## 4 Modified Functional Observer with Unknown Input
### 5 Observer-Kalman-Filter Identification for the Unknown Singular System  
We consider the continuous-time impulsive-mode-free singular system (Equations (1a) and(1b)), i.e.,  
\[E_{r}\dot{x}_{c}(t)=A_{r}x_{c}(t)+B_{r}u_{c}(t), \tag{59a}\] \[y_{c}(t)=C_{r}x_{c}(t), \tag{59b}\]  
where matrices \(E_{r}\), \(A_{r}\), \(B_{r}\), and \(C_{r}\) are assumed to be unknown. In theory, the above singular system can always be transformed into a corresponding low-order regular system with a direct transmission term from input to output  
\[\dot{\hat{x}}_{c,s}(t)=\hat{A}_{s}\hat{x}_{c,s}(t)+\hat{B}_{s}u_{c}(t), \tag{60a}\] \[y_{c}(t)=\hat{C}_{s}\hat{x}_{c,s}(t)+\hat{D}_{s}u_{c}(t). \tag{60b}\]  
Here, it is assumed that \((\hat{A}_{s},\ \hat{B}_{s},\ \hat{C}_{s},\ \hat{D}_{s})\) are unknown system matrices of appropriate dimensions. Step 1: Implement the off-line OKID method [17] to the singular system to determine the appropriate (low-) order of the singular system and the corresponding system matrices, with the corresponding sampling time \(T_{s}\). Step 2: Transform the discrete-time system-observer models obtained in Step 1 to the continuous-time system-observer models. Based on the identified discrete-time model \((G_{\text{dd}},\ H_{\text{ok}},\ C_{\text{ok}},\ D_{\text{ok}})\), the corresponding continuous-time model \((A_{\text{ok}},\ B_{\text{ok}},\ C_{\text{ok}},\ D_{\text{ok}})\) can be determined as  
\[\dot{x}(t)=A_{\text{ok}}x(t)+B_{\text{ok}}u_{c}(t), \tag{61a}\] \[y_{c}(t)=C_{\text{ok}}x(t)+D_{\text{ok}}u_{c}(t), \tag{61b}\]where \(x(t)\in\Re^{n}\) is the state vector, \(u_{c}(t)\in\Re^{m}\) is the control input vector, \(y_{c}(t)\in\Re^{p}\) is the measurable output vector, \(A_{ok}=\frac{1}{T_{s}}\ln G_{ok},\ B_{ok}=A_{ok}(G_{ok}-I_{n})^{-1}H_{ok}\), and \(T_{s}\) is the sampling time. **Part 2: Observer Design for Noisy Singular System**  
For servo control, it is required to have more control inputs than outputs. However, for EID estimation, it is desirable to have more outputs than control inputs. Considering that the main purpose of this paper is to propose a robust tracker for a system subject to unknown disturbances, it is naturally assumed that the given system has more control inputs than outputs. To achieve this, the non-square system with \(m>p\) is transformed into a square system with \(m=p\) using an \"artificial\" coordinate [14], and finally transformed back to non-square and implemented in the original coordinate. These steps are now briefly shown. Step 1: Assume that the corresponding regular system (Equations (60a) and (60b)) of order \(n\), with \(m\) inputs and \(p\) outputs and subject to an unknown disturbance, is non-square \((m>p)\), described by  
\[\dot{x}_{c,s}(t)=\hat{A}_{s}\hat{x}_{c,s}(t)+\hat{B}_{s}u_{c}(t)+d_{i}(t), \tag{62a}\] \[y_{c}(t)=\hat{C}_{s}\hat{x}_{c,s}(t)+\hat{D}_{s}u_{c}(t)+d_{o}(t), \tag{62b}\]  
where \(\hat{A}_{s}\in\Re^{n\times n}\), \(\hat{B}_{s}\in\Re^{n\times m}\), \(\hat{C}_{s}\in\Re^{p\times n}\), and \(\hat{D}_{s}\in\Re^{p\times n}\), are constant matrices. The mismatched input-output disturbances are given as \(d_{i}(t)=G_{i}\overline{d}_{i}(t)\in\Re^{n}\), \(d_{o}(t)=G_{o}\overline{d}_{o}(t)\in\Re^{p}\), respectively; with \(l_{i}\) and \(l_{o}\) representing the number of input and output disturbances, respectively; and \(G_{i}\in\Re^{n\times l_{i}}\), \(G_{o}\in\Re^{p\times l_{o}}\), \(\overline{d}_{i}(t)\in\Re^{l_{i}}\), and \(\overline{d}_{o}(t)\in\Re^{l_{o}}\) being unknown. Additionally, \(l_{i}\) and \(l_{o}\) may be greater than \(m\) and \(p\) and may even be no less than \(n\).