## 2 Design of a New Adaptive Fuzzy Controller
### Formulation of Control Strategy
#### 2.2.2 Proposed Switching Control: Small Disturbance Condition  
When the disturbance phenomenon has small effects on the response of the system, the following control \(u_{3}\) is proposed. \[u_{32}=-\frac{1}{M_{3\
abla}\tilde{\theta}_{g\
abla}\tilde{\xi}_{g\
abla}} \Big{(}K_{0}\sigma_{s\
abla}^{2}\Big{)} \tag{28}\]  
where, \(K_{0}=\mathbf{Z}_{0}(\mathbf{W}-(\gamma+1)\mathbf{I}_{n})^{-1},\mathbf{Z}_{0} \in R^{m\times n},\mathbf{W}\in R^{n\times n}\) with the system constraints are defined as follows:  
\[\mathbf{A}_{P}\mathbf{W}-\mathbf{W}\mathbf{A}_{P}^{T}-\mathbf{A}_{P}+ \mathbf{B}_{P}\mathbf{Z}_{0}=0 \tag{29}\]  
\[(\mathbf{A}_{P}+\mathbf{B}_{P}\mathbf{K}_{0})\mathbf{X}+\mathbf{X}(\mathbf{A }_{P}+\mathbf{B}_{P}\mathbf{K}_{0})^{T}+\mathbf{B}_{P}\mathbf{Z}+\mathbf{Z}^{ T}\mathbf{B}_{P}^{T}+\delta\mathbf{X}\leq 0 \tag{30}\]  
It is noted that constraints (29) and (30) can also be derived by using the Schur's complement method like the Equation (16). Using Equations (20) and (21) with a new updated control (28), the following equation is obtained. \[ \begin{array}{rl}\dot{\sigma}_{s\
abla}=&M_{1\
abla}+M_{2\
abla}+M_{3 \
abla}\widetilde{\Upsilon}_{f\
abla}\tilde{\xi}_{f\
abla}+M_{3\
abla} \widetilde{\Upsilon}_{g\
abla}\tilde{\xi}_{g\
abla}u_{\
abla}-M_{3\
abla} \tilde{x}_{d\
abla}+c_{s\
abla}\dot{\varphi}+w_{\
abla}+M_{3\
abla}\hat{ \theta}_{f\
abla}\tilde{\xi}_{f\
abla}-M_{1\
abla}-\\ &-M_{2\
abla}-M_{3\
abla}\hat{\theta}_{f\
abla}\tilde{\xi}_{f\
abla}+M_{3 \
abla}\tilde{x}_{d\
abla}-c_{s\
abla}\dot{\varphi}-\sum\limits_{i=1}^{n-1}P_ {(n-1)i}x_{i\
abla}+\frac{1}{2}M_{3\
abla}\Upsilon^{x}_{z\
abla}\tilde{x}_{ \
abla}\mathbf{P}\mathbf{S}_{2\
abla}S_{2\
abla}^{T}P_{\
abla}^{T}+\\ &\qquad+K_{P}M_{3\
abla}S_{2\
abla}^{T}\mathbf{E}\mathbf{E}^{T}\mathbf{P} \mathbf{B}_{P}+K_{I}M_{3\
abla}\int\limits_{0}^{\infty}\mathbf{S}_{2\
abla}^{T} \mathbf{E}\mathbf{E}^{T}\mathbf{P}\mathbf{B}_{P}dt+K_{D}M_{3\
abla}S_{2 \
abla}^{T}\dot{\mathbf{E}}\dot{\mathbf{E}}^{T}\mathbf{P}\mathbf{B}_{P}-K_{0 }\sigma_{s\
abla}^{2}\end{array}  \tag{31}\]  
**Theorem 2. **_The stability and robustness of the proposed control system are guaranteed if the main control function is designed by Equations (10), (11) and (28), and the adaptation laws are suggested like Equations (22)-(27). The proof of Theorem 2 can be completed using the same approach as the proof of Theorem 1 shown in Appendix A._  
**Remark 2. **_Definition for probability of switching system was shown in [20]. For all \(\Omega_{p}\in(0,1)\), there is a class \(\kappa\) function \(\Im(. )\) such that \(\forall\mathbb{C}>0,t\geq t_{0},\,|x(t_{0})|\leq\Im(\mathbb{C})\)_  
\[P(|x(t)|\leq\mathbb{C})\geq 1-\Omega_{p} \tag{32}\]  
_The value \(\Im(\mathbb{C})\) is chosen based on the dynamic property of the system. Normally, it is chosen following maximal displacement/or vibration. The group \(\mathbb{C}\) is also belonged to the dynamic parameters system. The value \(\Omega_{p}\) is defaulted as 0.2 for small vibration, and \(1\) for large vibration. This remark will be used for the switching control._  
**Remark 3**. : _The fuzzy model-based controller in [21] did not consider disturbance removal in control design process and hence its robustness could not be guaranteed in severe disturbance case. On the contrary, the proposed controller (15) associated with LMI (16) shows a close relationship with the parameters of the system and hence directly solves the disturbance following the desired boundaries. It is noteworthy that the prescribed performance in the sliding surface (5) also can improve the response of the system in case of the boundaries are not found exactly._  
**Remark 4**. : _The similar LMI used in the controller [21] has been applied in [22] where the fuzzy model is independent with the LMI whose role is to adjust control gains. Therefore, this controller could not bring robust control and then the convergence time to obtain stable states is long as shown in [22].