*2.1. Fractional Calculus*

There are several definitions used to regard the fractional differentiation operator, such as the Gr*u*¨nwald–Letnikov, Riemann–Liouville and Caputo definitions introduced in [31]. Among them, the Caputo definition is the most frequently used and has many applications in the engineering field. In the Caputo definition, the initial value of the differentiation is considered.

The Caputo fractional derivative for a function *f*(*t*) is

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

where *<sup>C</sup> t*0 *D<sup>α</sup> <sup>t</sup>* is the fractional integration differentiation operator, *t*<sup>0</sup> represents the initial time, and *α* ∈ [*n* − 1, *n*] is the order of the system. Γ(·) is a gamma function that is introduced in Definition 1. *f*(*t*) is a differentiable function with the *n*th derivative. In this paper, we mainly focus on the fractional-order system with *n* = 1 and *t*<sup>0</sup> = 0, then we have 0 < *α* < 1. When *n* = 1, *f*(*t*) is only required to have the first derivative. For the sake of simplicity, the differentiation operator *<sup>C</sup> t*0 *D<sup>α</sup> <sup>t</sup>* is replaced by *D<sup>α</sup> t* .

The Gr*u*¨nwald-Letnikov fractional derivative for a function *f*(*t*) is

$$\mathbf{f}\_{t\_0}^{GL} D\_t^a f(t) = \lim\_{N \to \infty} [\frac{t - t\_0}{N}]^{-a} \boldsymbol{\Sigma}\_{j=0}^{N-1} (-1)^j f(t - j[\frac{t - t\_0}{N}]). \tag{2}$$

The Riemann-Liouville fractional derivative for a function *f*(*t*) is

$${}^{RL}\_{t\_0}D\_t^\alpha f(t) = \begin{cases} \frac{1}{\Gamma(-\alpha)} \int\_{t\_0}^t (t-\tau)^{-\alpha-1} f(\tau)d\tau, & \alpha < 0\\ \qquad f(t), & \alpha = 0\\ \qquad D\_t^\alpha [\prescript{RL}{t\_0}{D}\_t^{\alpha-n} f(t)], & \alpha > 0 \end{cases} \tag{3}$$

with *n* − 1 ≤ *α* < *n*.

For a wide class of functions, the Gr*u*¨nwald–Letnikov and Riemann–Liouville definitions are equivalent [32]. However, it is difficult for the Gr*u*¨nwald-Letnikov definition to have a Laplace transform. The Laplace transform of the Riemann–Liouville definition is

$$L[\_0^{RL}D\_t^a f(t)] = \begin{cases} s^a F(s)\_{\prime} & q \le 0\\ s^a F(s) - \sum\_{k=0}^{n-1} s^{kR}\_0 D\_t^{a-k-1} f(0), & n-1 \le q < n \end{cases} \tag{4}$$

where *F*(*s*) is the Laplace transform of *f*(*t*).

The Laplace transform of the Caputo definition is

$$L[^{\mathbb{C}}\_0 D^a\_t] = s^a F(s) - \sum\_{k=0}^{n-1} s^{a-1-k} f^{(k)}(0). \tag{5}$$

Comparing (4) and (5), it is obvious that the Riemann–Liouville fractional derivative is unsuitable for the Laplace transform technique because it requires the knowledge of noninteger-order derivatives of the function at *t* = 0, while the Caputo fractional derivative only requires the knowledge of integer-order derivatives of the function. This is why Caputo fractional derivative was chosen.

**Definition 1** ([33])**.** *Gamma function is an important element for the Caputo fractional differentiation operator, which is defined by*

$$
\Gamma(a) = \int\_0^\infty e^{-t} t^{a-1} dt,\tag{6}
$$

*where value* Γ(*a*) *is in the convergence of the right-hand side of the complex plane.*
