## 3 Main Results
\right]<0 \tag{24}\]  
Then, pre- and post-multiplying the matrix \(\widetilde{\mathbf{Z}}_{1}^{\mathsf{T}}\) and its transpose matrix on both sides of (24), one can obtain the following equation by defining \(\mathbf{X}=\mathbf{Z}_{1}^{\mathsf{T}}\mathbf{P}_{1}\mathbf{Z}_{1}\), \(\mathbf{Y}_{pj}=\mathbf{F}_{pj}\mathbf{Z}_{1}\), \(\mathbf{Y}_{dj}=\mathbf{F}_{dj}\mathbf{Z}_{1}\), \(\mathbf{K}_{pi}=\mathbf{Z}_{2}^{\mathsf{T}}\mathbf{L}_{pi}\) and \(\mathbf{K}_{di}=\mathbf{Z}_{1}^{\mathsf{T}}\mathbf{L}_{di}\). Where \(\widetilde{\mathbf{Z}}_{1}=diag\big{[} \begin{array}{ccccc}\mathbf{Z}_{1}& \mathbf{Z}_{1}&\mathbf{Z}_{1}&\mathbf{Z}_{1}&\mathbf{Z}_{1}\end{array} \end{array}\). \[\left[ \begin{array}{ccccc}\mathbf{X}-He\Big{\{}\mathbf{E}\mathbf{Z}_{1}+ \mathbf{B}_{i}\mathbf{Y}_{dj}\Big{\}}&\mathbf{B}_{i}\mathbf{Y}_{dj}&\mathbf{ A}_{i}\mathbf{Z}_{1}-\mathbf{B}_{i}\mathbf{Y}_{pj}&\mathbf{B}_{i}\mathbf{Y}_{pj} \\ *&\mathbf{Z}_{1}^{\mathsf{T}}\big{(}\mathbf{P}_{2}+\mathbf{\Theta}_{3 ij}\big{)} \mathbf{Z}_{1}&0&\mathbf{Z}_{1}^{\mathsf{T}}\mathbf{\Theta}_{4 ij}\mathbf{Z}_{1 }\\ *&*&-\mathbf{X}&0\\ *&*&-\mathbf{Z}_{1}^{\mathsf{T}}\mathbf{P}_{2}\mathbf{Z}_{1}\end{array} \right]<0 \tag{25}\]  
Notice that the right-bottom block matrix in (25) is still not an LMI framework. Thus, the following fact is proposed via Lemma 2. \[\left[ \begin{array}{ccccc}\mathbf{Z}_{1}^{\mathsf{T}}\big{(}\mathbf{P}_{2}+ \mathbf{\Theta}_{3 ij}\big{)}\mathbf{Z}_{1}&0&\mathbf{Z}_{1}^{\mathsf{T}}\mathbf{ \Theta}_{4 ij}\mathbf{Z}_{1}\\ *&-\mathbf{X}&0\\ *&*&-\mathbf{Z}_{1}^{\mathsf{T}}\mathbf{P}_{2}\mathbf{Z}_{1}\end{array} \right]\]  
\[=\left[ \begin{array}{ccccc}\mathbf{Z}_{1}^{\mathsf{T}}&0&0\\ 0&\mathbf{I}&0\\ 0&0&\mathbf{Z}_{1}^{\mathsf{T}}\end{array} \right]\left[ \begin{array}{ccccc} \mathbf{P}_{2}+\mathbf{\Theta}_{3 ij}&0&\mathbf{\Theta}_{4 ij}\\ *&-\mathbf{X}&0\\ *&*&-\mathbf{P}_{2}\end{array} \right]\left[ \begin{array}{ccccc}\mathbf{Z}_{1}& 0&0\\ 0&\mathbf{I}&0\\ 0&0&\mathbf{Z}_{1}\end{array} \right]\leq-\alpha He\{\mathbf{\Gamma}_{1}\}-\alpha^{ 2}\mathbf{\Gamma}_{2 ij}^{-1} \tag{26}\]  
\[\text{where }\mathbf{\Gamma}_{1}=\left[ \begin{array}{ccccc}\mathbf{Z}_{1}&0&0\\ 0&\mathbf{I}&0\\ 0&0&\mathbf{Z}_{1}\end{array} \right]\text{ and }\mathbf{\Gamma}_{2 ij}=\left[ \begin{array}{ccccc} \mathbf{P}_{2}+\mathbf{\Theta}_{3 ij}&0&\mathbf{\Theta}_{4 ij}\\ *&-\mathbf{X}&0\\ *&*&-\mathbf{P}_{2}\end{array} \right].\]  
Thus, the following inequality can be inferred from the condition (25). \[\left[ \begin{array}{ccccc}-&-\\ \mathbf{\Gamma}_{1 ij}&\mathbf{\Gamma}_{2 ij}\\ *&\mathbf{\Gamma}_{3 ij}\end{array} \right]<0 \tag{27}\]  
for \(i=j=1,2,\ldots,n\)  
\[\mathbf{C}_{1}=\mathbf{C}_{2}=\left[ \begin{array}{cc}1&1&0\\ 0&0&1\end{array} \right]\text{. The membership functions are }h_{1}(x(k))=\frac{1}{2}\Big{(}1-\frac{x_{2}(k)}{\mu}\Big{)}\text{, }h_{2}(x(k))=\frac{1}{2}\Big{(}1+\frac{x_{2}(k)}{\mu}\Big{)}\text{, }\sum_{i=1}^{2}h_{i}(x(k))=1\text{ and stated in Figure 1}.\]  
To build the observer-based fuzzy controller (8) via PD feedback scheme, the following gains can be calculated by using LMI techniques with \(\alpha=20\). \[\mathbf{F}_{p1} = \left[ \begin{array}{cc}-0.2601&17.6914&4.2555\\ -0.0001&-0.0017&29.4621\end{array} \right]\text{, }\mathbf{F}_{p2} = \left[ \begin{array}{cc}0.1049&16.4441&-4.1554\\ -0.0000&-0.0014&21.3512\end{array} \right]\text{, }\mathbf{F}_{d1} = \left[ \begin{array}{cc}-0.9505&386.4357&-1.2783\\ -0.0012&0.0670&-33.5144\end{array} \right]\text{, }\mathbf{F}_{d2} = \left[ \begin{array}{cc}-0.7365&386.7741&1.9942\\ -0.0017&-0.0830&-25.0700\end{array} \right]\text{, }\mathbf{L}_{p1}^{ \mathbf{T}} = \left[ \begin{array}{cc}-0.4700&1.4193&0.0151\\ -0.1264&0.3761&-41.0435\end{array} \right]\text{, }\mathbf{L}_{p2}^{ \mathbf{T}} = \left[ \begin{array}{cc}0.8163&0.0429&-0.0053\\ 0.1429&-0.3622&-30.4227\end{array} \right]\text{, }\mathbf{L}_{d1}^{ \mathbf{T}} = \left[ \begin{array}{cc}567.0738&-597.9506&-8.3359\\ 1.1348&-0.9925&-138.3697\end{array} \right]\text{and }\mathbf{L}_{d2}^{ \mathbf{T}} = \left[