ \hline True values & 0.15000 & 0.25000 & 0.99000 & \(-0.78000\) & \(-0.10000\) & 0.35000 & \(-0.50000\) & \(-0.80000\) & 0.20000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The F-RGLS estimates and their errors for Example 1.