## 4 Modified Functional Observer with Unknown Input
### 5 Observer-Kalman-Filter Identification for the Unknown Singular System
To fulfill the required assumptions for the linear functional observer (see Part 3), the transformation matrix \(\eta\in\Re^{n\times p}\) is determined, such that the system described in terms of \(\left(\hat{A}_{s},\hat{B}_{a},\hat{C}_{s},\hat{D}_{a}\right)\) for the control input \(u_{a}(t)\)  
\[\dot{x}_{c,s}(t)=\hat{A}_{s}\hat{x}_{c,s}(t)+\hat{B}_{s}\eta u_{a}(t)+d_{i}(t) =\hat{A}_{s}\hat{x}_{c,s}(t)+\hat{B}_{a}u_{a}(t)+d_{i}(t), \tag{63a}\] \[y_{c}(t)=\hat{C}_{s}\hat{x}_{c,s}(t)+\hat{D}_{s}\eta u_{a}(t)+d_{o}(t)=\hat{C}_{ s}\hat{x}_{c,s}(t)+\hat{D}_{a}u_{a}(t)+d_{o}(t), \tag{63b}\]  
is square, controllable, and of the minimum phase, where \(\hat{B}_{a}=\hat{B}_{s}\eta\in\Re^{n\times p}\), \(\hat{D}_{a}=\hat{D}_{s}\eta\in\Re^{p\times p}\), \(u_{a}(t)\in\Re^{p}\), and  
\[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]0  
Step 2: Represent the system (Equations (63a) and (63b)) with mismatched input and output disturbances as a net EID, i.e.,  
\[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]1 \[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]2  
where \(d_{\alpha}(t)\in\Re^{p}\) is the squared-down EID. Step 3: Obtain the estimate \(\hat{d}_{\alpha}(t)\) of \(d_{\alpha}(t)\) in Part 3. To get \(\overline{d}_{\alpha}(t)\), construct the low-pass filter  
\[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]3 \[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]4  
where \(\hat{d}_{\alpha}(t)\) is the input, \(\overline{d}_{\alpha}(t)\in\Re^{p}\) is the output that estimates the equivalent unknown input, \(x_{e}(t)\) is the state of the Luenberger observer [25], \(A_{f}\in\Re^{n_{f}\times n_{f}}\), \(B_{fa}\in\Re^{n_{f}\times p}\), and \(C_{f}\in\Re^{p\times n_{f}}\). In the frequency domain, the closed-loop gain \(\mathrm{G}_{\mathrm{yd}}(s)\) from the disturbance \(d_{\alpha}(t)\) to the output \(y_{c}(t)\) cannot be minimized ideally through \(\omega\in[0,\infty)\). For this reason, it is assumed that the natural frequencies of the unknown disturbances to be suppressed are below the cut-off frequency \(\omega_{c}\). Step 4: Construct the Luenberger observer  
\[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]5 \[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]6  
where \(\hat{x}_{e}(t)\) and \(\hat{y}_{c}(t)\) are the estimated state and output vectors, respectively. Here, \(B_{\text{ok},a}=B_{\text{ok}}\eta\), \(D_{\text{ok},a}=D_{\text{ok}}\eta\), and \(L_{c}\) is the state observer gain. To further reduce the estimation errors, let \(\overline{A}=A+hl_{n}\). Then, apply Lemma 1 with appropriate \((Q_{o},\ R_{o})\) to determine  
\[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]7  
**Part 3: Design of the Linear Functional Observer**  
We use the linear functional observer to estimate the \(F\omega(t)\) in Equation (12). To achieve this goal based on the concept of EID and OKID methods, Equations (65a) and (65b) are represented as  
\[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]8 \[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]9  
Observe that \((A,\ B,\ C,\ D)\) and \(u_{f}(t)\in\Re^{m}\) in Equations (10a) and (10b) are now replaced by \((A_{\text{ok}},\ B_{\text{ok},a},\ C_{\text{ok}},\ D_{\text{ok},a})\) and \(u_{fa}(t)=[u_{a}(t)+d_{\text{an}}(t)]\in\Re^{p}\), respectively. Assumptions associated with the linear functional observer [11] include that \(m\leq p\), and that the number of inputs of the equivalent system is equal to the number of outputs of the system after transforming it to the square in part 2. The procedure to construct the linear functional observer is now described in detail. Step 1: Define the augmented state vector \(\omega(t)=[ \begin{array}{cc}x_{e}^{T}(t)&u_{fa}^{T}(t)\end{array} ]^{T}\in \Re^{(n+p)}\), such that the system (Equation (69a) and (69b)) can be expressed as  
\[\overline{E}_{\omega}\dot{\omega}(t)=\overline{A}_{\omega}\omega(t), \tag{70a}\] \[y_{c}(t)=\overline{C}_{\omega}\omega(t), \tag{70b}\]  
where \(\overline{E}_{\omega}=\left[ \begin{array}{cc}I_{\text{lx}\text{x}\text{n}}&0 _{\text{lx}\text{p}\text{p}}\\ 0_{\text{px}\text{n}}&0_{\text{px}\text{p}\text{p}}\end{array} \right]\in \Re^{(n+p)\times(n+p)}\), \(\overline{A}_{\omega}=\left[