## 2 The Existing Algorithms Based on Pole and Zero Recursive (Geometric) Distribution  
This section reviews the algorithms based on the recursive (geometric) pole and zero distribution found in the literature for the synthesis of a fractional order integrator or differentiator respectively described by the transfer functions:  
\[I^{\
u}(s)=\frac{1}{s^{\
u}}\quad\text{and}\quad D^{\
u}(s)=s^{\
u}\quad\text{ with}\quad 0<\
u<1. \tag{1}\]  
Approximation of a Fractional Integrator and Differentiator by a Recursive (Geometric) Distribution of Pole and Zeros: A Graphical Approach  
The demonstration of the approximation of a fractional order integrator operator frequency response is here done graphically. This method appeared for the very first time in the literature in the 1960s in two studies, apparently carried out independently by the respective authors [23; 24]. Some years later, this demonstration was used for the analog implementation of fractional integrators [25]. Subsequent work by other authors also contributed to this synthesis method [26; 27; 28]. For real orders, to obtain an approximation of a fractional integration operator, a solution consists in limiting the frequency band on which the fractional behavior is required. This first leads to the approximation:  
\[I^{\
u}(s)\approx I^{\
u}_{a}(s)=C_{0}\left(\frac{1+\frac{s}{\omega_{h}}}{1+ \frac{s}{\omega_{h}}}\right)^{\
u}\qquad\text{with}\qquad C_{0}=\left(\frac{ \omega_{h}}{\omega_{b}}\right)^{\
u}\left(\frac{{\omega_{b}}^{2}+1}{{\omega_{ h}}^{2}+1}\right)^{\frac{\
u}{2}}. \tag{2}\]  
As shown by Figure 1, the Bode plots of relation (2) exhibit:  
* a gain whose slope is a fractional multiple of \(-20\) dB/decade,
* a constant phase whose value is a fractional multiple of \(-90^{\circ}\). These Bode plots can be approximated using a distribution of poles and zeros. This leads to the following algorithm (Algorithm 1). As shown in [26], with this algorithm, the corner frequencies \(\omega_{k}\) and \({\omega^{\prime}}_{k}\) (respectively the poles and zeros of the transfer function \(I^{v}_{N}(s)\)) are geometrically distributed to obtain the required frequency behavior. Figure 1 highlights this distribution and also compares the asymptotic Bode plots of \(I^{v}(s)\), \(I^{v}_{a}(s)\) and \(I^{v}_{N}(s)\). It also shows that the high and low asymptotic frequency behaviors are constant. In [22], to obtain an integer integral like asymptotic behavior at low and high frequency, relation (2) is replaced by:  
\[I^{v}(s)\approx I^{v}_{a}(s)=C_{0}\frac{1}{s}\left(\frac{1+\frac{s}{\omega_{h }}}{1+\frac{s}{\omega_{b}}}\right)^{v-1}\qquad\text{with}\qquad C_{0}=\left( \frac{\omega_{h}}{\omega_{b}}\right)^{v-1}\left(\frac{{\omega_{b}}^{2}+1}{ \omega_{h}^{2}+1}\right)^{\frac{v-1}{2}}. \tag{7}\]  
This leads to the following algorithm (Algorithm 2). ```
1. Initialize \[r=\alpha\eta=\left(\frac{\omega_{h}}{\omega_{b}}\right)^{\frac{1}{N}}\quad \alpha=r^{1-v}\quad\eta=r^{v}\quad{\omega^{\prime}}_{1}=\eta^{1/2}\omega_{b} \quad\omega_{1}=\alpha{\omega^{\prime}}_{1}\] (8)
2. For\(i\in[1\dots N]\)do \[{\omega^{\prime}}_{i+1}=r{\omega^{\prime}}_{i}\qquad\omega_{i+1}=r\omega_{i}\] (9) End for
3. Compute \(C^{\prime}_{0}\) with relation (5)
4. Define the fractional integrator (1) approximation in the frequency band \([\omega_{b},\omega_{h}]\), by the transfer function

\[I^{v}_{a}(s)=C_{0}\frac{1}{s}\left(\frac{1+\frac{s}{\omega_{h}}}{1+\frac{s}{ \omega_{h}}}\right)^{v-1}\approx I^{v}_{N}(s)=C^{\prime}_{0}\frac{1}{s}\frac{ \prod\limits_{k=1}^{N}\left(1+\frac{s}{\omega^{\prime}_{k}}\right)}{\prod \limits_{k=1}^{N}\left(1+\frac{s}{\omega_{h}}\right)}. \tag{10}\]

**Algorithm 2** Fractional integrator approximation--second method

In [22], with this algorithm, the corner frequencies \(\omega_{k}\) and \({\omega^{\prime}}_{k}\) (respectively the poles and zeros of the transfer function \(I^{v}_{N}(s)\)) are geometrically distributed to obtain the required frequency behavior.