## 6 Illustrative Examples
\right],\]  
with the same mismatched input disturbances (but without state-dependent disturbances) and output disturbances \(d_{so}(t)\) described in Example 1,  
\[d_{s}(t)=G_{i}\overline{d}_{s}(t)=\left[ \begin{array}{rrrr}-1&2&5\\ 4&-5&3\\ 3&3&4\\ 5&4&-1\end{array} \right]\overline{d}_{s}(t),\] \[\overline{d}_{s}(t)=\left[ \begin{array}{rrrr}0.1(20\sin(4\pi t )+\cos(5\pi t)+\sin(\pi t)+\sin(5\pi t)-1)\\ 0.1(20\sin(5\pi t)+\cos(2\pi t)+\sin(\pi t)+\sin(2\pi t)-1)\\ 0.1(20\sin(3\pi t)+\cos(4\pi t)+\sin(3\pi t)+\sin(2\pi t)-1)\end{array}  \right]\in\Re^{I_{1}},\]  
for \(t=10\sim 30\) sec, or \(\overline{d}_{s}(t)=0_{I_{1}\times 1}\) otherwise. Similarly, it is assumed that \(G_{i},\ G_{o},\ \overline{d}_{s}(t)\) and \(\overline{d}_{so}(t)\) are unknown. Eigenvalues and \"control zeros\" [26] of the open-loop singular system are \(\{-9.1996,-3.4145,3.6372,7.0590\}\) and \(\{-3.4606\times 10^{5},\)\(-30.9692,\)\(-1.5141\pm 1.8765i\}\), respectively. Also, the desired output trajectory \(r(t)\) is the same as in Example 1. Now, assume that the singular system is unknown. The objective is to design a robust optimal OKID-based LQAT with a modified functional observer-based EID estimator, such that the tracking performance of the controlled non-square system is improved. Step 1: Identification of the singular system. Apply the off-line OKID method to determine the appropriate order as 4, and the converted identified system matrices in the continuous-time domain are given as  
\[A_{\text{ok}}=\left[ \begin{array}{cccc}212.1358&-2.6065&-206.2029&1.861\\ 2.526&202.0124&-1.9511&-208.864\\ 206.2035&1.8947&-200.1689&-2.7866\\ -1.7579&208.8631&2.8165&-215.8972\\ \end{array} \right]B_{\text{ok}}=\left[ \begin{array}{cccc}10.5226&-19.5544&-14. 9769\\ 40.2286&-6.32&41.3521\\ 10.7314&-19.3488&-14.6376\\ 40.2016&-6.5768&40.5922\\ \end{array} \right],\] \[C_{\text{ok}}=\left[ \begin{array}{cccc}-2.7742&2.3193&-2.5354&2. 1565\\ -2.3362&-2.7567&-2.1735&-2.5158\\ \end{array} \right],\]  
Eigenvalues and \"control zeros\" of the open-loop OKID model are \(\{-9.2013,-3.3976,3.5768,7.1041\}\) and \(\{-3.4777\times 10^{5},-30.9407,-1.512\pm 1.8739i\}\), respectively. Appropriately choose a transformation matrix \(\eta\) such that the system \((A_{\text{ok}},\ B_{\text{ok}}\eta,\ C_{\text{ok}},\ D_{\text{ok}}\eta)\) is square and of the minimum phase. Here, \(\eta\) is chosen as \(\eta=\left[ \begin{array}{cccc}-8&-9\\ 6&1\\ 3&4\\ \end{array} \right]\), while eigenvalues and \"control zeros\" of the transformed system are \(\{-9.2013,-3.3976,3.5768,7.1041\}\) and \(\{-265.2835,\ -15.646,\ -1.1069\pm 0.9538i\}\), respectively. Therefore, \(B_{\text{ok}}\) and \(D_{\text{ok}}\) are replaced by  
\[B_{\text{ok},a}=\eta B_{\text{ok}}=\left[ \begin{array}{cccc}-246.4379&-174. 1652\\ -235.6925&-202.9687\\ -245.8564&-174.4816\\ -239.2965&-206.022\\ \end{array} \right]\]  
and \(D_{\text{ok},a}=\eta D_{\text{ok}}=\left[ \begin{array}{cccc}20.4077&10.6199 \\ 34.9912&21.3096\\ \end{array} \right]\), respectively. Step 2: Determination of the state observer. Let the low-pass filter \(\left(A_{f},B_{f},C_{f}\right)\) be the same as in Example 1. Also, based on the same parameters specified in Step 2 of Example 1  
\[L_{\text{c}}=\left[ \begin{array}{cccc}-142.854&120.2801\\ 11.1893&-138.0508\\ -31.9198&-27.344\\ 27.7948&-32.4321\\ \end{array} \right].\]  
Step 3: Design the modified functional observer according to Section 4 to estimate the linear combination of the state and unknown input \(F\omega(t)\) in Equation (71). \[\text{Choose}\,F=\left[ \begin{array}{cccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ \end{array} \right],\] such that the ranks of those matrices to be checked in Theorem 2 are all equal to 12 and the unknown input \(\hat{n}_{fa}(t)\) can be estimated. According to Section 4, all theorems hold due to the condition given in this example, and thus in the linear functional observer \[\hat{w}(t)=Nw(t)+Jy_{c}(t), \tag{82a}\] \[\hat{z}(t)=w(t)+Qy_{c}(t)\approx F\omega(t), \tag{82b}\]  
where  
\[\mathcal{L}=\left[