## 4 Modified Functional Observer with Unknown Input
### 5 Observer-Kalman-Filter Identification for the Unknown Singular System
\begin{array}{cc}A_{\text{ok}}&B_{\text{ok},a}\\ 0_{\text{px}\text{n}}&0_{\text{px}\text{p}}\end{array} \right]\in\Re^{(n+p) \times(n+p)}\), and \(\overline{C}_{\omega}=[ \begin{array}{cc}C_{\text{ok}}&D&\text{ok},a\end{array} ] \in\Re^{p\times(n+p)}\). Define the functional state vector \(z(t)\in\Re^{\kappa}\) that must be reconstructed (or estimated) as  
\[z(t)=F\omega(t)=\left[ \begin{array}{cc}F_{1}&0\\ 0&I_{p}\end{array} \right]\left[ \begin{array}{c}x_{e}(t)\\ u_{fa}(t)\end{array} \right]=\left[ \begin{array}{c}F_{1}x_{e}(t)\\ u_{fa}(t)\end{array} \right], \tag{71}\]  
where \(F_{1}\in\Re^{(\kappa-p)\times n}\), and \(F\in\Re^{\kappa\times(n+p)}\) is a given constant matrix, which satisfies \(rank(F)=\kappa\) and \(\text{rank}\left[ \begin{array}{c}\overline{C}_{\omega}\\ F\end{array} \right]=(p+\kappa)\leq(n+p)\). To estimate \(z(t)\), we construct the reduced-order functional observer of order \(\kappa\) for the system (Equations (69a) and (69b))  
\[\dot{w}(t)=Nw(t)+Jy_{c}(t), \tag{72a}\] \[\dot{z}(t)=w(t)+Qy_{c}(t). \tag{72b}\]  
Therefore, the estimated unknown input vector becomes  
\[\hat{u}_{fa}(t)=[ \begin{array}{cc}0_{p\times(\kappa-p)}&I_{p}\end{array} ] \dot{z}(t)=[ \begin{array}{cc}0_{p\times(\kappa-p)}&I_{p}\end{array} ]\left[ \begin{array}{c}F_{1}\hat{x}_{e}(t)\\ \hat{u}_{fa}(t)\end{array} \right]\in\Re^{p}. \tag{73}\]  
Thus, the estimated EID \(\hat{d}_{\alpha}(t)=\hat{u}_{fa}(t)-u_{a}(t)\) can be derived. Step 2: Check the existence of Condition 1 of Theorem 2 to see whether  
\[rank\left[ \begin{array}{ccc}F\overline{A}_{\omega}&F\\ \overline{C}_{\omega}\overline{A}_{\omega}&\overline{C}_{\omega}\\ \overline{C}_{\omega}&0\\ 0&\overline{E}_{\omega}\\ F&0\end{array} \right]=rank\left[ \begin{array}{ccc}\overline{C}_{\omega} \overline{A}_{\omega}&\overline{C}_{\omega}\\ \overline{C}_{\omega}&0\\ 0&\overline{E}_{\omega}\\ F&0\end{array} \right]\] (74a) is satisfied or not. If not, repeat Step 1 by choosing another \[F\] in terms of \[F_{1}\] in Equation ( 71 ). Step 3: Systematically derive all observer parameters. (i) Check the existence of Condition 2 of Theorem 2 to see whether  
\[rank\left[ \begin{array}{ccc}sF-F\overline{A}_{\omega}&-F\\ \overline{C}_{\omega}\overline{A}_{\omega}&\overline{C}_{\omega}\\ \overline{C}_{\omega}&0\\ 0&\overline{E}_{\omega}\\ \end{array} \right]=rank\left[ \begin{array}{ccc}\overline{C}_{\omega} \overline{A}_{\omega}&\overline{C}_{\omega}\\ \overline{C}_{\omega}&0\\ 0&\overline{E}_{\omega}\\ F&0\end{array} \right]\forall s\in C,\ \Re(s)\geq 0 \tag{74b}\]  
is satisfied or not. If yes, use Equation (34) to obtain \(\Phi\) and \(\Omega\). If not, which implies the pair \((\Phi,\ \Omega)\) is not detectable, stop, since a stable observer does not exist;  
(ii) Using Equation (33), derive \(Z\), such that the matrix \(N\) is stable;  
(iii) From Equation (32), obtain matrices \(Q\), \(T\), and \(\mathcal{L}\). Substitute \(Q\), \(T\), and \(\mathcal{L}\) into Equation (27) and check the necessary and sufficient condition to ensure that the matrix \(N\) is a Hurwitz function. (iv) Finally, obtain matrix \(J\) from Equation (27), i.e., \(J=T+NQ\). As a result, the unknown matrices \(N\), \(\mathcal{L}\), \(J\), \(Q\), are determined and the observer design is, thus, completed.