*4.2. The Frequency-Distributed Model of the Fractional Integrator*

As shown in Section 3, the Riemann–Liouville integral of a function *v*(*t*) is defined as the convolution of *v*(*t*) with the impulse response *hn*(*t*) of the fractional integrator.

Another expression of *hn*(*t*) can be derived from the inverse Laplace transform of 1 *<sup>s</sup><sup>n</sup>* [31,32,34], for 0 < *<sup>n</sup>* < 1, i.e., *hn*(*t*) = *<sup>L</sup>*−<sup>1</sup> <sup>1</sup> *sn* .

Using a Bromwich contour, we can write (see [34] and the references therein):

$$h\_{\mathbf{u}}(t) = \begin{cases} \frac{1}{2\pi\mathfrak{r}} \int\_{-\gamma}^{\gamma+j\omega} \frac{1}{s^{\pi}} e^{st} \, ds & \text{for } t > 0\\ 0 & \text{for } t < 0 \end{cases}$$

Thus, we obtain

$$\begin{aligned} h\_n(t) &= \int\_0^\infty \mu\_n(\omega) e^{-\omega t} d\omega = \frac{t^{n-1}}{\Gamma(n)} \quad \text{for } 0 < n < 1\\ \text{with } \mu\_n(\omega) &= \frac{\sin(n\pi)}{\pi} \omega^{-n} \end{aligned} \tag{26}$$

Note that, in the particular case where *v*(*t*)=(*t*) is an impulse function, the output *x*(*t*) corresponds to the impulse response *hn*(*t*) and is provided by the following distributed integer-order differential system

$$\begin{cases} \frac{\partial z(\omega, t)}{\partial t} = -\omega z(\omega, t) + \delta(t) \,\omega \in [0, +\infty) \\\ h\_n(t) = \int\_{\mathbb{D}} \mu(\omega) z(\omega, t) d\omega \end{cases} \tag{27}$$

$$\text{Thus } z(\omega, t) = e^{-\omega t}, \text{ which leads to } h\_{\text{ll}}(t) = \int\_0^{\infty} \mu(\omega) e^{-\omega t} d\omega.$$

More generally, for any input *v*(*t*), the corresponding output *x*(*t*) of the fractional integrator is provided by the following distributed frequency system:

$$\begin{cases} \frac{\partial z(\omega, t)}{\partial t} = -\omega z(\omega, t) + v(t) & \omega \in [0, \infty) \\ \qquad x(t) = \int \mu\_n(\omega) z(\omega, t) d\omega \\ \qquad \text{with } \mu\_n(\omega) = \frac{\sin(n\pi)}{\pi} \omega^{-n} \end{cases} \tag{28}$$

It is fundamental to notice that the original model of the fractional integrator has been transformed into an infinite-dimension integer-order differential system (28), where integer-order differentiation *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* has been substituted to fractional-order differentiation, and where the fractional order *n* appears in the weighting function μ*n*(*ω*). This means that the fractional integrator <sup>1</sup> *<sup>s</sup><sup>n</sup>* is an infinite-dimension linear system [54]. These models are the two "faces" of the fractional integrator.

In fact, according to linear system theory [52], the fractional integrator has two types of models, as does any other linear system:


**Remark 1**: let us consider the Laplace transform of (27):

$$\mathcal{L}\{h\_{\boldsymbol{n}}(t)\} = \mathcal{L}\left\{\int\_{0}^{\infty} \mu(\omega)e^{-\omega t}d\omega\right\} = \int\_{0}^{\infty} \mu\_{\boldsymbol{n}}(\omega)\mathcal{L}\{e^{-\omega t}\}d\omega = \int\_{0}^{\infty} \mu\_{\boldsymbol{n}}(\omega)\frac{1}{s+\omega}d\omega$$

The previous equality and Equation (11) lead to

$$\frac{1}{s^n} = \bigcap\_{0}^{\infty} \mu\_n(\omega) \frac{1}{s+\omega} d\omega \quad 0 < n < 1. \tag{29}$$

This relation exhibits that the fractional integrator is composed of an infinity of modes *ω*, ranging from 0 to +∞, whereas the integer-order integrator corresponds to only one mode situated at *ω* = 0. Figure 1 displays the graphic representation of Equation (28). Note that the distributed differential Equations of (28) correspond to the first-order systems displayed in Figure 1. Due to the distributed nature of the differential system, the graph of Figure 1 is composed of an infinity of first-order systems.

**Figure 1.** The frequency-distributed model of the fractional integrator.

**Remark 2**: The classical Laplace transform of the integer order derivative is known as *L dx*(*t*) *dt* <sup>=</sup> *sX*(*s*) <sup>−</sup> *<sup>x</sup>*(0), which corresponds in fact to the relation <sup>X</sup>(*s*) = <sup>1</sup> *s L dx*(*t*) *dt* + *<sup>x</sup>*(0) *s* i.e., in the time domain, to *<sup>x</sup>*(*t*) = #*<sup>t</sup>* 0 *dx*(*τ*) *<sup>d</sup><sup>τ</sup> dτ* + *x*(0), which means that *x*(0) is not the initial condition of the derivative but is, in fact, the initial condition of the integrator which has memorizing capability.
