## 2 System Descriptions and Problem Statements
The overall closed-loop T-S fuzzy singular system is represented as follows:  
\[\mathbf{E}x(k+1)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}h_{i}h_{j}\Big{\{} \mathbf{A}_{i}x(k)-\mathbf{B}_{i}\mathbf{F}_{pj}\hat{x}(k)-\mathbf{B}_{i} \mathbf{F}_{dj}\hat{x}(k+1)\Big{\}} \tag{9}\]  
According to motivations as mentioned above, the observer-based control scheme is built in this work. One of the purposes of this paper is to estimate the systems' states and to ensure the stability of the system (9). As the estimation errors converge to zero, the estimated states can replace the actual states. Next, define the estimation error as \(e(k)=x(k)-\hat{x}(k)\), the following equation can be inferred from (9). \[\mathbf{E}x(k+1)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}h_{i}h_{j}\Big{\{} \mathbf{A}_{i}x(k)-\mathbf{B}_{i}\mathbf{F}_{pj}x(k)-\mathbf{B}_{i}\mathbf{F} _{dj}x(k+1)+\mathbf{B}_{i}\mathbf{F}_{pj}e(k)+\mathbf{B}_{i}\mathbf{F}_{dj}e( k+1)\Big{\}} \tag{10}\]  
The closed-loop T-S fuzzy singular system is presented as aforementioned. Subsequently, the errors systems have to be built to guarantee estimation errors converge to zero as \(t\rightarrow\infty\). Now multiplying the matrix \(\mathbf{E}\) on the left side of the difference function \(e(k+1)=x(k+1)-\hat{x}(k+1)\), one can obtain the following closed-loop error systems. \[\mathbf{E}e(k+1)=\mathbf{E}x(k+1)-\mathbf{E}\hat{x}(k+1)\]  
\[=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}h_{i}h_{j}\big{\{}\mathbf{A}_{i}x( k)+\mathbf{B}_{i}u(k)-\left(\mathbf{A}_{i}\hat{x}(k)+\mathbf{B}_{i}u(k)+ \mathbf{L}_{pi}\big{(}\mathbf{C}_{j}x(k)-\mathbf{C}_{j}\hat{x}(k)\big{)}\ +\mathbf{L}_{di}\big{(}\mathbf{C}_{j}x(k+1)-\mathbf{C}_{j}\hat{x}(k+1) \big{)}\right)\big{\}}\]  
\[y(k)=\sum\limits_{i=1}^{n}h_{i}(\Im(k))\{\mathbf{C}_{i}x(k)\} \tag{4}\]0  
The following augmented systems can be further obtained from the system (10) and (11). \[y(k)=\sum\limits_{i=1}^{n}h_{i}(\Im(k))\{\mathbf{C}_{i}x(k)\} \tag{4}\]1  
where \(\widetilde{x}(k)=\left[ \begin{array}{cc}x^{\intercal}(k)&e^{\intercal}(k) \end{array} \right]^{\intercal}\), \(\mathbf{E}_{Rij}(h)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}h_{i}h_{j} \mathbf{E}_{Rij}\), \(\mathbf{A}_{Rij}(h)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}h_{i}h_{j} \mathbf{A}_{Rij}\), \(\mathbf{E}_{Rij}=\left[ \begin{array}{cc}\mathbf{E}+\mathbf{B}_{i}\mathbf{F}_ {dj}&-\mathbf{B}_{i}\mathbf{F}_{dj}\\ 0&\mathbf{E}+\mathbf{L}_{di}\mathbf{C}_{j}\end{array} \right]\) and \(\mathbf{A}_{Rij}=\left[ \begin{array}{cc}\mathbf{A}_{i}-\mathbf{B}_{i} \mathbf{F}_{pj}&\mathbf{B}_{i}\mathbf{F}_{pj}\\ 0&\mathbf{A}_{i}-\mathbf{L}_{pi}\mathbf{C}_{j}\end{array} \right]\). Furthermore, the following assumptions concerning controllability and observability of T-S fuzzy singular systems are proposed. Similar assumptions can be found in [26;33]. In addition, the lemmas are presented to convert stability conditions into LMI form. **Assumption 1**. : _System (3) is completely controllable if the following conditions are satisfied._  
\[rank\big{[} \begin{array}{cc}s\mathbf{E}-\mathbf{A}_{i}&\mathbf{B}_{i}\end{array}  \end{array}\]  
\[=m_{x}\text{ and }rank\big{[} \begin{array}{cc}\mathbf{E}&\mathbf{B}_{i}\end{array}  \end{array}\]  
\[=m_{x}\text{, }\forall s\in\mathbb{C}_{+}.\]_where \(\mathbb{C}_{+}\) is the open right-half of the complex plane._  
**Assumption 2**. : _The system (3) is completely observable if the following conditions are held._  
\[rank\bigg{[} \begin{array}{c}s\mathbb{E}-\mathbf{A}_{i}\\ \mathbf{C}_{i}\end{array} \bigg{]}=m_{x}\text{ and }rank\bigg{[} \begin{array}{c} \mathbb{E}\\ \mathbf{C}_{i}\end{array} \bigg{]}=m_{x},\forall s\in\mathbb{C}_{+}.\]  
**Remark 1**. : _Under the necessary assumptions as mentioned previously, one knows that the matrix \(\mathbb{E}_{Rij}\) is a full rank matrix. (i.e., \(\mathbb{E}_{Rij}\) is invertible.)