## 2 The Integro-Differential Operator  
In FC the same continuous integro-differential operator \({}_{a}D_{t}^{a}\) represents both integration and differentiation to a non-integer order:  
\[{}_{a}D_{t}^{a}=\left\{ \begin{array}{rl}d^{a}/dt^{a}&\text{Re}(\alpha)>0\\ 1&\text{Re}(\alpha)=0\\ \int\limits_{a}^{t}(d\tau)^{-\alpha}&\text{Re}(\alpha)<0\end{array} \right. \tag{1}\]  
In Equation (1) \(a\) and \(t\) are the limits of the operation and \(\alpha\) is the order, which can be real or complex; in the following, \(\alpha\in\mathbb{R}\). In the scientific literature several definitions of the integro-differential operator have been proposed (Grunwald-Letnikov, Riemann-Liouville, [PERSON], [PERSON], [PERSON], among the others) [2], but all of these are proved to be equivalent [44]. In the following the Grunwald-Letnikov definition is adopted, since it leads to a robust discrete-time implementation [45]. According to the Grunwald-Letnikov definition, the differentiation of fractional order \(\alpha\) (if \(\alpha>0\)) or the integration of fractional order \(-\alpha\) (if \(\alpha<0\)) of a function of time \(x(t)\) is defined as:  
\[{}_{a}D_{t}^{a}x(t)=\lim\limits_{h\to 0}\left[\frac{1}{h^{h}}\sum\limits_{k=0}^{ \lfloor\frac{t-a}{h}\rfloor}(-1)^{k}\,\frac{\Gamma(\alpha+1)}{\Gamma(k+1)\Gamma (\alpha-k+1)}x(t-kh)\right] \tag{2}\]  
where \(h\) is the time increment and \(\Gamma\) is the Gamma function, which extends the factorial function to real and complex numbers and is defined by the following equation:  
\[\Gamma(z)=\int\limits_{0}^{\infty}t^{z-1}e^{-t}dt \tag{3}\]In order to understand intuitively the meaning of FO derivatives and integrals, independently of their mathematical definition, we can consider the following properties:  
* similarly to IO derivatives and integrals, if an FO derivative/integral of order \(\alpha\) is applied twice to a function of time, the resulting function is the derivative of order \(2\alpha\); for example, the derivative of order \(1/2\) of the derivative of order \(1/2\) is the first-order derivative, and the integral of order \(1/2\) of the integral of order \(1/2\) is the first-order integral;
* for sinusoidal functions, similarly to IO derivatives/integrals, FO derivatives/integrals of order \(\alpha\) produce a phase shift of \(\alpha\pi/2\): for example, the first-order derivative causes a positive phase shift of \(\pi/2\), while the derivative of order \(1/2\) causes a positive phase shift of \(\pi/4\); the first-order integral causes a negative phase shift of \(\pi/2\), while the integral of order \(1/2\) causes a negative phase shift of \(\pi/4\). In Equation (2) the number of terms of the sum tends towards infinity, since \(h\) tends towards zero; for a discrete-time numerical computation, Equation (2) can be rewritten adopting a small but finite sampling time \(T_{s}\), in order to obtain the following discrete-time approximation [46]:  
\[D_{t}^{\alpha}x(t)\cong D^{\alpha}x_{k}=\left[\frac{1}{T_{s}^{\alpha}}\sum_{j= 0}^{k}w_{j}^{\alpha}x(t-jT_{s})\right] \tag{4}\]  
where \(k=(t-a)/T_{s}\) is the current step and:  
\[w_{0}^{\alpha}=1 \tag{5}\]  
For real-time implementation on a digital controller, for \(t>>a\) the number of addends becomes too large; therefore, it is necessary to limit the number of considered steps, in order to have a computational burden compatible with the controller CPU.