*3.1. Fundamentals of Fractional Calculus*

Fractional calculus, essentially the non-integer order calculus, has the same history as integer order calculus. The three frequently used definitions of fractional calculus are the Grunwald–Letnikov definition, the Riemann–Liouville definition and the Caputo definition [25].

We consider the Caputo definition in this study because of its wide applications in engineering problems, the fractional integral and derivative by Caputo definition are as follows:

$$\_{t\_0}D\_t^{-\alpha}f(t) = \frac{1}{\Gamma(\alpha)} \int\_{t\_0}^t \frac{f(\tau)}{(t-\tau)^{1-\alpha}}d\tau\tag{5}$$

$$\_{t\_0}D\_t^\alpha f(t) = \frac{1}{\Gamma(m-\alpha)} \int\_{t\_0}^t \frac{f^{(m)}(\tau)}{(t-\tau)^{1+\alpha-m}} d\tau \tag{6}$$

where *<sup>t</sup>*<sup>0</sup> *D<sup>α</sup> <sup>t</sup>* represents the fractional calculus operator, *f*(*t*) is a continuous function and *t*<sup>0</sup> denotes the initial time. *α* represents the fractional-order, *m* = *α*. Γ(·) denotes the Gamma function as in (7).

$$
\Gamma(z) = \int\_0^\infty e^{-t} t^{z-1} dt\tag{7}
$$

In the numerical simulations, we adopt the standard Oustaloup approximation method to obtain the consistent frequency characteristics as fractional differential operator. A rational transfer function in the form of zero-pole type is described according to the Oustaloup method, that *N* is the order of filter, [*ωb*, *ωh*] is the selected frequency bound. The zero and pole are defined as *ω <sup>k</sup>* and *ωk*, which divide the frequency band into 2*N* + 1 intervals.

The Oustaloup rational approximation is described as

$$\mathbf{s}^{\mathfrak{a}} \approx \mathbf{G}(\mathbf{s}) = K \prod\_{k=1}^{N} \frac{\mathbf{s} + \omega\_{\mathbf{k}}^{\mathfrak{f}}}{\mathbf{s} + \omega\_{\mathbf{k}}} \tag{8}$$
  $\text{where } \omega\_{\mathbf{k}}^{\mathfrak{f}} = \omega\_{\mathfrak{b}} \omega\_{\mathfrak{u}}^{(2k-1-a)/N}, \omega\_{\mathfrak{u}} = \sqrt{\omega\_{\mathfrak{h}}/\omega\_{\mathfrak{b}}}, \omega\_{\mathbf{k}} = \omega\_{\mathfrak{b}} \omega\_{\mathfrak{u}}^{(2k-1+a)/N}, K = \omega\_{\mathfrak{h}}^{\mathfrak{a}}.$ 
