## 3 Main Results  
The main object of this paper is to discuss the stabilization issues of T-S fuzzy singular systems. Via the observer-based PD fuzzy control strategy and Lyapunov stability theory, some sufficient stability conditions for the system (13) are derived as follows:  
**Theorem 1**. : _If there exist matrices \(\mathbb{P}_{1}=\mathbb{P}_{1}^{\mathrm{T}}>0\), \(\mathbb{P}_{2}=\mathbb{P}_{2}^{\mathrm{T}}>0\), \(\mathbb{F}_{pj}\), \(\mathbb{F}_{dj}\), \(\mathbb{L}_{pi}\), \(\mathbb{L}_{di}\), \(\mathbb{Z}_{1}\) and \(\mathbb{Z}_{2}\) such that the following condition is held, the augmented system (13) is stable._  
\[\bigg{[} \begin{array}{cc}\widetilde{\mathbb{P}}-He\big{\{}\mathbf{\Pi}^{ \mathrm{T}}\mathbb{E}_{Rij}\big{\}}&\mathbf{\Pi}^{\mathrm{T}}\mathbf{A}_{Rij} \\ *&-\widetilde{\mathbb{P}}\end{array} \bigg{]}<0 \tag{17}\]  
_where \(\widetilde{\mathbb{P}}=\bigg{[} \begin{array}{cc}\mathbb{P}_{1}&0\\ 0&\mathbb{P}_{2}\end{array} \bigg{]}\), \(\mathbf{\Pi}=\bigg{[} \begin{array}{cc}\mathbb{Z}_{1}^{-1}&0\\ 0&\mathbb{Z}_{2}\end{array} \bigg{]}\) and \(He\{\bullet\}\) denotes the shorthand of \(\bullet+\bullet^{\mathrm{T}}\). The \(\mathbb{E}_{Rij}\) and \(\mathbb{A}_{Rij}\) are pre-defined in (12)._  
**Proof. ** Choosing the Lyapunov quadratic function \(V(\widetilde{x}(k))=\widetilde{x}^{T}(k)\widetilde{\mathbf{P}}\widetilde{x}(k)\), one can obtain the first forward difference of \(V(\widetilde{x}(k))\) as following equation. \[\Delta V(\widetilde{x}(k))=\widetilde{x}^{\mathrm{T}}(k+1)\widetilde{\mathbf{P} }\widetilde{x}(k+1)-\widetilde{x}^{\mathrm{T}}(k)\widetilde{\mathbf{P}} \widetilde{x}(k)\]  
\[=\widetilde{x}^{\mathrm{T}}(k)\Big{(}\mathbf{E}_{Rij}^{-1}(h)\mathbf{A}_{Rij}( h)\Big{)}^{\mathrm{T}}\widetilde{\mathbf{P}}\Big{(}\mathbf{E}_{Rij}^{-1}(h) \mathbf{A}_{Rij}(h)\Big{)}\widetilde{x}(k)-\widetilde{x}^{\mathrm{T}}(k) \widetilde{\mathbf{P}}\widetilde{x}(k)\]  
\[=\widetilde{x}^{\mathrm{T}}(k)\Big{(}\mathbf{A}_{Rij}^{\mathrm{T}}(h)\mathbf{E }_{Rij}^{-\mathrm{T}}(h)\widetilde{\mathbf{P}}\mathbf{E}_{Rij}^{-1}(h) \mathbf{A}_{Rij}(h)-\widetilde{\mathbf{P}}\Big{)}\widetilde{x}(k) \tag{18}\]  
Based on \(\sum\limits_{i=1}^{n}h_{i}=1\) and \(0\leq h_{i}\leq 1\), the condition (17) can be rewritten as follows:  
\[\left[ \begin{array}{cc}\widetilde{\mathbf{P}}-He\big{\{}\mathbf{\Pi}^{ \mathrm{T}}\mathbf{E}_{Rij}(h)\big{\}}&\mathbf{\Pi}^{\mathrm{T}}\mathbf{A}_{Rij }(h)\\ *&-\widetilde{\mathbf{P}}\end{array} \right]=\Xi+He\Big{\{}\mathbf{\Psi}_{ij}^{ \mathrm{T}}\mathbf{\Pi}\mathbf{\Lambda}\Big{\}}<0 \tag{19}\]  
where \(\mathbf{\Psi}_{ij}=\left[ \begin{array}{cc}-\mathbf{E}_{Rij}(h)&\mathbf{A}_{ Rij}(h)\end{array} \right]\), \(\mathbf{\Lambda}=\left[ \begin{array}{cc}\mathbf{I}&0\end{array} \right]\), \(\mathbf{\Xi}=\left[ \begin{array}{cc}\widetilde{\mathbf{P}}&0\\ 0&-\widetilde{\mathbf{P}}\end{array} \right]\), \(\mathbf{I}\) is identify matrix and \({}^{*}\) denotes the transposed elements of the symmetric position. According to Lemma 1, the following inequalities can be found from (19). \[\mathbf{A}_{Rij}^{\mathrm{T}}(h)\mathbf{E}_{Rij}^{-\mathrm{T}}(h)\widetilde{ \mathbf{P}}\mathbf{E}_{Rij}^{-1}(h)\mathbf{A}_{Rij}(h)-\widetilde{\mathbf{P}} =\mathbf{\Psi}_{ij\perp}^{\mathrm{T}}\Xi\mathbf{\Psi}_{ij\perp}<0 \tag{20}\]  
and  
\[-\widetilde{\mathbf{P}}=\mathbf{\Lambda}_{\perp}^{\mathrm{T}}\Xi\mathbf{ \Lambda}_{\perp}<0 \tag{21}\]  
where \(\mathbf{\Psi}_{ij\perp}^{\mathrm{T}}=\left[ \begin{array}{cc}\mathbf{A}_{Rij }^{\mathrm{T}}(h)\mathbf{E}_{Rij}^{-\mathrm{T}}(h)&\mathbf{I}\end{array} \right]\) and \(\mathbf{\Lambda}_{\perp}^{\mathrm{T}}=\left[ \begin{array}{cc}0&\mathbf{I} \end{array} \right]\). Clearly, the \(\Delta V(\widetilde{x}(k))<0\) if \(\mathbf{\Psi}_{ij\perp}^{\mathrm{T}}\Xi\mathbf{\Psi}_{ij\perp}<0\). Based on [PERSON] theory, it further means that the systems (13) is stable. The proof of Theorem 1 is completed. \(\square\)  
**Remark 2. **_According to [32], a similar Lyapunov function is chosen to derive the stability conditions for the controlled systems.