## 6 Illustrative Examples
In the LQAT design, the weighting matrix pair \(\left\{Q_{c},R_{c}\right\}=\left\{10^{4}I_{2},I_{2}\right\}\) is chosen to obtain the optimal control law \(u_{c}(t)=-K_{c}\dot{\chi}_{e}(t)+E_{c}r(t)-\overline{d_{e}}(t)\), where  
\[K_{c}=\left[ \begin{array}{cccc}-0.0501&2.5368&-0.0444&2.3302\\ 0.7365&3.4298&0.6812&3.1562\end{array} \right]\text{and }E_{c}=\left[ \begin{array}{cccc}-0.5648&-0.4303\\ -0.8952&-0.4056\end{array} \right].\]  
Step 5: Simulation results. First, to show the superiority of the proposed tracker, Figure 3 plots the input disturbance \(d_{s1}(t)\)(state-dependent disturbance inside) and the pure non-linear state-dependent disturbance. All of these disturbances (shown by parts) indicate a highly time-varying phenomenon. Figure 4a shows the time response of the controlled system and Figure 4b shows the output tracking errors under the control action of the traditional optimal OKID-based tracker without an EID estimator. Second, Figure 5 shows the input disturbance \(d_{s1}(t)\) (state-dependent disturbance inside) and the pure non-linear state-dependent disturbance. Here, \(d_{s1}(t)\) is slightly different from the optimal OKID-based LQAT without the EID estimator, because the differences between the two states \(x_{s}(t)\) are induced by different controllers. Next, the robustness of the optimal OKID-based LQAT integrated with the functional observer-based EID estimator is shown in Figure 6a,b. Also, Figure 7 plots the control inputs corresponding to the proposed approach. Notice that \(u_{c}(t)=u_{a}(t)\) for this square system, where \(\eta=I_{p}\). A comparison with Figure 4 shows that the proposed approach (Figure 6) significantly improves the tracking errors of the noisy singular system. Figure 4: (**a**) Time responses of the closed-loop noisy square singular system for the optimal OKID-based LQAT without the EID estimator, (**b**) tracking errors of output \(e(t)=r(t)-y_{c}(t)\). Figure 3: (**a**) Input disturbance \(d_{s1}(t)\)(state-dependent disturbance inside) and (**b**) state-dependent disturbance in \(d_{s1}(t)\). Figure 5: (**a**) Input disturbance \(d_{s1}(t)\) (state-dependent disturbance inside) and (**b**) state-dependent disturbance in \(d_{s1}(t)\). Figure 6: (**a**) Time responses of the proposed method and (**b**) errors of output \(e(t)=r(t)-y_{c}(t)\). Figure 7: Control inputs of the OKID-based LQAT tracker with the proposed EID compensation for the square system with mismatched input and output disturbances. **Example 2**. : _Non-Square MIMO System_  
Consider the continuous-time non-square singular system without impulsive mode described by  
\[E_{r}\dot{x}(t)=A_{r}x(t)+B_{r}u_{c}(t)+d_{r}(t), \tag{81a}\] \[y_{c}(t)=C_{r}x(t)+d_{ro}(t), \tag{81b}\]  
where \(E_{r}\) and \(A_{r}\) are given in Example 1,  
\[B_{r}=\left[ \begin{array}{rrrr}8&-9&-4\\ 2&-1&3\\ 4&3&8\\ 5&-4&7\\ -2&0&-6\\ -5&-5&-2\end{array} \right],C_{r}=\left[ \begin{array}{rrrrrr}8&9&-7&3&-5&-5\\ -5&2&7&1&-9&-3\end{array} \right],\] \[x(0)=\left[ \begin{array}{rrrrrr}0.15&0.015&0&0.2661&0.3593&-0. 0675\end{array} \right]^{T}.\]  
The singular system can be converted into the equivalent regular system with an input-output direct feedthrough term  
\[\dot{x}_{s}(t)=\hat{A}_{s}x_{s}(t)+\hat{B}_{s}u_{c}(t)+d_{s}(t),\] \[y_{c}(t)=\hat{C}_{s}x_{s}(t)+\hat{D}_{s}u_{c}(t)+d_{so}(t),\,x_{ s}(0)=\left[ \begin{array}{rrrrrr}0.15&0.015&0&-0.0675\end{array} \right]^{T},\]  
where  
\[\hat{A}_{s}=\left[ \begin{array}{rrrr}-4.4403&0.9627&-2.7015&2.7164\\ 3.1940&-0.7463&-2.0299&3.3284\\ 7.3284&-2.9552&1.6418&5.9403\\ 15.0522&2.5299&4.7612&1.6269\end{array} \right]\hat{B}_{s}=\left[ \begin{array} []{rrrr}12.3881&-3.6045&9.2313\\ -0.2388&-1.0896&-2.3731\\ -2.8657&-3.0746&-11.4776\\ -12.9104&-8.2164&-22.4851\end{array} \right],\] \[\hat{C}_{s}=\left[ \begin{array}{rrrr}1.0672&6.3955&-11.1642&-6. 194\\ -25.097&-0.1269&1.0149&-4.1642\end{array} \right]\hat{D}_{s}=\left[ \begin{array} []{rrrr}2.6866&3.0075&7.9478\\ 8.1194&5.5448&22.1866\end{array}