## 4 Modified Functional Observer with Unknown Input
### Existence Conditions
\begin{array}{cc}\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.\] (24)  
**Proof. ** The part of the proof related to the design procedure is now briefly summarized. The complete proof is omitted here and can be obtained by appropriately modifying the ones in [11]. Through some mathematical manipulations, Equations (21) and (22) induce  
\[NF=F\overline{A}_{\omega}-\left[ \begin{array}{cc}Q&T&\mathcal{L}\\ \end{array} \right]\left[ \begin{array}{cc}\overline{C}_{\omega}\overline{A}_ {\omega}\\ \overline{C}_{\omega}\\ \overline{E}_{\omega}\end{array} \right]. \tag{25}\]  
More precisely, the proof can be obtained from Equation (21) by  
(i) substituting \(\overline{E}=\overline{E}_{\omega}-I_{(n+m)}\) into Equation (22) to get \(\mathcal{L}=F-Q\overline{C}_{\omega}-\mathcal{L}\overline{E}\);  
(ii) substituting Equation (22) into Equation (21) to get \(N\mathcal{L}\overline{E}_{\omega}+J\overline{C}_{\omega}=N(F-Q\overline{C}_{ \omega})+J\overline{C}_{\omega}=\mathcal{L}\overline{A}_{\omega}\) and \(NF=NQ\overline{C}_{\omega}-J\overline{C}_{\omega}+\mathcal{L}\overline{A}_{ \omega}=(-T)\overline{C}_{\omega}+\mathcal{L}\overline{A}_{\omega}\), where \(T=(J-NQ)\);  
(iii) using (i) and (ii) yields \(NF=-T\overline{C}_{\omega}+(F-Q\overline{C}_{\omega}-\mathcal{L}\overline{E}) \overline{A}_{\omega}=F\overline{A}_{\omega}-\left[ \begin{array}{cc}Q&T& \mathcal{L}\\ \end{array} \right]\left[ \begin{array}{c}\overline{C}_{\omega}\overline{A}_ {\omega}\\ \overline{C}_{\omega}\\ \overline{E}A_{\omega}\end{array} \right]\). Since \(F\) is a known matrix, we define the full-row rank matrix  
\[\left[ \begin{array}{cc}H_{1}&E_{1}\end{array} \right]=\left[ \begin{array}{ cc}F^{+}&(I_{n+m}-F^{+}F)\end{array} \right], \tag{26}\]  
where \(F^{+}\) is the Moore-Penrose inverse of \(F\), where \(FH_{1}=I_{\kappa}\) and \(FE_{1}=0_{\kappa\kappa(n+m)}\). Post-multiplying both sides of Equation (25) by Equation (26) gives  
\[N=F\overline{A}_{\omega}H_{1}-\left[ \begin{array}{cc}Q&T&\mathcal{L}\\ \end{array} \right]\left[ \begin{array}{c}\overline{C}_{\omega}\overline{A}_{ \omega}H_{1}\\ \overline{C}_{\omega}H_{1}\\ \overline{E}\overline{A}_{\omega}H_{1}\end{array} \right], \tag{27}\]  
where \(T=(J-NQ)\) and  
\[F\overline{A}_{\omega}E_{1}=\left[ \begin{array}{cc}Q&T&\mathcal{L}\\ \end{array} \right]\left[ \begin{array}{c}\overline{C}_{\omega}\overline{A}_{ \omega}E_{1}\\ \overline{C}_{\omega}E_{1}\\ \overline{E}\overline{A}_{\omega}E_{1}\end{array} \right], \tag{28}\]  
In Equation (28), \(F,\ \overline{A}_{\omega},\ E_{1},\ \overline{C}_{\omega}\), and \(\overline{E}\) are known matrices, while matrices \(Q\), \(T\), and \(\mathcal{L}\) are unknown and need to be determined to find matrix \(N\) in Equation (27). Now, we augment Equation (28) with Equation (22) as  
\[\left[ \begin{array}{cc}Q&T&\mathcal{L}\\ \end{array} \right]\Delta=\psi, \tag{29}\]where \(\Delta\in\Re^{(2p+n+m)\times(2n+2m)}\) and \(\psi\in\Re^{\times(2n+2m)}\) are known matrices defined by  
\[\Delta=\left[ \begin{array}{cc}\overline{C}_{\omega}\overline{A}_{\omega}E_{1}& \overline{C}_{\omega}\\ \overline{C}_{\omega}E_{1}&0\\ \overline{E}\overline{A}_{\omega}E_{1}&\overline{E}_{\omega}\end{array} \right]\] (30a) and \[\psi=\left[ \begin{array}{cc}F\overline{A}_{\omega}E_{1}&F\end{array} \right]\]. (30b)  
A necessary and sufficient condition for the existence of a solution \(\{Q,\ T,\) and \(\mathcal{L}\}\) can be derived from Equation (29). Then, a necessary and sufficient condition for ensuring that matrix \(N\) is a Hurwitz function can be derived by substituting \(Q,\ T,\) and \(\mathcal{L}\) into Equation (27). At last, the matrix \(J\) is obtained as \(J=T+NQ\). As a result, the unknown matrices \(N,\mathcal{L},J,Q\) and \(F\) satisfy matrix Equations (21) and (22) of Theorem 1. Based on the general solution of the linear matrix equations [21], a solution to Equation (29) exists only if  
\[\text{rank}\left[