## 4 Modified Functional Observer with Unknown Input
### Existence Conditions  
As the functions to be utilized have been defined, this section presents the reconstruction of a linear function problem. Assuming that any unobservable state can be eliminated, by defining an observer state vector of a lower dimension, the order \(\kappa\) of the observer defined in Equations (13a) and (13b) should be less than or equal to the reduced-order state observer, i.e., \(\kappa\leq n+m-p\). In Equation (13b), the output \(\hat{z}(t)\) provides an asymptotic estimate of \(F\omega(t)\) if  
\[\lim_{t\rightarrow\infty}[\hat{z}(t)-F\omega(t)]=0. \tag{16}\]  
Given a full-row rank matrix \(\mathcal{L}\in\mathfrak{R}^{\times\times(n+m)}\), the two error vectors \(\varepsilon(t)\in\mathfrak{R}^{\kappa}\) and \(e(t)\in\mathfrak{R}^{\kappa}\) can be described by  
\[\varepsilon(t)=w(t)-\mathcal{L}\overline{E}_{\omega}\omega(t), \tag{17}\]  
\[e(t)=\hat{z}(t)-z(t), \tag{18}\]  
where \(w(t)\) in Equation (13a) estimates a different linear combination of \(\omega(t)\). Then,  
\[ \begin{array}{ll}\dot{\varepsilon}(t)&=\dot{w}(t)-\mathcal{L}\overline{E}_{ \omega}\dot{\omega}(t)=\left(Nw(t)+Jy_{c}(t)\right)-\mathcal{L}\Big{(} \overline{A}_{\omega}\omega(t)\Big{)}\\ &=Nw(t)-N\mathcal{L}\overline{E}_{\omega}\omega(t)+N\mathcal{L}\overline{E}_ {\omega}\omega(t)+J\overline{C}_{\omega}\omega(t)-\mathcal{L}\overline{A}_{ \omega}\omega(t)\\ &=N\varepsilon(t)+(N\mathcal{L}\overline{E}_{\omega}+J\overline{C}_{\omega}- \mathcal{L}\overline{A}_{\omega})\omega(t).\end{array}  \tag{19}\]  
\[ \begin{array}{ll}e(t)&=\hat{z}(t)-z(t)=\hat{z}(t)-F\omega(t)=\left(w(t)+Qy _{c}(t)\right)-F\omega(t)\\ &=w(t)-\mathcal{L}\overline{E}_{\omega}\omega(t)+\mathcal{L}\overline{E}_{ \omega}\omega(t)+Q\overline{C}_{\omega}\omega(t)-F\omega(t)\\ &=\varepsilon(t)+(\mathcal{L}\overline{E}_{\omega}+Q\overline{C}_{\omega}-F) \omega(t).\end{array}  \tag{20}\]  
This introduces the following theorem. **Theorem 1. **_From Equations (19) and (20), \(\hat{z}(t)\) in (13b) is an asymptotic estimate of \(z(t)\) if matrix \(\mathcal{L}\in\mathfrak{R}^{\times(n+m)}\) exists, such that the following equations_  
\[N\mathcal{L}\overline{E}_{\omega}+J\overline{C}_{\omega}-\mathcal{L}\overline {A}_{\omega}=0,\;N\;\text{is Hurwitz} \tag{21}\]  
_and_  
\[\mathcal{L}\overline{E}_{\omega}+Q\overline{C}_{\omega}-F=0 \tag{22}\]  
_are satisfied [11]. Matrices \(N\), \(J\), and \(Q\) are shown in Equations (13a) and (13b); and \(\overline{E}_{\omega}\) is given in Equation (11a). Upon satisfaction of the conditions in Equations (21) and (22), the dynamic equation in Equation (19) can be reduced to \(\dot{e}(t)=N\varepsilon(t)\), which implies that \(\varepsilon(t)\to 0\) as \(t\rightarrow\infty\), and consequently \(e(t)\to 0\)._  
To derive the parameters of the linear functional observer (Equations (13a) and (13b)), it is necessary to solve matrix Equations (21) and (22) for the unknown matrices \(N(\text{Hurwitz})\), \(\mathcal{L}\), \(Q\), and \(J\). The following theorem provides necessary and sufficient conditions for the solvability of matrix Equations (21) and (22), while ensuring that matrix \(N\) is a Hurwitz matrix  
**Theorem 2. **_Matrix Equations (21) and (22) are completely solvable with, \(N\) being a Hurwitz matrix, only if the following two conditions hold:_  
**Condition 1**. : \[\text{rank}\left[ \begin{array}{cc}F\overline{A}_{\omega}&F\\ \overline{C}_{\omega}\overline{A}_{\omega}&\overline{C}_{\omega}\\ \overline{C}_{\omega}&0\\ 0&\overline{E}_{\omega}\\ F&0\end{array} \right]=\text{rank}\left[ \begin{array}{cc}\overline{C}_{ \omega}\overline{A}_{\omega}&\overline{C}_{\omega}\\ \overline{C}_{\omega}&0\\ 0&\overline{E}_{\omega}\\ F&0\end{array} \right].\] (23)  
**Condition 2**. : \[\text{rank}\left[ \begin{array}{cc}sF-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]=\text{rank}\left[