## 6 Illustrative Examples
Now, assume that the singular system is unknown. The objective of this paper is to design a robust optimal OKID-based LQAT with a modified functional observer-based EID estimator, such that the controlled square system has an improved tracking performance. Step 1: Identification of the singular system. It is suitable to assume that the unknown disturbances occur during some unexpected periods, such that a set of disturbance-free input-output data of the open-loop system can be collected and an appropriate-order (4 this example) open-loop model can be identified by applying the off-line OKID method. As a consequence, the converted identified system matrices in the continuous-time domain are given as  
\[A_{ok}=\left[ \begin{array}{ccc}224.4093&20.3493&-203.0721&-1.1996\\ -20.2149&189.8678&0.7314&-212.3123\\ 203.0742&-1.3015&-181.2974&20.4771\\ 0.4529&212.3102&-20.5253&-234.8976\end{array} \right],B_{ok}=\left[ \begin{array} []{ccc}-36.393&29.3443\\ 26.0444&-15.3636\\ -36.9952&30.6218\\ 26.8896&-16.8004\end{array} \right],C_{ok}=\left[ \begin{array}{ccc}-2.1609 &-2.862&-1.9911&-2.6455\\ 2.9487&-2.1395&2.7195&-1.9425\end{array} \right],D_{ok}=\left[ \begin{array}{ cccc}-36.393&-2.9443\\ 26.0444&-15.3636\\ -36.9952&30.6218\\ 26.8896&-16.8004\end{array} \right],\]  
The eigenvalues and zeros of the open-loop OKID model are \(\{-9.1909,-3.4346,\ 3.7011,\ 7.0066\}\) and \(\{-33.3957\pm 23.7553i,\ -2.0351\pm 1.8661i\}\), respectively. Step 2: Determination of the state observer. Error equations are constructed by choosing the weighting matrix pairs \(\{\ Q_{o},\ R_{o}\}=\left\{ \begin{array}{ccc}10^{4}I_{4},\ I_{2}\end{array} \right\}\) for the determination of \(L_{c}\) with \(h=50\). This results in  
\[L_{c}=\left[ \begin{array}{ccc}-115.3191&156.8902\\ -138.9246&-103.4300\\ -24.5142&32.8421\\ -35.1095&-25.3732\end{array} \right].\]  
Figure 2: (a) Input disturbance \(d_{s1}(t)\) (without state-dependent disturbance); (b) output disturbance \(d_{s01}(t)\). The filter matrices \(\left(A_{f},B_{fa},C_{f}\right)\) are chosen as \(A_{f}=\left[ \begin{array}{cc}-100&0\\ 0&-100\end{array} \right]\), \(B_{fa}=\left[ \begin{array}{cc}8&0\\ 0&8\end{array} \right]\), and \(C_{f}=\left[ \begin{array}{cc}12.5&0\\ 0&12.5\end{array} \right]\), so that \(C_{f}\left(sl-A_{f}\right)^{-1}B_{fa}\approx I_{m}\) for \(s=j\omega=0\). Note that the selected eigenvalues of \(A_{f}\) are sufficiently negative to filter out high-frequency disturbances. Step 3: Design of the functional observer according to Section 4 to estimate the linear combination of state and unknown input \(F\omega(t)\) in Equation (71). \[\text{Choose }F=\left[ \begin{array}{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right]\text{, such that the ranks of those matrices checked in Theorem 2 are all equal to 12 and the unknown input \(\hat{u}_{fa}(t)\) can be estimated. According to Section 4, the condition given in this example implies that all theorems hold, thus the linear functional observer  
\[\dot{w}(t)=Nw(t)+Jy_{c}(t),w(0)=0_{4\times 1}, \tag{80a}\] \[\dot{w}(t)=Nw(t)+Jy_{c}(t),\dot{z}(t)=w(t)+Qy_{c}(t)\approx F\omega(t), \tag{80b}\]  
is obtained, where  
\[\mathcal{L}=\left[ \begin{array}{cccccc}1.0000&0.0000&0.0000&0.0000&0&0\\ 0.0000&1.0000&0.0000&0.0000&0&0\\ 0.0486&-2.5373&0.0458&-2.3302&0&0\\ -0.7384&-3.4302&-0.6794&-3.1565&0&0\end{array} \right]\in\Re^{\times\times(n+m)}\]  
(mentioned in Theorem 1),  
\[N=\left[ \begin{array}{cccccc}444.673&19.807&-406.926&303.26\\ -20.905&421.012&110.864&-10.718\\ 39.759&-2158.599&-508.808&14.632\\ -600.263&-2918.42&-82.242&-427.738\end{array} \right]\]  
is a stable matrix,  
\[J=\left[ \begin{array}{cccc}-5.7167&3.7585\\ -0.9552&-4.9763\\ 2.1877&24.0545\\ 12.389&26.9334\end{array} \right],\]  
and  
\[Q=\left[ \begin{array}{cccc}0&0\\ 0&0\\ -0.5648&-0.4304\\ -0.8953&-0.4057\end{array} \right].\]  
Step 4: Design of the linear-quadratic state feedback tracker integrated with the EID estimator.