## 4 Examples  
**Example 1**. : _Consider the following state-space system with moving average noise,_  
\[\mathbf{x}(k+1) =\left[ \begin{array}{cc}0&1\\ g_{1}&g_{2}\end{array} \right]\mathbf{x}(k)+\left[ \begin{array}{c}h_{1}\\ h_{2}\end{array} \right]u(k)=\left[ \begin{array}{cc}0&1\\ -0.90&0.80\end{array} \right]\mathbf{x}(k)+\left[ \begin{array}{c}1.10\\ -1.60\end{array} \right]u(k),\] \[y(k) =[1,0]\mathbf{x}(k)+e(k),\] \[e(k) =(1+e_{1}z^{-1}+e_{2}z^{-2})v(k)=(1+0.20z^{-1}-0.60z^{-1})v(k).\]  
The parameter vector to be estimated is:  
\[\mathbf{\theta}=[g_{1},g_{2},h_{1},h_{2},e_{1},e_{2}]^{\tau}=[-0.90,0.80,1.10,-1.60,0.20,-0.60]^{\tau}.\]  
In simulation, the input \(\{u(k)\}\) is set as a persistent excitation sequence and \(\{v(k)\}\) as a zero-mean noise sequence with variance \(\delta=0.25\). Take the data length \(L=1200\). Use the first 1000 data and apply the BC-PSE algorithm to get the estimates of the parameter vector \(\mathbf{\theta}\) and the system state \(\mathbf{x}(k)\). The parameter estimates and their errors under different noise variances are displayed in Table 2 and Figure 1, where the estimation error \(\tau:=\|\hat{\mathbf{\theta}}(k)-\mathbf{\theta}\|/\|\mathbf{\theta}\|\times 100\%\). The parameter estimate \(\hat{\mathbf{\theta}}(k)\) versus \(k\) is plotted in Figure 2, and the state estimate \(\hat{\mathbf{x}}(k)\) computed by the BC-PSE algorithm is illustrated in Figures 3 and 4. The remaining 200 data from \(k=1001\)-1200 are taken to test the effectiveness of the estimated model. As a comparison, the curves of the estimated output \(\hat{y}(k)\) and the predicted output \(y(k)\) are depicted in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\delta\) & \(k\) & \(g_{1}\) & \(g_{2}\) & \(h_{1}\) & \(h_{2}\) & \(e_{1}\) & \(e_{2}\) & \(\tau\)**(\%)** \\ \hline
1.00 & 20 & \(-0.90672\) & 0.58805 & 2.47895 & \(-2.33308\) & 0.12542 & \(-1.41696\) & 74.94801 \\  & 50 & \(-0.91146\) & 0.77864 & 1.56458 & \(-1.45439\) & 0.06347 & \(-0.68669\) & 21.66474 \\  & 100 & \(-0.89769\) & 0.78062 & 1.45065 & \(-1.56252\) & 0.11477 & \(-0.61555\) & 15.33989 \\  & 200 & \(-0.89947\) & 0.79122 & 1.18977 & \(-1.58852\) & 0.18174 & \(-0.62250\) & 4.02568 \\  & 500 & \(-0.90289\) & 0.79326 & 1.19050 & \(-1.56703\) & 0.17911 & \(-0.62993\) & 4.35605 \\  & 1000 & \(-0.89502\) & 0.79742 & 1.14729 & \(-1.63721\) & 0.18966 & \(-0.61945\) & 2.71347 \\ \hline
0.25 & 20 & \(-0.89902\) & 0.68391 & 1.87036 & \(-1.86695\) & 0.37006 & \(-0.70475\) & 35.74537 \\  & 50 & \(-0.90830\) & 0.79067 & 1.34730 & \(-1.49092\) & 0.18690 & \(-0.51892\) & 11.92791 \\  & 100 & \(-0.90003\) & 0.79044 & 1.28652 & \(-1.56142\) & 0.20501 & \(-0.50891\) & 8.91773 \\  & 200 & \(-0.90083\) & 0.79596 & 1.15147 & \(-1.58293\) & 0.22360 & \(-0.53086\) & 3.84174 \\  & 500 & \(-0.90199\) & 0.79695 & 1.14809 & \(-1.57856\) & 0.20617 & \(-0.56528\) & 2.67739 \\  & 1000 & \(-0.89776\) & 0.79869 & 1.12493 & \(-1.61634\) & 0.20773 & \(-0.57641\) & 1.63983 \\ \hline True values & \(-0.90000\) & 0.80000 & 1.10000 & \(-1.60000\) & 0.20000 & \(-0.60000\) & 0.00000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The bias compensation-based parameter estimates and errors for Example 1. Figure 1: The estimation errors \(\tau\) versus \(k\) with different noise variances for Example 1. Figure 2: The parameter estimates \(\hat{\theta}(k)\) versus \(k\) for Example 1. Figure 4: The state \(x_{2}(k)\) and the estimated state \(\hat{x}_{2}(k)\) versus \(k\) for Example 1 (\(\delta=0.25\)). Figure 5: The actual output \(y(k)\) and the estimated output \(\hat{g}(k)\) versus \(k\) for Example 1. Figure 3: The state \(x_{1}(k)\) and the estimated state \(\hat{x}_{1}(k)\) versus \(k\) for Example 1 (\(\delta=0.25\)). Figures 9-11 plot the dynamics of three state estimates \(\hat{\pi}_{i}(k),\ i=1,2,3\). Figure 12 describes the output comparison of \(\hat{y}(k)\) with \(y(k)\). From Tables 2 and 3 and Figures 1-12, we can draw some conclusions as follows.