## 2 FOSMC of an Integer Second Order Perturbed Nonlinear System
The derivative of (8) to the order \((-\beta)\) is now taken (this is equivalent to integrate (8) to the order \(\beta\)), leading to  
\[\dot{s_{\chi}}=-\sigma_{\chi}(D^{-\beta}sgn(s_{\chi}))-\mu_{\chi}s_{\chi}. \tag{9}\]  
On the other hand, from (5) and the nominal nonlinear system (7), one has that  
\[\dot{s_{\chi}}=\ddot{\chi}-\ddot{x_{d}}+\lambda_{\chi}\dot{e}=f_{x}+g_{x}u- \ddot{x_{d}}+\lambda_{\lambda}\dot{e}. \tag{10}\]  
Thus by combining Equations (9) and (10), the following FOSMC is obtained:  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]0  
where  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]1  
In fact, the FOSMC Equations (11) and (12) can achieve the attraction of the perturbed dynamics (2) to the sliding surface (6) under a sufficient condition on the size of the perturbation term \(\Delta_{\chi}\), the parameters \(\sigma_{\chi}\mu_{\chi}\) and the discontinuous term \(D^{-\beta}sgn(s_{\chi})\), as stated in the following result. Theorem 1: _Consider the perturbed nonlinear system (2) together with Assumption A1. If the parameters \(\sigma_{\chi}\), \(\mu_{\chi}\) and the bound \(\kappa_{\Delta}\) satisfy_  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]2_then the FOSMC Equations (11) and (12) assures the attractiveness of the perturbed system to the switching surface \(s_{\chi}=0\) defined by (6)._  
Proof. : Let us consider the following Lyapunov function candidate:  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]3  
which is positive definite. The time derivative of \(V_{\chi}\) has the form  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]4  
substituting the dynamics of the perturbed system (2) into (15) together with the FOSMC Equations (11) and (12) leads to  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]5  
Since \(s_{\chi}=|s_{\chi}|sgn(s_{\chi})\), \(\dot{V_{\chi}}\) can also be written as  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]6  
Also \(0<\beta<1\), thus \(sgn(D^{-\beta}sgn(s_{\chi}))=sgn(s_{\chi})\)[25], then (17) takes the form  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]7  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]8  
Majoring this last expression and using the bound \(\kappa_{\Delta}\) in (3), one gets  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]9  
Thus if Condition (3) is satisfied, \(\dot{V_{\chi}}<0\) and the convergence to the surface \(s_{\chi}=0\) is accomplished. **Remark 1**. : _The FOSMC (11)-(12) compensate the effect of additive perturbations in a second order nonlinear system described by (2). In fact, the main idea to define a sliding surface (6) together with its \((1+\beta)\) fractional order derivative is that as soon as the disturbance appears, \(s_{\chi}\) becomes nonzero and, at that time, the discontinuous control term becomes active and rejects the disturbance. Also, and similar to the control scheme proposed in [26], expression (19) shows that a stronger attraction to \(s_{\chi}=0\) is achieved and that the attraction is higher for any \(s_{\chi}\) with \(\mu_{\chi}\
eq 0\) than with \(\mu_{\chi}=0\). In particular, the parameter \(\mu_{\chi}\) is not used in [26] where a fractional order linear system is controlled._