## 3 A Fractiona-Order Sliding Mode Controller (FOSMC)
The nominal system associated to the perturbed system in Equation (10) with \(\delta(x)=0\) for all \(x\) is given by  
\[\dot{x}(t)=f(x)+g(x)u(t). \tag{14}\]  
In order to attract the dynamics of the system in Equation (10) to the sliding surface in Equation (13) and based on the strategy proposed in Reference [39], the first-order time derivative of \(s\) is set to be the following (see, for example, Reference [38])  
\[\dot{s}=-k_{2}1^{\beta}sgn(s)-k_{3}s, \tag{15}\]  
with \(k_{2}\) and \(k_{3}\) being nonzero positive real parameters and \(sgn(\cdot)\) being the signum function. On the other hand, from the switching function in Equation (12) and the nominal system in Equation (14), \(\dot{s}\) is given by  
\[\dot{s}=\dot{x}_{r}-f(x)-g(x)u+k_{1}e. \tag{16}\]  
Combining Equations (15) and (16) leads to the following fractional-order sliding mode controller:  
\[u=-\frac{1}{g(x)}(f(x)+\eta(x,x_{r})), \tag{17}\]  
with  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]0  
A sufficient condition that assures the attraction of the perturbed nonlinear system in Equation (10) to the sliding surface in Equation (13) can be given. This is stated in the following theorem. **Theorem 1**. : _Let us consider the perturbed first-order nonlinear system in Equation (10) with a bounded disturbance \(\delta(x)\) that satisfies Equation (4). That is, Assumption 1 holds. If the parameter \(k_{2}\) and the bound \(\delta_{max}\) satisfy_  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]1_then the fractional order sliding mode controller in Equation (17) assures the attractiveness of the perturbed system to the sliding surface \(s=0\)._  
Proof. : Consider the Lyapunov function candidate  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]2  
which is positive definite. The time derivative of \(V\) along the trajectories of the perturbed system dynamics together with the fractional-order controller in Equation (17) takes the following form:  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]3  
Since \(s=\left|s\right|sgn(s)\) and \(sgn(D^{-\beta}sgn(s))=sgn(s)\), for \(0<\beta<1\)[39], \(\dot{V}\) can be written as  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]4  
Now majoring Equation (21) and using the bound \(\delta_{max}\) in Assumption 1, one obtains  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]5  
Notice that the term \(k_{3}s^{2}\) is always positive. Thus, if condition (18) is satisfied, \(\dot{V}<0\) and the convergence to the surface \(s=0\) is assured. **Remark 1**. : _The main idea of the fractional-order sliding mode controller design proposed here is to define a switching function (Equation (12)), together with its first-order derivative (Equation (15)), such that, when the disturbance appears, \(s\) is different from zero and, at that time, the discontinuous control term becomes active and rejects the disturbance._  
**Remark 2**. : _From the proof of Theorem 1, it can be noticed that the following sufficient condition allows us to obtain the attraction of the perturbed nonlinear system to the sliding surface \(s=0\)._  
\[l^{\beta}\phi(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}\frac{\phi(\tau)}{(t-\tau) ^{1-\beta}}d\tau, \tag{9}\]6  
In fact, the condition in Equation (23) shows that a stronger attraction to \(s=0\) is achieved and that the attraction is higher for any value of \(s\) with \(k_{3}\
eq 0\). However, the condition in Equation (23) is more restrictive since it requires the knowledge of \(\left|s\right|\) at every time \(t\).