# Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac

^{*}

## Abstract

**:**

## 1. Introduction

^{®}(The MathWorks, Natick, MA, USA)/Simulink and similar frameworks, which allow using block diagrams for a controller synthesis.

## 2. Diffusive Realization of Non-Integer Order Integrator

**Theorem**

**1.**

**Proof.**

- Contour EHWe substitute$$s=x{\mathrm{e}}^{\pi i}$$Then,$$\begin{array}{cc}\hfill \mathrm{d}s=& -\mathrm{d}x\hfill \end{array}$$$$\begin{array}{cc}\hfill {s}^{\alpha}=& {x}^{\alpha}(cos\alpha \pi +isin\alpha \pi )\hfill \end{array}$$$$\begin{array}{cc}\hfill \underset{EH}{\int}\frac{{\mathrm{e}}^{st}}{{s}^{\alpha}}\mathrm{d}s=& \underset{-R}{\overset{-\epsilon}{\int}}\frac{{\mathrm{e}}^{st}}{{s}^{\alpha}}\mathrm{d}s=-\underset{R}{\overset{\epsilon}{\int}}\frac{{\mathrm{e}}^{-xt}}{{x}^{\alpha}}(cos\alpha \pi -isin\alpha \pi )\mathrm{d}x=\hfill \\ \hfill =& \underset{\epsilon}{\overset{R}{\int}}\frac{{\mathrm{e}}^{-xt}}{{x}^{\alpha}}(cos\alpha \pi -isin\alpha \pi )\mathrm{d}x\hfill \end{array}$$
- Contour HJKWe substitute$$s=\epsilon {\mathrm{e}}^{\theta i},\phantom{\rule{4pt}{0ex}}\theta \in [-\pi ,\pi ]$$Then,$$\begin{array}{cc}\hfill \mathrm{d}s=& i\epsilon {\mathrm{e}}^{i\theta}\mathrm{d}\theta \hfill \end{array}$$$$\begin{array}{cc}\hfill {s}^{\alpha}=& {\epsilon}^{\alpha}{\mathrm{e}}^{i\alpha \theta}\hfill \end{array}$$$$\underset{HJK}{\int}\frac{{\mathrm{e}}^{st}}{{s}^{\alpha}}\mathrm{d}s=\underset{-\pi}{\overset{\pi}{\int}}\frac{exp\left(\epsilon {\mathrm{e}}^{i\theta}t\right)}{{\epsilon}^{\alpha}{\mathrm{e}}^{i\alpha \theta}}i\epsilon {\mathrm{e}}^{i\theta}\mathrm{d}\theta =i{\epsilon}^{1-\alpha}\underset{-\pi}{\overset{\pi}{\int}}exp(i(1-\alpha )\theta +\epsilon {\mathrm{e}}^{i\theta}t)\mathrm{d}\theta $$
- Contour KLWe substitute$$s=x{\mathrm{e}}^{-\pi i}$$$$\mathrm{d}s=-\mathrm{d}x$$$${s}^{\alpha}={x}^{\alpha}(cos\alpha \pi -isin\alpha \pi )$$$$\underset{KL}{\int}\frac{{\mathrm{e}}^{st}}{{s}^{\alpha}}\mathrm{d}s=\underset{-\epsilon}{\overset{-R}{\int}}\frac{{\mathrm{e}}^{st}}{{s}^{\alpha}}\mathrm{d}s==-\underset{\epsilon}{\overset{R}{\int}}\frac{{\mathrm{e}}^{-xt}}{{x}^{\alpha}}(cos\alpha \pi +isin\alpha \pi )\mathrm{d}x$$

## 3. Approximation of Diffusive Realizations

- Create approximation directly from integral on an unbounded domain using Gauss–Laguerre quadrature.
- Create approximation from on an unbounded domain using variable change, which results in Fourier–Chebyshev quadrature.
- Create approximation on a bounded domain, but on a logarithmic scale using Clenshaw–Curtis and Gauss–Legendre quadratures.

## 4. Quadrature Formulas

#### 4.1. Gauss–Laguerre Quadrature

#### 4.2. Clenshaw Curtis and Fourier–Chebyshev Quadrature

#### 4.3. Gauss–Legendre Quadrature

## 5. SoftFRAC Library to Realization Fractional Order Dynamic Elements

^{®}, Python and C. Currently it offers functionality to create state-space fractional model and transfer function fractional model of the non-integer dynamic element $\frac{1}{{s}^{\alpha}}$ based on approximations described in the previous section.

`ssf`) inherits from the MATLAB

^{®}Control Systems Toolbox

^{TM}state-space model class (

`ss`) and has all functionality of this class. Similarly, transfer function fractional model class (

`ssf`) inherits form transfer function model. In both cases variables describing approximation extend standard object properties:

- ss - base MATLAB
^{®}object, - A, B, C, D - matrix containing approximation parameters
- alpha - derivative order,
- omega_min - approximation frequency lower band,
- omega_max - approximation frequency upper band,
- approximation_order - approximation order
- method - name of approximation method

- Inheritance is the concept allowing defining subclasses of data objects that share some or all of the main class characteristics. This property of OOP reduces development time and ensures more accurate coding.
- Class defines only the data it needs to be concerned with. This protects running instance of the class (an object) from accidentally accessing other program data.
- The concept of data classes allows the programmer to create any new data type that is not already defined in the language itself.

