## 3 Design of a New FSMC
### Structure FSMC
\right..\)  
In order to avoid the chattering phenomenon, the function of \(\text{sat}(. )\) is used instead of \(\text{sgn}(.)\). Besides, to reduce the calculating cost, an adaptive gain \(\rho_{ad}\) is determined as follows. \[\rho_{ad}=\left\{ \begin{array}{ll}\max\Big{(}\eta_{1}\Omega_{0},\,\eta_{2} \Big{(}\varepsilon+\exp\Big{(}\eta_{3}S(\mathbf{x})\text{sgn}(\dot{S}(\mathbf{ x}))\Big{)}\Big{)}\Big{)}&if\quad|S(\mathbf{x})|\geq\Phi\\ \max\Big{(}\eta_{1}\frac{\Omega_{0}\Phi}{|S(\mathbf{x})|+\varepsilon^{\prime}}, \,\eta_{2}\Big{(}\varepsilon+\exp\Big{(}\eta_{3}S(\mathbf{x})\text{sgn}(\dot{ S}(\mathbf{x})\Big{)}\Big{)}\Big{)}&if\quad|S(\mathbf{x})|<\Phi\,,\end{array} \right. \tag{11}\]where, \(\eta_{1}>1\), \(\eta_{2,3}\geq 1\), and \(0<\epsilon<<1\) are adaptive coefficients chosen by the designer. The building update law (11) is summarized in Theorem 1. To establish \(\hat{g}_{1}(\mathbf{x},t)\), \(\hat{f}(\mathbf{x},t)\), a fuzzy system MISO is used as following:  
\[R^{(i)}:\quad\text{IF }x_{1}\text{ is }A_{1}^{i},\text{ AND }\ldots,\text{ AND }x_{n}\text{ is }A_{n}^{i} \tag{12}\] \begin{tabular}{l l}  & THEN \(y\) is \(B^{i}\quad(i=1\ldots m)\) \\ \end{tabular}   
where, \(A_{j}^{i}\), \(j=1\ldots n\), is the fuzzy set in the input space related to the physical parameter \(x_{j}\) and the \(i\)-th fuzzy law, while \(B^{i}\) is the corresponding fuzzy set in the output space. Then, using the center-average defuzzification, the output is calculated as below. \[y(\mathbf{x})=\left(\sum_{  
_where, \(0\leq\varepsilon_{0}<<1\). It is the sampling time of the control system if \(\hat{g}_{1}(\mathbf{x},\varphi_{g})=0\), otherwise \(\varepsilon_{0}=0\); parameters \(\varphi_{g}\), \(\varphi_{f}\), and \(\rho_{ad}\) are updated as follows:_  
\[\dot{\varphi}_{f}=S(\mathbf{x})\lambda(\mathbf{x}) \tag{20}\]  
\[\dot{\varphi}_{g}=\left\{ \begin{array}{l}S(\mathbf{x})\lambda(\mathbf{x})u_{ s}(t)-\Delta\lambda(\mathbf{x})\quad if\quad\hat{g}_{1}(\mathbf{x},\varphi_{g})=0\\ S(\mathbf{x})\lambda(\mathbf{x})u_{s}(t)\quad\text{otherwise}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(\Delta\lambda( \mathbf{x})=\lambda(\mathbf{x})^{(r)}-\lambda(\mathbf{x})^{(r-1)}\end{array} \right. \tag{21}\]  
\[\rho_{ad}=\max\Bigl{(}\eta_{1}\Omega_{0},\,\eta_{2}\Bigl{(}\varepsilon+\exp \Bigl{(}\eta_{3}S(\mathbf{x})\mathrm{sgn}(\hat{S}(\mathbf{x})\Bigr{)}\Bigr{)} \Bigr{)}\quad if\quad|S(\mathbf{x})|\geq\Phi\]  
\[\rho_{ad}=\max\Bigl{(}\eta_{1}\frac{\Omega_{0}\Phi}{|S(\mathbf{x})|+\varepsilon},\,\eta_{2}\Bigl{(}\varepsilon+\exp\Bigl{(}\eta_{3}S(\mathbf{x})\mathrm{sgn} (\hat{S}(\mathbf{x})\Bigr{)}\Bigr{)}\Bigr{)}\quad if\quad|S(\mathbf{x})|< \Phi\,, \tag{22}\]  
_In the above, \(\eta_{1}>1\), \(\eta_{2,3}\geq 1\), and \(0<\varepsilon<<1\) are adaptive coefficients chosen by the designer; \(r\) denotes the \(r\)-th sampling time. If the control system (1) is controlled by the control law \(u_{s}(t)\) in (19) with the updated laws of (20)-(22), then the process of \(\mathbf{e}(t)\to 0\) when \(t\to\infty\). This indicates that the control system is asymptotically stable in the sense of Lyapunov._  
**Proof. ** Using the time derivative of (4) and \(x^{(n)}\) from (1) with a note that in this phase, the controller is designed without any considered disturbance, \(d(t)=0\); \(u(t)=u_{s}(t)\), and then the followings conditions are given. \[\dot{S}(\mathbf{x})=\sum_{i=1}^{n-1}a_{i}e^{(i)}+x^{(n)}-x_{d}{}^{(n)} \tag{23}\]  
\[\dot{S}(\mathbf{x})=\sum_{i=1}^{n-1}a_{i}e^{(i)}+f(\mathbf{x},t)+g_{1}( \mathbf{x},t)\,u_{s}(t)-x_{d}{}^{(n)}\]  
From (19), (22), and (23), the following equation is obtained.