*2.2. Fractional-Order Approximation of Laplacian Operator*

In this section, the approximation of the fractional-order Laplacian operator is described.

The integro-differential operator *s* ±*α* can be approximated as follows [22,23],

$$s^{\mathfrak{a}} \approx T\left(\frac{s}{\omega\_{\mathbb{C}}}\right) = \frac{a\_0 \left(\frac{s}{\omega\_{\mathbb{C}}}\right)^2 + a\_1 \left(\frac{s}{\omega\_{\mathbb{C}}}\right) + a\_2}{a\_2 \left(\frac{s}{\omega\_{\mathbb{C}}}\right)^2 + a\_1 \left(\frac{s}{\omega\_{\mathbb{C}}}\right) + a\_0}, \quad 0 < \mathfrak{a} < 1,\tag{9}$$

which is a biquadratic module that exhibits a flat phase response, where *ω<sup>c</sup>* stands for the center frequency and *a*0, *a*1, *a*<sup>2</sup> are alpha-dependent real constants defined as follows,

$$\begin{array}{rcl} a\_0 &=& \alpha^{\alpha} + 3\alpha + 2, \\ a\_2 &=& \alpha^{\alpha} - 3\alpha + 2, \\ a\_1 &=& 6\alpha \tan{\frac{(2-\alpha)\pi}{4}}. \end{array} \tag{10}$$

By assuming *ω* = *ω<sup>c</sup>* and considering constants (10), the integro-differential operator (9) will be described as

$$s^a \approx T\left(\frac{j\omega\_c}{\omega\_c}, a\right) = \frac{(a\_2 - a\_0) + ja\_1}{-(a\_2 - a\_0) + ja\_1} = \frac{-6a + j6a \tan\frac{(2-a)\pi}{4}}{6a + j6a \tan\frac{(2-a)\pi}{4}}\tag{11}$$

whose phase contribution will be given by arg{*T*(*j*1, *<sup>α</sup>*)} <sup>=</sup> <sup>−</sup><sup>2</sup> tan−<sup>1</sup> tan (2−*α*)*<sup>π</sup>* 4 . Thus, the phase contribution of approximation (9) will be given by [22,23],

$$\arg\{T(s/\omega\_{\mathcal{C}})|\_{j\omega\_{\mathcal{C}}}\} = \pm a\pi\ell/2,\tag{12}$$

which means that depending on the value of *α*, the biquadratic approximation of *s* ±*α* contributes with 0 < arg{*T*(*s*/*ωc*)|*jω<sup>c</sup>* } < ±90◦ .

Equation (9) will behave as a fractional-order differentiator around *ω<sup>c</sup>* as long as *a*<sup>0</sup> > *a*<sup>2</sup> > 0, i.e., arg{*T*(*s*/*ωc*)} > 0. Conversely, the effect of fractional-order integrator can be produced by ensuring that 0 < *a*<sup>0</sup> < *a*2, which produces arg{*T*(*s*/*ωc*)|*jω<sup>c</sup>* } < 0. The latter is consistent with the location of zeros/poles of (9) in the complex plane, where zeros lead poles for *a*<sup>0</sup> > *a*<sup>2</sup> and poles lead zeros for *a*<sup>0</sup> < *a*2, which confirms derivative/integral effects.

In Figure 2 the frequency response of *s* <sup>±</sup>0.5 is shown, where *a*<sup>0</sup> = 4.2071, *a*<sup>1</sup> = 7.2427 and *a*<sup>2</sup> = 1.2071 to ensure derivative (Figure 2a) and integral (Figure 2b) effects.

**Figure 2.** Frequency response of *s* <sup>±</sup>0.5, where arg{*T*(*j*1)} <sup>=</sup> <sup>±</sup>*απ*/2 <sup>=</sup> <sup>±</sup>45◦ for both (**a**) derivative and (**b**) integral effect of (9).

The synthesis of controller structure, using the fractional-order approximation of Laplacian operator, is described in the following section.
