# Coupled Least Squares Identification Algorithms for Multivariate Output-Error Systems
## 4 Examples  
**Example 1**. : _Consider the following multivariate output-error system:_  
\[\mathbf{y}(t) =\frac{\mathbf{\Phi}_{s}(t)\mathbf{\theta}}{A(z)}+\mathbf{v}(t),\] \[\mathbf{\Phi}_{s}(t) =\left[ \begin{array}{cc}y_{1}(t-2)u_{2}(t-2)&y_{1}(t-2)\sin(t/ \pi)&u_{1}(t-1)+u_{2}(t-2)&u_{2}(t-1)\sin(u_{2}(t-2))\\ y_{1}(t-2)\sin(y_{2}(t-2))&y_{2}(t-2)u_{1}(t-2)&u_{1}(t-2)u_{2}(t-2)&u_{2}(t-1 )\cos(t/\pi)\end{array} \right],\] \[\mathbf{\theta} =[\theta_{1},\theta_{2},\theta_{3},\theta_{4}]^{\intercal}\] \[=[-0.25,0.47,-0.50,0.57]^{\intercal}\in\mathbb{R}^{4},\] \[A(z) =1+a_{1}z^{-1}+a_{2}z^{-2}=1+0.30z^{-1}+0.64z^{-2},\] \[\mathbf{\theta} =[a_{1},a_{2},\theta_{1},\theta_{2},\theta_{3},\theta_{4}]^{\intercal}\] \[=[0.30,0.64,-0.25,0.47,-0.50,0.57]^{\intercal}\in\mathbb{R}^{6}.\]  
_In simulation, we generate two persistent excitation signal sequences with zero mean and unit variances as the inputs \(\{u_{1}(t)\}\) and \(\{u_{2}(t)\}\), and take \(v_{1}(t)\) and \(v_{2}(t)\) to be two white noise sequences with zero mean and variances \(\sigma_{1}^{2}\) for \(v_{1}(t)\), and \(\sigma_{2}^{2}\) for \(v_{2}(t)\). Taking \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}=0.50^{2}\), the data length \(L\) = 3000, and applying the AM-RLS, C-S-AM-RLS and C-AM-RLS algorithms to estimate the parameters of this system, respectively, the parameter estimates are shown in Tables 1-3, and the estimation errors \(\delta:=\|\hat{\mathbf{\theta}}(t)-\mathbf{\theta}\|/\|\mathbf{\theta}\|\) versus \(t\) are shown in Figures 1 and 2._  
From Tables 1-3 and Figures 1 and 2, we can draw the following conclusions. * The parameter estimation errors by the presented algorithms become smaller and smaller and go to zero with the increasing of time \(t\). * In contrast to the AM-RLS algorithm, the proposed C-S-AM-RLS and C-AM-RLS algorithms have faster convergence rates and more accurate parameter estimates with the same simulation conditions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(t\) & \(a_{1}\) & \(a_{2}\) & \(\theta_{1}\) & \(\theta_{2}\) & \(\theta_{3}\) & \(\theta_{4}\) & \(\delta\) (\%) \\ \hline
100 & 0.29097 & 0.63357 & \(-\)0.27254 & 0.45781 & \(-\)0.49700 & 0.52648 & 4.44497 \\
200 & 0.29760 & 0.62737 & \(-\)0.26354 & 0.45916 & \(-\)0.50241 & 0.54735 & 2.69357 \\
500 & 0.29689 & 0.63747 & \(-\)0.25217 & 0.46685 & \(-\)0.50539 & 0.55960 & 1.11229 \\
1000 & 0.29887 & 0.63910 & \(-\)0.25196 & 0.46698 & \(-\)0.50283 & 0.55826 & 1.08848 \\
2000 & 0.30151 & 0.63810 & \(-\)0.24935 & 0.46850 & \(-\)0.50098 & 0.55961 & 0.92996 \\
3000 & 0.30313 & 0.63900 & \(-\)0.24647 & 0.46872 & \(-\)0.50302 & 0.56643 & 0.58689 \\ \hline True values & 0.30000 & 0.64000 & \(-\)0.25000 & 0.47000 & \(-\)0.50000 & 0.57000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The C-S-AM-RLS estimates and their errors for Example 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(t\) & \(a_{1}\) & \(a_{2}\) & \(\theta_{1}\) & \(\theta_{2}\) & \(\theta_{3}\) & \(\theta_{4}\) & \(\delta\) (\%) \\ \hline
100 & 0.29097 & 0.63357 & \(-\)0.27254 & 0.45781 & \(-\)0.49700 & 0.52648 & 4.44497 \\
200 & 0.29760 & 0.62737 & \(-\)0.26354 & 0.45916 & \(-\)0.50241 & 0.54735 & 2.69357 \\
500 & 0.29689 & 0.63747 & \(-\)0.25217 & 0.46685 & \(-\)0.50539 & 0.55960 & 1.11229 \\
1000 & 0.29887 & 0.63910 & \(-\)0.25196 & 0.46698 & \(-\)0.50283 & 0.55826 & 1.08848 \\
2000 & 0.30151 & 0.63810 & \(-\)0.24935 & 0.46850 & \(-\)0.50098 & 0.55961 & 0.92996 \\
3000 & 0.30313 & 0.63900 & \(-\)0.24647 & 0.46872 & \(-\)0.50302 & 0.56643 & 0.58689 \\ \hline True values & 0.30000 & 0.64000 & \(-\)0.25000 & 0.47000 & \(-\)0.50000 & 0.57000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The C-AM-RLS estimates and their errors for Example 1. Figure 1: The AM-RLS and the C-S-AM-RLS estimation errors \(\delta\) versus \(t\) for Example 1. Figure 4: The C-S-AM-RLS estimation errors \(\delta\) versus \(t\) with different \(\sigma^{2}\) for Example 2. Figure 5: The C-AM-RLS estimation errors \(\delta\) versus \(t\) with different \(\sigma^{2}\) for Example 2.