## 3 Sub-Optimality of Algorithms 1-4 and Beyond Geometric Distribution
Moreover, if it is imposed that distribution \(\mathrm{D}_{3}\) fits the geometric distribution in the middle of the interval \([\omega_{b},\ldots,\omega_{h}]\), the following constraint must be added: \(\mathrm{P}(N/2)=1/2\) and thus \(\frac{a_{1}}{2}+\frac{a_{2}N}{4}+\frac{a_{3}N^{2}}{8}=1/2\). Finally, parameters \(a_{2}\) and \(a_{3}\), meet the following system of equations:  
\[\left[ \begin{array}{cc}N&N^{2}\\ \frac{N}{4}&\frac{N^{2}}{8}\end{array} \right]\left[ \begin{array}{c}a_{2}\\ a_{3}\end{array} \right]=\left[ \begin{array}{c}1-a_{1}\\ \frac{1-a_{1}}{2}\end{array} \right]. \tag{35}\]  
By varying parameter \(a_{1}\), Figure 7 shows the large number of distributions that can be obtained in both cases. In the two cases, it must be pointed out that only one parameter needs to be tuned to obtain a wide variety of distributions. For the geometric distribution, according to Algorithm 1, the poles, and zeros in relation (3) are defined by relation (34) with  
Figure 6: Two examples of pole distributions compared to the geometric one. Figure 7: Two examples of pole distributions with \(\mathrm{D}_{3}\): \(M=2\) (**left**), \(M=3\) (**right**). \[\varphi=k-1+\frac{1-\
u}{2},\qquad\text{for pole}\qquad\omega_{k} \tag{36}\]  
and  
\[\varphi=k-1+\frac{1+\
u}{2}\qquad\text{for zero}\qquad\omega^{\prime}{}_{k}. \tag{37}\]  
To define the poles and the zeros using the distribution introduced, the same values of \(\varphi\) are used, but the values of \(\omega_{k}\) and \(\omega^{\prime}{}_{k}\) are given by relation (34) leading to the following algorithm (Algorithm 5). 1. Initialize \[P(\varphi)=\sum_{m=0}^{M}a_{m}\varphi^{m}\quad r=\left(\frac{\omega_{h}}{ \omega_{b}}\right)^{\frac{1}{N}}\] (38)
2. **For**\(i\in[1\dots N]\)**do** \[\log(\omega_{k})=\log(\omega_{b})+P(\varphi)\log(r)\quad\text{with}\quad \varphi=k-1+\frac{1-\
u}{2}\] (39) \[\log(\omega^{\prime}{}_{k})=\log(\omega_{b})+P(\varphi)\log(r)\quad\text{ with}\quad\varphi=k-1+\frac{1+\
u}{2}\] (40)  
**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  
\[I_{a}^{\
u}(s)=C_{0}\bigg{(}\frac{1+\frac{s}{\omega_{h}}}{1+\frac{s}{\omega_{ h}}}\bigg{)}^{\
u}\approx I_{N}^{\
u}(s)=C^{\prime}_{0}\sum_{ \begin{subarray}{c}k=1 \\ k=1\end{subarray} }^{N}\Big{(}1+\frac{s}{\omega_{h}}\Big{)}. \tag{41}\]  
The efficiency of this algorithm is highlighted on the example used at the beginning of Section 3, but with a distribution \(D_{3}\) and \(M=3\). The parameter \(a_{1}\) is computed to reduce the relative error given in relation (27). Given that only one parameter has to be found, it is done by a gridding of parameter \(a_{1}\) on the interval [\(-0.1\), \(3\)]. A comparison of relative and absolute errors obtained with Algorithms 1 and 5 is given in Figure 8. The maximum values of these absolute and relative errors are given by:  
\[\max_{\omega}\lvert E_{rel}(\omega)\rvert=1.0133\times 10^{-2}\qquad\quad \max_{\omega}\lvert E_{abs}(\omega)\rvert=3.1903\times 10^{-3}. \tag{42}\]  
Figure 8: Comparison of relative (**left**) and absolute (**right**) errors between Algorithms 1 and 5 with distribution \(\text{D}_{3}\). In comparison with Algorithm 1, these values are greatly reduced with Algorithm 5. **4. Fractional Model Approximation and Stability Issues**  
We now consider a fractional order model described by the pseudo state space description,  
\[\left\{ \begin{array}{c}\frac{d^{x}}{dt^{x}}x(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)+Du(t)\end{array} \right., \tag{43}\]  
where \(x(t)\in\mathsf{R}^{n}\) is the pseudo state vector, \(v\) is the fractional order of the system and \(A\in\mathsf{R}^{nxn}\), \(B\in\mathsf{R}^{nxm}\), \(C\in\mathsf{R}^{pxn}\) and \(D\in\mathsf{R}^{pxm}\) are constant matrices. A solution for the implementation of such a model consists in the approximation of the fractional order derivative by the approximation given by Algorithm 3 or Algorithm 4. However, such an approximation has an impact on model stability.