## 2 FOSMC of an Integer Second Order Perturbed Nonlinear System  
Fractional calculus is mainly the calculus of derivatives and integrals of fractional order that can be real or complex [23] and it has many applications in science and engineering. In this article a Sliding Mode Control of Fractional Order is proposed for the trajectory tracking of a quadrotor. Among the definitions of fractional derivative the most cited are the Riemman-Liouville and Caputo definitions. Here, the Caputo definition is used which is expressed as  
\[D^{\alpha}\phi=\phi^{\alpha}:=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}\frac{\phi ^{(n)}(\xi)}{(t-\xi)^{\alpha+n-1}}d\xi, \tag{1}\]  
where \(\Gamma\) is the so called gamma function defined as \(\Gamma(n)=\int_{0}^{\infty}t^{z-1}e^{-t}dt\) and \(n-1\leq\alpha\leq n\) with n being an integer. From this definition, one can notice that when \(\alpha<0\) one has integration and when \(\alpha>0\) one has differentiation; when \(\alpha=1.0\) one has the first order derivative of a function. The gamma function generalizes the notion of a number raised to an arbitrary real number (or complex number). Thus the gamma function somehow involves the computation of a fractional derivative. The definition of a fractional derivative, as stated above, can not be used in practice, thus numeric methods such as the one based on the Grunwald-Letnikov approach is commonly used. Let us consider a perturbed nonlinear system described by a second order differential equation of the form  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t)+\Delta_{\chi}(\chi,t), \tag{2}\]  
where \(\chi(\cdot)\in\mathbb{R}\), \(f_{\chi}(\cdot)\) and \(g_{\chi}(\cdot)\) are bounded functions of their arguments and \(u(\cdot)\in\mathbb{R}\) is the system input. \(\Delta_{\chi}(\cdot,\cdot)\in\mathbb{R}\) is a disturbance term which includes uncertainties associated to the modelling of a process and external perturbations. The following assumption is made on \(\Delta_{\chi}(\cdot,\cdot)\)  
**Assumption A1**. _The disturbance term \(\Delta_{\chi}(\cdot,\cdot)\) is unknown but bounded, more precisely \(\Delta_{\chi}(\cdot,\cdot)\) satisfies_  
\[|\Delta_{\chi}(\chi,t)|\leq\kappa_{\Delta}, \tag{3}\]  
_for all \(\chi(\cdot)\) which are solutions of (2) and all \(t\geq 0\) with \(\kappa_{\Delta}\) being a real positive constant different from 0._  
It is desired that the signal \(\chi\) tracks a given reference signal \(\chi_{d}\), despite the presence of the disturbance term \(\Delta_{\chi}\). Thus a tracking error \(e\) is defined as follows:  
\[e=\chi-\chi_{d}. \tag{4}\]  
In accordance to the sliding mode control methodology [24] a so called switching function \(s_{\chi}\) is chosen as  
\[s_{\chi}=\dot{e}+\lambda_{\chi}e, \tag{5}\]  
where \(\lambda_{\chi}\) is a real constant parameter. The switching function (5) defines the sliding surface  
\[s_{x}=0=\dot{e}+\lambda_{\chi}e. \tag{6}\]  
Thus, \(\lambda_{\chi}\) is selected in such a way that the first order differential Equation (6) has a solution that exponentially converges to zero; as a consequence \(\chi\) also converges to \(\chi_{d}\) exponentially. Let us also consider the _nominal nonlinear system_ associated to the perturbed system (2), this is with \(\Delta_{\chi}(\chi,t)=0\), for all \(\chi\in\mathbb{R}\) and all \(t>0\), given by  
\[\ddot{\chi}=f_{\chi}(\chi,t)+g_{\chi}(\chi,t)u(t). \tag{7}\]  
In order to attract the dynamics of system (2) to the sliding surface (6) and based on the strategy proposed in [25], the \((1+\beta)\) fractional order derivative of \(s_{\chi}\) is set to be  
\[s_{\chi}^{1+\beta}=-\sigma_{\chi}sgn(s_{\chi})-\mu_{\chi}s_{\chi}^{\beta}, \tag{8}\]  
where \(\sigma_{\chi}\) and \(\mu_{\chi}\) are positive real constants and \(sgn(\cdot)\) is the signum function.