## 2 System Descriptions and Problem Statements  
To represent the nonlinear singular systems, the T-S fuzzy modeling approach was applied in this paper. The T-S fuzzy singular system can be represented as follows:  
Plant Rule _i_:  
IF \(\Im_{1}(k)\) is M\({}_{i1}\) and \(\dots\) and \(\Im_{q}(k)\) is M\({}_{iq}\) THEN  
\[\mathbf{E}x(k+1)=\mathbf{A}_{i}x(k)+\mathbf{B}_{i}u(k) \tag{1}\]  
\[y(k)=\mathbf{C}_{i}x(k) \tag{2}\]  
where \(\Im(k)=\left[ \begin{array}{cc}\Im_{1}(k)&\Im_{2}(k)&\cdots&\Im_{q}(k)\end{array}  \right]\), \(i=1,\ \dots,\ n\) and \(n\) is the number of rules, M\({}_{iq}\) are fuzzy sets, \(q\) is the number of premise variables, \(x(k)\in\Re^{m_{x}}\), \(u(k)\in\Re^{m_{u}}\) and \(y(k)\in\Re^{m_{y}}\) are the vector of state, control input, and output, respectively. \(\mathbf{A}_{i}\), \(\mathbf{B}_{i}\) and \(\mathbf{C}_{i}\) are constant matrices with appropriate dimensions. \(\mathbf{E}\) is a matrix with \(rank(\mathbf{E})<m_{x}\). By blending all the fuzzy rules, one can infer the T-S fuzzy singular system as follows:  
\[\mathbf{E}x(k+1)=\sum\limits_{i=1}^{n}h_{i}(\Im(k))\{\mathbf{A}_{i}x(k)+ \mathbf{B}_{i}u(k)\} \tag{3}\]  
\[y(k)=\sum\limits_{i=1}^{n}h_{i}(\Im(k))\{\mathbf{C}_{i}x(k)\} \tag{4}\]  
where \(h_{i}(\Im(k))=\prod\limits_{l=1}^{q}\text{M}_{il}(\Im_{l}(k))/\sum\limits_{i= 1}^{n}\prod\limits_{l=1}^{q}\text{M}_{il}(\Im_{l}(k))\), \(h_{i}(\Im(k))\geq 0\) and \(\sum\limits_{i=1}^{n}h_{i}(\Im(k))=1\). Due to the limitation of space, the abbreviation \(h_{i}(\Im(t))\triangleq h_{i}\) is defined in the following context. Considering the negative effects caused by algebraic equations, regularity and causality are the premise criteria and need to be discussed via model transformations [7,8]. In recent years, the PD feedback scheme is applied for singular systems to eliminate the impulse terms [25,26]. Because singular systems are transformed into normal state-space models, the above properties can be naturally satisfied. Referring to [25], however, in order to reasonably use the PD control method for discrete-time singular systems, the observer should be constructed to estimate system's states. As same as [25,26], it is assumed that the current time step is instant \(k+1\) and the previous time step is instant \(k\). For online implementation, it is expected that the time taken to estimate the state \(x(k+1)\) is smaller compared with the sampling period. The T-S fuzzy PD observer can be constructed as follows:  
Observer Rule _i_:  
IF \(\Im_{1}(k)\) is M\({}_{i1}\) and \(\dots\) and \(\Im_{q}(k)\) is M\({}_{iq}\) THEN  
\[\mathbf{E}\hat{x}(k+1)=\mathbf{A}_{i}\hat{x}(k)+\mathbf{B}_{i}u(k)+\mathbf{L} _{pi}\big{(}y(k)-\mathbf{C}_{j}\hat{x}(k)\big{)}+\mathbf{L}_{di}\big{(}y(k+1)- \mathbf{C}_{j}\hat{x}(k+1)\big{)} \tag{5}\]  
where \(\hat{x}(k)\in\mathbb{R}^{m_{x}}\) is the estimated state vector, \(\mathbf{L}_{pi}\) and \(\mathbf{L}_{di}\) are the observer gains with appropriate dimensions. The overall T-S fuzzy PD observer as follows:  
\[\mathbf{E}\hat{x}(k+1)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}h_{i}h_{j}\big{\{} \mathbf{A}_{i}\hat{x}(k)+\mathbf{B}_{i}u(k)+\mathbf{L}_{pi}\big{(}y(k)-\mathbf{ C}_{j}\hat{x}(k)\big{)}+\mathbf{L}_{di}\big{(}y(k+1)-\mathbf{C}_{j}\hat{x}(k+1) \big{)}\big{\}} \tag{6}\]  
With output observations, the system states can be estimated via PD fuzzy observer. Next, to ensure the stability of the system (3), the observer-based PD controller is represented as follows:  
Controller Rule _i_:IF \(\Im_{1}(k)\) is M\({}_{i1}\) and... and \(\Im_{q}(k)\) is M\({}_{iq}\) THEN  
\[u(k)=-\mathbf{F}_{pi}\hat{x}(k)-\mathbf{F}_{di}\hat{x}(k+1) \tag{7}\]  
where \(\mathbf{F}_{pi}\) and \(\mathbf{F}_{di}\) are the controller gains. The final control output is constructed as  
\[u(k)=\sum\limits_{i=1}^{n}h_{i}\{-\mathbf{F}_{pi}\hat{x}(k)-\mathbf{F}_{di} \hat{x}(k+1)\} \tag{8}\]  
Subsequently, the closed-loop T-S fuzzy singular system is obtained by substituting the observer-based PD fuzzy controller (8) into the system (3).