## 4 Modified Functional Observer with Unknown Input
### Stability Analysis in the Frequency Domain  
Here, it is assumed that the only input to the system is the net EID. Therefore, from Equations (13a), (13b), (14), (37a), (37b), (38a), (38b), and (40), one has  
\[\overline{d}_{e}(s)=C_{f}(sl_{n_{f}}-A_{f})^{-1}B_{f}\;\hat{d}_{e}(s)\equiv G_ {dd}(s)\hat{d}_{e}(s), \tag{42}\]  
\[\hat{s}_{e}(s)=(sl_{n}-A+L_{c}C)^{-1}L_{c}y_{c}(s)+(sl_{n}-A+L_{c}C)^{-1}(B-L_{ c}D)u_{f}(s), \tag{43}\]  
\[u_{f}(s)=-K_{c}\hat{s}_{e}(s), \tag{44}\]  
\[\hat{u}_{f}(s)=\Big{[} \begin{array}{cc}0_{m\kappa(\kappa-m)}&I_{m}\end{array}  \Big{]}\big{[}(sl_{k}-N)^{-1}+Q\Big{]}y_{c}(s)\equiv G_{ufo}(s)y_{c}(s), \tag{45}\]  
in the frequency domain, where \(\overline{d}_{e}(s)\), \(\hat{d}_{e}(s)\), \(\hat{y}_{c}(s)\), \(y_{c}(s)\), \(u_{f}(s)\), and \(\hat{x}_{e}(s)\) are the Laplace transforms of \(\overline{d}_{e}(t)\), \(\hat{d}_{e}(t)\), \(\hat{y}_{c}(t)\), \(u_{f}(t)\), \(\hat{u}_{f}(t)\), and \(\hat{x}_{e}(t)\), respectively. Substituting Equation (44) into Equation (43) yields  
\[ \begin{array}{l}\hat{s}_{e}(s)=(sl_{n}-A+L_{c}C)^{-1}L_{c}y_{c}(s)-(sl_{n}- A+L_{c}C)^{-1}(B-L_{c}D)K_{c}\hat{s}_{e}(s),\\ \Big{[}I_{n}+(sl_{n}-A+L_{c}C)^{-1}(B-L_{c}D)K_{c}\Big{]}\hat{x}_{e}(s)=(sl_{n} -A+L_{c}C)^{-1}L_{c}y_{c}(s),\\ \hat{s}_{e}(s)=\Big{[}I_{n}+(sl_{n}-A+L_{c}C)^{-1}(B-L_{c}D)K_{c}\Big{]}^{-1}( sl_{n}-A+L_{c}C)^{-1}L_{c}y_{c}(s)\equiv G_{xy}(s)y_{c}(s),\end{array} \]  
\[u_{f}(s)=-K_{c}\hat{s}_{e}(s)=-K_{c}G_{xy}(s)y_{c}(s), \tag{46}\]  
where \(G_{xy}(s)=\Big{[}I_{n}+(sl_{n}-A+L_{c}C)^{-1}(B-L_{c}D)K_{c}\Big{]}^{-1}(sl_{n }-A+L_{c}C)^{-1}L_{c}\). Thus, from Equation (36) and Equation (39), the filtered disturbance estimate \(\overline{d}_{e}(t)\) can be chosen as  
\[\overline{d}_{e}(t)=f_{d}(t)\otimes\hat{d}_{e}(t)=f_{d}(t)\otimes(\hat{u}_{f} (t)-u_{c}(t))=f_{d}(t)\otimes(\hat{u}_{f}(t)-u_{f}(t)+\overline{d}_{e}(t)), \tag{47}\]  
where \(\otimes\) denotes \"convolution\" and the Laplace transform of \(f_{d}(t)\) is \(G_{dd}(s)\), such that  
\[\overline{d}_{e}(s)=G_{dd}(s)\hat{u}_{f}(s)-G_{dd}(s)u_{f}(s)+G_{dd}(s) \overline{d}_{e}(s), \tag{48}\]  
\[\overline{d}_{e}(s)=(I_{m}-G_{dd}(s))^{-1}G_{dd}(s)(\hat{u}_{f}(s)-u_{f}(s)). \tag{49}\]  
Substituting Equations (45) and (46) into Equation (49) yields  
\[\overline{d}_{  
which, from Equation (52), implies that  
\[ \begin{array}{ll}\mathrm{y_{c}}(s)&=\big{\{}I_{p}+\mathrm{G_{P}}(s)\big{[}K_{c} \mathrm{G_{xy}}(s)+\big{(}I_{m}-\mathrm{G_{dd}}(s)\big{)}^{-1}\mathrm{G_{dd}}( s)\big{(}\mathrm{G_{ufo}}(s)+K_{c}\mathrm{G_{xy}}(s)\big{)}\big{]}\big{\}}^{-1} \mathrm{G_{P}}(s)\mathrm{d_{e}}(s)\\ &=\big{[}I_{p}+\mathrm{G_{P}}(s)K(s)\big{]}^{-1}\mathrm{G_{P}}(s)\mathrm{d_{e}} (s)\equiv\mathrm{G_{yd}}(s)\mathrm{d_{e}}(s),\end{array}  \tag{53}\]  
where \(\mathrm{G_{P}}(s)=C(sI_{n}-A)^{-1}B+D\), \(\mathrm{K}(s)=K_{c}\mathrm{G_{xy}}(s)+\big{(}I_{m}-\mathrm{G_{dd}}(s)\big{)}^{ -1}\mathrm{G_{dd}}(s)\big{(}\mathrm{G_{ufo}}(s)+K_{c}\mathrm{G_{xy}}(s)\big{)}\), \(\mathrm{G_{yd}}(s)=\big{[}I_{p}+\mathrm{G_{P}}(s)K(s)\big{]}^{-1}\mathrm{G_{ P}}(s)\) is the transfer function from the EID to the system output, and \(I_{m}\), \(I_{n}\), \(I_{p}\), and \(I_{n_{f}}\) are identity matrices of dimensions \(m\times m\), \(n\times n\), \(p\times p\), and \(n_{f}\times n_{f}\), respectively. Note that Equation (53) is a multi-input-multi-output (MIMO) expression. To simplify the discussion, we denote \(\mathrm{G_{yd}^{ij}}(s)\) and \(\mathrm{G_{dd}^{ij}}(s)\) as the elements in the \(i^{th}\) rows and \(j^{th}\) columns of \(\mathrm{G_{yd}}(s)\) and \(\mathrm{G_{dd}}(s)\), respectively. From Equation (53), the effects of the unknown inputs can be effectively suppressed by minimizing \(\|\mathrm{G_{yd}^{ij}}(s)\|_{\infty}\)\((i=1,\cdots,p;\ j=1,\cdots,m)\), and \(\|\mathrm{G_{yd}^{ij}}(s)\|_{\infty}\) is sufficiently small with appropriate design of \(\mathrm{G_{dd}}(s)\).