^{®}language enable developing complex technical computing applications. In the MATLAB

^{®}environment we can define classes and apply standard object-oriented design patterns, inheritance, encapsulation, and reference behavior without engaging in the low-level housekeeping tasks required by other languages.

`ss`and

`tf`. In particular, the user can use this realization to Simulink

^{®}simulation, system behavior analysis, and easy plotting of dynamic characteristics. We have added to the classes a construction method for easy conversion between those types, from

`ssf`to

`tff`, and vice versa. We present the UML diagram of the

`ssf`class implementation in Figure 2.

#### Implementation Analysis

^{®}.

## 6. Approximation Analysis

#### 6.1. Infinite Intervals

#### 6.2. Finite Intervals

## 7. Discussion of Low Frequency Behavior

**Theorem**

**1**

**.**Let a function f analytic in $[-1,1]$ be analytically continuable to the open Bernstein ellipse ${E}_{\rho}$, where it satisfies $\left|f\right(x\left)\right|\le M$ for some M. Chebyshev interpolants ${p}_{n}$ of f satisfy

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**

`ssf`class inherits from MATLAB

^{®}Control Systems Toolbox

^{TM}

`ss`. Its methods allow model conversion and typical functions.

**Figure 3.**Analysis of ${\parallel e\parallel}_{\infty}$ norm with increasing order of quadrature, for methods defined on unbounded domain. Gauss–Laguerre quadrature rules for order >180 could not be obtained because of ill conditioning of that problem. Convergence of both methods is very slow.

**Figure 6.**Comparison of time constants and gains of elements of approximating sum for Fourier–Chebyshev approximation.

**Figure 7.**Comparison of time constants and gains of elements of approximating sum for Gauss–Laguerre approximation.

**Figure 8.**Analysis of ${\parallel e\parallel}_{\infty}$ norm with increasing order of quadrature for the integration interval of $[-5,5]$. Isolated low error is an artifact of phase oscillation.

**Figure 9.**Analysis of ${\parallel e\parallel}_{\infty}$ norm with increasing order of quadrature for the integration interval of $[-7,7]$.

**Figure 10.**Analysis of frequency responses approximations with Clenshaw–Curtis quadrature and the integration interval of $[-5,5]$.

**Figure 11.**Analysis of frequency responses approximations with Clenshaw–Curtis quadrature and the integration interval of $[-7,7]$.

**Figure 12.**Analysis of frequency responses approximations with Gauss–Legendre quadrature and the integration interval of $[-5,5]$.

**Figure 13.**Analysis of frequency responses approximations with Gauss–Legendre quadrature and the integration interval of $[-7,7]$.

**Figure 14.**Comparison of time gains and time constants for bounded domain quadratures. For both methods distribution of gain-time constant pairs is similar with concentration of time constants ant the integration interval boudary (

**a**) Clenshaw–Curtis quadrature on the [−5, 5] interval; (

**b**) Gauss–Legendre quadrature on the [−5, 5] interval; (

**c**) Clenshaw–Curtis quadrature on the [−7, 7] interval; (

**d**) Gauss–Legendre quadrature on the [−7, 7] interval.

**Figure 15.**Bernstein ellipses for different points at the frequency response. According to Bernstein theorem convergence speed of analytic functions is controlled by the ρ coefficient which is the sum of lengths of both semiaxes. Approximated function is analytic for all frequencies in the integration interval, but for low frequencies analyticity region is vanishingly small. (

**a**) Bernstein ellipse for integrated function in (22) for the frequency ω = 10

^{5}rad/s. ρ coefficient is approximately 3 ensuring fast convergence. (

**b**) Bernstein ellipse for integrated function in (22) for the frequency ω = 1 rad/s. ρ coefficient is approximately 2, which is still fast converging. (

**c**) Bernstein ellipse for integrated function in (22) for the frequency ω = 10

^{−5}rad/s. ρ coefficient is approximately 1, which causes convergence to be very slow.

Approximation Order | 10 | 100 | 1000 | 10,000 | |
---|---|---|---|---|---|

Gauss–Laguerre quadrature | time [s] | 0.40 | 0.44 | 11.04 | 869.73 |

memory [Kb] | 3,162,112 | 3,158,016 | 8,019,968 | 801,566,720 | |

Clenshaw–Curtis quadrature | time [s] | 0.03 | 0.007 | 0.028 | 1.90 |

memory [Kb] | 3,162,112 | 3,158,016 | 8,019,968 | 801,566,720 |

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**MDPI and ACS Style**

Baranowski, J.; Bauer, W.; Mularczyk, R. Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac. *Electronics* **2021**, *10*, 1767.
https://doi.org/10.3390/electronics10151767

**AMA Style**

Baranowski J, Bauer W, Mularczyk R. Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac. *Electronics*. 2021; 10(15):1767.
https://doi.org/10.3390/electronics10151767

**Chicago/Turabian Style**

Baranowski, Jerzy, Waldemar Bauer, and Rafał Mularczyk. 2021. "Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac" *Electronics* 10, no. 15: 1767.
https://doi.org/10.3390/electronics10151767