## 2 Preliminaries and Motivation
### Model Conversion for Singular Systems
\]  
and \(P_{s}>0\) satisfies the algebraic Riccati equation  
\[\hat{A}_{s}^{T}P_{s}+P_{s}\hat{A}_{s}-\Big{(}P_{s}\hat{B}_{s}+N_{s}\Big{)} \overline{R_{c}^{-1}}\Big{(}\hat{B}_{s}^{T}P_{s}+N_{s}^{T}\Big{)}+\hat{C}_{s}^ {T}Q_{c}\hat{C}_{s}=0. \tag{6}\]  
Our previous work [4] shows that theoretically, the tracker-design-oriented algebraic Riccati equation (ARE) for a regular system can be directly generalized for (i) a singular system in terms of (\(E_{r}\), \(A_{r}\), \(B_{r}\), \(C_{r}\)) in Equations (1a) and (1b); (ii) its equivalent model in terms of (\(\hat{E}\), \(\hat{A}\), \(\hat{B}\), \(\hat{C}\)), with a non-symmetric singular matrix \(\hat{E}\) for the singular system with impulsive modes, where \(\hat{E}=\left[ \begin{array}{cc}I_{\xi}&0\\ 0&\hat{E}_{f}\end{array} \right]_{m\times m}\), \(\hat{A}=\left[ \begin{array}{cc}\hat{A}_{s}&0\\ 0&I_{n-\xi}\end{array} \right]_{m\times m}\), \(\hat{B}=\left[ \begin{array}{c}\hat{B}_{s}\\ \hat{B}_{f}\end{array} \right]_{m\times m}\), \(\hat{C}=\left[ \begin{array}{cc}\hat{C}_{s}&\hat{C}_{f}\end{array} \right]_{p \times m}\); or (iii) its equivalent model in terms of (\(E\), \(\hat{A}\), \(\hat{B}\), \(\hat{C}\)), with a symmetric singular matrix \(\hat{E}\) for the impulsive-mode-free singular system, where \(\hat{E}=\left[ \begin{array}{cc}I_{\xi}&0\\ 0&0_{n-\xi}\end{array} \right]_{m\times m}\). It is worthwhile noting that if \(\hat{C}_{f}\
eq 0\), then the generalized Riccati equation for a singular system might have no solution \(P\in\Re^{m\times m}\), even if \(Q_{c}\in\Re^{p\times p}\), and \(R_{c}\in\Re^{m\times m}\) are chosen as positive definite matrices. For the known linear continuous-time singular system described in Equations (1a) and (1b), through a series of coordinate transforms, the impulsive-mode-free singular system can be decomposed into a reduced-order regular subsystem with a direct input-output feedthrough term \(\hat{D}_{s}\), described in Equations (3a) and (3b). As a result, the corresponding Riccati equation in Equation (6) is solvable; consequently, an optimal tracker for the singular system can be designed. As for the unknown linear singular system, the aforementioned series of transforms are not available. To overcome this issue, we are motivated to take the merits of the OKID method to directly identify the equivalent discrete-time reduced-order regular subsystem with a direct input-output feedthrough term in the general coordinate form for the impulsive mode-free singular system. Then, we obtain the corresponding continuous-time model through the discrete-to-continuous model conversion. With this approach, the aforementioned series of transforms can be avoided, even for the given linear continuous-time singular system. For the unknown square-non-square singular sampled data systems subject to unknown input and output disturbances, a new functional observer-based discrete equivalent input disturbance (EID) estimator was presented in our previous work [4]. The main objective of this paper is to propose a new functional observer-based EID estimator for the unknown square-non-square singular continuous-time systems subject to unknown input and output disturbances.