## 3 Design of a New FSMC  
### Structure FSMC  
Consider control system (1) without external disturbance. Let \(\mathbf{e}(t)\) be the error vector expressing the difference between the state vector and the corresponding desired state vector as follows. \[\mathbf{e}(t)=\mathbf{x}(t)-\mathbf{x}_{d}(t)=\left[e,\dot{e},\dots,e^{(n-1)} \right]^{T}\in R^{n} \tag{3}\]  
Figure 1: The solution structure: the \(B\)-controller, including the estimating uncertainties related to the model error, while the disturbance estimator (DE) for the estimating external disturbances \(d(t)\) only. Then, a sliding surface \(S(\mathbf{x})\) can be defined using the error vector as follows. \[S(\mathbf{x})=a_{1}e+a_{2}\dot{e}+,\ldots,+a_{n-1}e^{(n-2)}+e^{(n-1)} \tag{4}\]  
where, \(\mathbf{a}=[a_{1},\ldots,a_{n-1}]^{T}\) is the vector of Hurwitz polynomial \(H=s^{n-1}+a_{n-1}s^{n-2}+\ldots+a_{1}\), which has all poles that are to be located in the left half of the complex coordinate plane. In the polynomial, \(s\) is the Laplace operator. Then, by assuming the initial error is zero, the control issue can be considered as determining the control law \(u(t)\) to which \(S(\mathbf{x}(t))\to 0\) and then, \(\mathbf{e}(t)\) is remained on the sliding surface during system's operating process. Based on the state vector, the following Lyapunov candidate function is chosen:  
\[V_{0}(\mathbf{x})=\frac{1}{2}S(\mathbf{x})^{2} \tag{5}\]  
Then, the time derivation of this function along the error trajectory is expressed as follows. \[\frac{d}{dt}V_{0}(\mathbf{x})=S(\mathbf{x})\frac{d}{dt}S(\mathbf{x})=S( \mathbf{x})\dot{S}(\mathbf{x}) \tag{6}\]  
From the above equation, the control law \(u(t)\) is supposed to design so that the following equation is satisfied. \[\dot{S}(\mathbf{x})=-\rho\text{sgn}(S(\mathbf{x})) \tag{7}\]  
In the above, \(\rho\) is the positive coefficient. This is the sufficient condition to guarantee that \(\mathbf{e}(t)\to 0\) or \(\mathbf{x}\rightarrow\mathbf{x}_{d}\) when \(t\rightarrow\infty\). As mentioned earlier, in the process of calculating \(u_{s}(t)\), the external disturbance \(d(t)\) is ignored. From Equations (7), (4), (3), and (1), the following equation is obtained. \[\dot{S}(\mathbf{x})=-\rho\text{sgn}\left(S(\mathbf{x})\right)=\sum\
olimits_{ i=1}^{n-1}a_{i}e^{(i)}+f(\mathbf{x},t)+g_{1}(\mathbf{x},t)u_{s}(t)-{x_{d}}^{(n)}. \tag{8}\]  
Hence, the feedback control signal is such that the state \(\mathbf{x}\) of the closed-loop system will follow the desired state \(\mathbf{x}_{d}\), and the following controller can be inferred from Equation (8). \[u_{s}(t)=g_{1}(\mathbf{x},t)^{-1}\Big{(}-\sum\
olimits_{i=1}^{n-1}a_{i}e^{(i )}-f(\mathbf{x},t)+{x_{d}}^{(n)}-\rho\text{sgn}(S(\mathbf{x},t))\Big{)}. \tag{9}\]  
To calculate \(u_{s}(t)\) in Equation (9), it is necessary to estimate \(g_{1}(\mathbf{x},t)\), \(f(\mathbf{x},t)\), and \(h(\mathbf{x},t)=\rho\text{sgn}(S(\mathbf{x}))\). In this work, instead of using the fuzzy appropriations given in [12; 18], an adaptively smooth approach for \(h(\mathbf{x},t)\) is used. It is noted here that since the coefficient \(a_{i}\) of the polynomial \(S(\mathbf{x})\) is chosen to satisfy the Hurwitz stability, \(S(\mathbf{x}(t))\to 0\), and hence the error becomes zero by an appropriate controller with the assumption that the initial error is zero. Now, to estimate \(g_{1}(\mathbf{x},t)\), \(f(\mathbf{x},t)\), fuzzy systems MISO (Multiple Inputs, Single Output), are used to establish \(\hat{g}_{1}(\mathbf{x},t)\), \(\hat{f}(\mathbf{x},t)\) with a proposed solution for avoiding the singular case of \(\hat{g}_{1}(\mathbf{x},\varphi_{g})^{-1}\). First, based on the value \(S(\mathbf{x})\) and its required boundary layer, \(\Phi\), the initial function of \(h(\mathbf{x},t)\) is converted as follows. \[h(\mathbf{x},t)=\rho\,\text{sgn}(S(\mathbf{x})),\,\text{or}\,h(\mathbf{x},t)= \rho_{ad}\text{sat}(S(\mathbf{x})/\Phi) \tag{10}\]  
where, \(\text{sat}(S/\Phi)=\left\{ \begin{array}{ll}\text{sgn}(S/\Phi)&if\quad|S/ \Phi|\geq 1\\ S/\Phi&if\quad|S/\Phi|<1\end{array}