## 6 Illustrative Examples  
This section presents two examples to illustrate the application of the proposed approach and to show that it outperforms the traditional approach. **Example 1**. : _Square MIMO System_  
Consider the continuous-time singular system without the impulsive mode described as  
\[E_{r}\dot{x}(t)=A_{r}x(t)+B_{r}u_{c}(t)+d_{r}(t), \tag{78a}\] \[y_{c}(t)=C_{r}x(t)+d_{ro}(t), \tag{78b}\]  
where  
\[E_{r}=\left[ \begin{array}{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 1&0&-1&0&0&1\\ 0&1&0&0&0&0\\ 0&0&0&0&0&1\end{array} \right]A_{r}=\left[ \begin{array}{cccccc}7&6&5&6&-9&5 \\ -2&1&-1&3&0&4\\ -10&-7&-6&-4&10&4\\ -4&1&-7&-7&8&-4\\ 7&-4&-5&-5&2&2\\ -4&4&3&4&5&2\end{array} \right],\]  
Figure 1: An observer–Kalman-filter (OKID)-based linear quadratic analog tracker (LQAT) with a functional observer for non-square singular systems. EID, equivalent input disturbance. \[B_{r}=\left[ \begin{array}{cc}6&4\\ -10&6\\ -6&-4\\ 10&3\\ 6&-9\\ 3&1\end{array} \right]C_{r}=\left[ \begin{array}{rrrrr}1&-6&10&2&-1&-3\\ 7&9&-3&1&-3&-8\end{array} \right],\]  
\[x(0)=\left[ \begin{array}{rrrrr}0.5&0.05&0&0.7862&1.2448&0.225\end{array}  \right]^{T}.\]  
The singular system can be transformed into the equivalent regular system with an input-output direct feedthrough term  
\[\dot{x}_{s}(t)=\hat{A}_{s}x_{s}(t)+\hat{B}_{s}u_{c}(t)+d_{s}(t), \tag{79a}\] \[y_{c}(t)=\hat{C}_{s}x_{s}(t)+\hat{D}_{s}u_{c}(t)+d_{so}(t), \tag{79b}\]  
where \(x_{s}(0)=\left[ \begin{array}{rrrrr}0.5&0.05&0&0.225\end{array} \right]^{T}\),  
\[\hat{A}_{s}=\left[ \begin{array}{rrrrr}-4.4403&0.9627&-2.7015&2.7164\\ 3.1940&-0.7463&-2.0299&3.3284\\ 7.3284&-2.9552&1.6418&5.9403\\ 15.0522&2.5299&4.7612&1.6269\\ \end{array} \right]\hat{B}_{s}=\left[ \begin{array}{rrr}-3.0448&7.6940\\ -1.8955&-1.1194\\ 11.2537&-14.4328\\ 27.8358&-18.4552\\ \end{array} \right]\] \[\hat{C}_{s}=\left[ \begin{array}{rrr}2.0373&-7.3358&8.6866&-3.55 22\\ 1.4552&7.9030&-5.2239&-8.5373\end{array} \right]\hat{D}_{s}=\left[ \begin{array} []{rrr}2.5970&-2.7537\\ -5.7164&3.6045\end{array} \right],\]  
with  
\[d_{s}(t)=G_{t}\overline{d}_{s}(t)=\left[ \begin{array}{rrr}-1&2&5\\ 4&-5&3\\ 3&3&4\\ 5&4&-1\end{array} \right]\overline{d}_{s}(t),\]  
where  
\[\overline{d}_{s}(t)=\left[ \begin{array}{rrr}0.1(20\sin(4\pi t)+\cos(5\pi t) +\sin(\pi t)+\sin(5\pi t)-1)+\sin(x_{s1}(t)+x_{s3}(t))\\ 0.1(20\sin(5\pi t)+\cos(2\pi t)+\sin(\pi t)+\sin(2\pi t)-1)+\cos(x_{s2}(t))\\ 0.1(20\sin(3\pi t)+\cos(4\pi t)+\sin(3\pi t)+\sin(2\pi t)-1)+\sin(x_{s4}(t)) \end{array} \right]\in\Re^{l_{i}}\]  
for \(t=10\sim 30\) sec, or \(\overline{d}_{s}(t)=0_{l,\times 1}\) otherwise  
\[d_{so}(t)=G_{o}\overline{d}_{so}(t)=\left[ \begin{array}{rrr}-4&7\\ 3&-1\end{array} \right]\overline{d}_{so}(t),\]  
where \(\overline{d}_{so}(t)=\left[ \begin{array}{rrr}0.1(\sin(\pi t)+\sin(5\pi t)-1 )\\ 0.1(\sin(2\pi t)-1)\end{array} \right]\in\Re^{l_{o}}\), for \(t=10\sim 30\) sec, or \(\overline{d}_{so}(t)=0_{l\times 1}\) otherwise. It is assumed that \(G_{i},\ G_{o},\ \overline{d}_{s}(t)\) and \(\overline{d}_{so}(t)\) are unknown. Additionally, there are more disturbances than outputs. The eigenvalues and zeros of the open-loop singular system are \(\{-9.1996,-3.4145,\ 3.6372,\ 7.0590\}\) and \(\{-33.4278\pm 23.8397i,\ -2.0359\pm 2.0429i\}\), respectively. The desired output trajectory is given by \(r(t)=\left[r_{1}(t),\ r_{2}(t)\right]^{T}\), where  
\[r_{1}(t)=\left\{ \begin{array}{rrr}0&,\ 9\ <t\sec\\ \sin(0.1\pi t)+\cos(0.2\pi t)-\sin(0.5\pi t)\times\cos(0.3\pi t)&,\ 9\leq t<30\sec\\ -0.5&,\ t\ \geq\ 30\sec\\ 1&,\ 9\ <t\sec\\ \end{array} \right.,\] \[r_{2}(t)=\left\{ \begin{array}{rrr}\sin(0.2\pi t)+\cos(0.1\pi t)+ \sin(0.5\pi t)\times\cos(0.5\pi t)&,\ 9\leq t<30\sec\\ 2&,\ t\ \geq\ 30\sec\end{array} \right..\]Figure 2 shows the significant time-varying features of \(d_{s1}(t)\) without state-dependent disturbance (bounded by \(\pm 13\)) and \(d_{s01}(t)\) (bounded by \(\pm 2\)) when compared with the smooth desired output trajectory \(r(t)\) (bounded by \(\pm 2\)), which makes this example more challenging.