## 5 Formulation of the LMI Stability Criterion
### Development of the Stability Condition
Taking \(\phi=\mathcal{C}_{1}\mathcal{X}\), with \(\mathcal{C}_{1}=\left[ \begin{array}{cc}1&0\end{array} \right]\). Therefore, Expression (44) becomes as follows:  
\[\Phi_{2}(\mathcal{X})^{2}\leq\mu_{1}\mathcal{X}^{\mathrm{T}}\mathcal{C}_{1}^{ \mathrm{T}}\mathcal{C}_{1}\mathcal{X}+2\mu_{2}\mathcal{C}_{1}\mathcal{X}+\mu_ {3} \tag{46}\]  
As a result, Expression (35) is recast like so:  
\[2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{D}_{2}\Phi_{2}(\mathcal{X})\leq \mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{D}_{2}m_{2}\mathcal{D}_{2}^{ \mathrm{T}}\mathcal{P}\mathcal{X}+\mu_{1}m_{2}^{-1}\mathcal{X}^{\mathrm{T}} \mathcal{C}_{1}^{\mathrm{T}}\mathcal{C}_{1}\mathcal{X}+2\mu_{2}m_{2}^{-1} \mathcal{C}_{1}\mathcal{X}+\mu_{3}m_{2}^{-1} \tag{47}\]  
Furthermore, since \(\Delta_{t}\) denotes a scalar function, then using the Young inequality (Lemma 1), the following equation is established:  
\[2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{B}\Delta_{t}\leq\mathcal{X}^{ \mathrm{T}}\mathcal{P}\mathcal{M}_{3}\mathcal{P}\mathcal{X}+\mathcal{B}^{ \mathrm{T}}\mathcal{M}_{3}^{-1}\mathcal{B}\Delta_{t}^{2} \tag{48}\]  
where \(\mathcal{M}_{3}=\mathcal{M}_{3}^{\mathrm{T}}>0\). Moreover, condition (48) may be recast in the following way based on Assumption 1:  
\[2\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{B}\Delta_{t}\leq\mathcal{X}^{ \mathrm{T}}\mathcal{P}\mathcal{M}_{3}\mathcal{P}\mathcal{X}+\tilde{\delta}^{2 }\mathcal{B}^{\mathrm{T}}\mathcal{M}_{3}^{-1}\mathcal{B} \tag{49}\]Replacing conditions (34), (47) and (49) in the resulting expression (33), the following inequality is hence obtained:  
\[2\mathcal{X}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{B} \mathcal{K}) \mathcal{X}+\zeta\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{X}+ \mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{M}_{1}\mathcal{P}\mathcal{X}+ \mathcal{D}_{1}^{\mathrm{T}}\mathcal{M}_{1}^{-1}\mathcal{D}_{1}+\mathcal{X}^{ \mathrm{T}}\mathcal{P}\mathcal{D}_{2}m_{2}\mathcal{D}_{2}^{\mathrm{T}} \mathcal{P}\mathcal{X}\] \[+\mu_{1}m_{2}^{-1}\mathcal{X}^{\mathrm{T}}\mathcal{C}_{1}^{ \mathrm{T}}\mathcal{C}_{1}\mathcal{X}+2\mu_{2}m_{2}^{-1}\mathcal{C}_{1} \mathcal{X}+\mu_{3}m_{2}^{-1}+\mathcal{X}^{\mathrm{T}}\mathcal{P}\mathcal{M}_{ 3}\mathcal{P}\mathcal{X}\] \[+\delta^{2}\mathcal{B}^{\mathrm{T}}\mathcal{M}_{3}^{-1}\mathcal{ B}<0\qquad\text{ such that}\qquad\mathcal{Z}(\mathcal{X})>0 \tag{50}\]  
This is another way to recast this condition:  
\[\mathcal{Y}(\mathcal{X})=\left[ \begin{array}{c}\mathcal{X}\\ 1\end{array} \right]^{\mathrm{T}}\!\!\left[ \begin{array}{c}\mathrm{Y}_{11}& \mathrm{Y}_{12}\\ \star&\mathrm{Y}_{22}\end{array} \right]\left[ \begin{array}{c}\mathcal{X}\\ 1\end{array} \right]<0\qquad\text{ such that}\qquad\mathcal{Z}(\mathcal{X})>0 \tag{51}\]  
where  
\[\mathrm{Y}_{11} =(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})+( \mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})^{\mathrm{T}}+ \mathcal{P}\mathcal{M}_{1}\mathcal{P}+m_{2}\mathcal{P}\mathcal{D}_{2}\mathcal{ D}_{2}^{\mathrm{T}}\mathcal{P}\] \[+\mu_{1}m_{2}^{-1}\mathcal{C}_{1}^{\mathrm{T}}\mathcal{C}_{1}+ \mathcal{P}\mathcal{M}_{3}\mathcal{P}+\zeta\mathcal{P} \tag{52a}\] \[\mathrm{Y}_{12} =\mu_{2}m_{2}^{-1}\mathcal{C}_{1}^{\mathrm{T}}\] (52b) \[\mathrm{Y}_{22} =\mathcal{D}_{1}^{\mathrm{T}}\mathcal{M}_{1}^{-1}\mathcal{D}_{1} +\mu_{3}m_{2}^{-1}+\delta^{2}\mathcal{B}^{\mathrm{T}}\mathcal{M}_{3}^{-1} \mathcal{B} \tag{52c}\]  
Utilizing the S-procedure (Lemma 3) along with Expressions (51) and (27), the subsequent stability criteria are developed:  
\[\left[ \begin{array}{cc}\mathrm{Y}_{11}+\xi_{1}\mathcal{I}_{n}& \mathrm{Y}_{12}\\ \star&\mathrm{Y}_{22}-\xi_{1}\epsilon^{2}\end{array} \right]<0 \tag{53}\]  
with \(\xi_{1}>0\)  
The resultant matrix inequality (53) is clearly defined with regard to the unknown matrix variables \(\mathcal{P}\), \(\mathcal{K}\), \(\mathcal{M}_{1}\), and \(\mathcal{M}_{3}\), and the scalar variables \(m_{2}\), \(\delta\) and \(\xi_{1}\), where the parameter \(\zeta\) will be fixed.