## 2 Design of a New Adaptive Fuzzy Controller
### Formulation of Control Strategy  
#### 2.2.1 Proposed Switching Control: High Disturbance Condition  
In this case, the disturbance phenomenon generates major effects on the response of the system. Hence, the control input term, \(u_{3}\), is proposed as follows:  
\[u_{31}=-\frac{1}{M_{3\
abla}\hat{\theta}_{g}\
abla\xi_{g}\
abla}\left(K_{0} \sigma_{s\
abla}^{2}+\|\sigma_{s\
abla}\|^{v+\epsilon}\mathbf{K}_{g}e^{\Theta_{ \xi}(-ln\|\sigma_{s\
abla}\|)}\sigma_{s\
abla}\right) \tag{15}\]  
where, \(\mathbf{K}_{0}=\mathbf{Z}_{0}(\gamma\mathbf{W}-\mathbf{I}_{n})^{-1}\in R^{m\times n}\) is a matrix related the properties of the system through the matrices \(\mathbf{A}_{P}\) and \(\mathbf{B}_{P}\); \(\mathbf{Z}_{0}\in R^{m\times n}\) is a chosen Lyapunov matrix; \(\mathbf{W}\in R^{n\times n}\) is a matrix which is found from solve the Lyapunov function; \(\gamma\in R,\alpha\in R^{+}\) are positive chosen constants; \(K_{\xi}\in R^{m\times n}\) is a matrix which is defined as \(\mathbf{K}_{\xi}=\mathbf{Z}\mathbf{P}_{\xi}\); the matrices \(\mathbf{Z}\in R^{m\times n}\) and \(\mathbf{P}_{\xi}\in R^{n\times n}\) are chosen matrices from the constraints system which will be described in next equations; \(\mathbf{G}_{\xi}\in R^{n\times n}\) is defined as \(\mathbf{G}_{\xi}=v\mathbf{W}+\epsilon\mathbf{I}_{n}\), where \(\mathbf{I}_{n}\) is the unit matrix; \(v\in R,\epsilon\in R\) are two positive constants. The value of \(\mathbf{Z}\), \(\mathbf{W}\), \(\mathbf{K}_{0}\) and \(\mathbf{K}_{\xi}\) are found from the system constraints which are defined using the linear matrix inequality (LMI in short) method as follows:\[\left[ \begin{array}{cc}\left(\mathbf{A}_{P}\mathbf{W}-\mathbf{WA}_{P}^{T}-\mathbf{A}_{P}+\mathbf{B}_{ P}\mathbf{Z}_{0}\right)&\sqrt{ln\frac{\delta_{dmx}}{\delta_{dmx}}}\mathbf{B}_{P}\mathbf{Z}_{0}& \mathbf{Z}_{0}ln\frac{\delta_{dmx}}{\delta_{dmx}}\\ \sqrt{ln\frac{\delta_{dmx}}{\delta_{dmx}}}\mathbf{B}_{P}^{T}\mathbf{Z}_{0}^{T}&( \gamma\mathbf{W}-\mathbf{I}_{n})\mathbf{B}_{P}&0\\ \mathbf{Z}_{0}ln\frac{\delta_{dmx}}{\delta_{dmx}}&0&\left[\begin{array}{c}(\mathbf{A }_{P}+\mathbf{B}_{P}\mathbf{K}_{0})\mathbf{X}+\mathbf{X}(\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{K}_{0})^{T}+ \\ \mathbf{B}_{P}\mathbf{Z}+\mathbf{Z}^{T}\mathbf{B}_{P}^{T}+\delta\mathbf{X}+\mathbf{D}_{s}\end{array} \right]\\ \Leftrightarrow\left(\begin{array}{c}\mathbf{A}_{P}\mathbf{W}-\mathbf{WA}_{P}^{T}-\\ -\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{Z}_{0}\end{array}\right)(\gamma\mathbf{W}-\mathbf{I}_{n})\bm {B}_{P}\left[\begin{array}{c}(\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{K}_{0})\mathbf{X}+\mathbf{X}( \mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{K}_{0})^{T}+\\ +\mathbf{B}_{P}\mathbf{Z}+\mathbf{Z}^{T}\mathbf{B}_{P}^{T}+\delta\mathbf{X}+\mathbf{D}_{s}\end{array} \right]-\\ -\mathbf{Z}_{0}^{2}ln^{2}\frac{\delta_{dmx}}{\delta_{dmx}}(\gamma\mathbf{W}-\mathbf{I}_{n}) \mathbf{B}_{P}-ln\frac{\delta_{dmx}}{\delta_{dmx}}\left[\begin{array}{c}(\mathbf{A} _{P}+\mathbf{B}_{P}\mathbf{K}_{0})\mathbf{X}+\mathbf{X}(\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{K}_{0})^{T}+ \\ +\mathbf{B}_{P}\mathbf{Z}+\mathbf{Z}^{T}\mathbf{B}_{P}^{T}+\delta\mathbf{X}+\mathbf{D}_{s}\end{array} \right]\leq 0\end{array} \right] \tag{16}\]  
where, \(\delta_{dmx},\delta_{dmx}\) are positive robustness parameters which are chosen based on the boundary of disturbance \(\mathbf{D}_{s}\) and \(\delta_{dmx}\leq\delta_{dmx}\). From the constraint Equation (16), the detailed constraints for the robustness control term are derived and given by:  
\[\mathbf{A}_{P}\mathbf{W}-\mathbf{WA}_{P}^{T}-\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{Z}_{0}=0 \tag{17}\]  
\[(\gamma\mathbf{W}-\mathbf{I}_{n})\mathbf{B}_{P}<0 \tag{18}\]  
\[(\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{K}_{0})\mathbf{X}+\mathbf{X}(\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{K}_{0} )^{T}+\mathbf{B}_{P}\mathbf{Z}+\mathbf{Z}^{T}\mathbf{B}_{P}^{T}+\delta\mathbf{X}+\mathbf{D}_{s}\leq 0 \tag{19}\]  
where, \(\mathbf{X}\in R^{n\times n}\) is chosen matrix; \(\delta\in R\) is a positive chosen constant; \(\mathbf{P}_{\xi}=\mathbf{X}^{-1}\).