# Influence of Methods Approximating Fractional-Order Differentiation on the Output Signal Illustrated by Three Variants of Oustaloup Filter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Materials and Methods

## 4. Theory/Calculation

#### 4.1. Variants of Oustaloup Filter Approximations

`oustafod.m`function):

`new_fod.m`function):

`new_fod.m`function [51]. It uses the product over indices starting from 1, as in the original Oustaloup notations. Nevertheless, both functions (from FOMCOM and FOTF toolboxes) are equivalent to each other and realize Equation (6), not Equation (7); the last index has, however, another meaning.

#### 4.2. Steady-State Value

#### 4.3. Nyquist Plot and Time Response

#### 4.4. Approximation Errors

## 5. Results and Discussion

#### 5.1. Steady-State Value

#### 5.2. Nyquist Plot

#### 5.3. Unit Step Response

#### 5.4. Closed-Loop Control System

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Nyquist plot of FO inertia with $\alpha =0.9$ approximated with Xue’s variant of Oustaloup filter.

**Figure 4.**Nyquist plot of FO inertia with $\alpha =0.6$ approximated with the refined Oustaloup filter.

**Figure 5.**Nyquist plot of FO inertia with $\alpha =0.6$ approximated with Xue’s variant of the Oustaloup filter.

**Figure 8.**Part of Simulink diagram used during simulations. The upper part presents the FO-PID block. The lower part presents the FO controller build from Discrete Fractional Transducer Functions (DFTFs).

**Figure 9.**Simulink dialog boxes showing available parameters. (

**a**) FO-PID allows imputing parameters of PID controller calculated using standard Oustaloup approximation. (

**b**) Discrete Fractional Transfer Functions allow imputing FO poles and zeros, as well as the choice between the standard and refined variant of Oustaloup approximation, additionally, discretization parameters: method, critical frequency, and sampling time.

**Figure 10.**Unit step responses of the closed-loop control system with PI${}^{\lambda}$D${}^{\mu}$. The system consists of the PI${}^{\lambda}$D${}^{\mu}$ controller implemented as: FO-PID—the Simulink the FO-PID block, DFTF Std—the Simulink circuit with the DFTF blocks with unselected refined filter, Octave Std—the Octave procedure with the standard Oustaloup filter approximation, DFTF Ref—the Simulink circuit with the DFTF blocks with selected refined filter, Octave Ref—the Octave procedure with the refined Oustaloup filter approximation, Octave Xue—the Octave procedure with Xue’s variant of Oustaloup filter approximation.

Parameter | Value | Parameter | Value |
---|---|---|---|

Inertia | Oustaloup Approximation | ||

k | 2 | ${N}_{\mathrm{F}}$ | 0 or 5 |

T | 10 s${}^{\alpha}$ | ${\omega}_{\mathrm{b}}$ | 0.001 rad/s |

$\alpha $ | 0.9 or 0.6 | ${\omega}_{\mathrm{h}}$ | 1000 rad/s |

f | $(0.001;1)$ Hz | b | 10 |

$\omega $ | $2\pi f$ | d | 9 |

**Table 2.**Errors of fractional-order (FO) approximation models calculated for FO inertia with $\alpha =0.9$ and ${N}_{\mathrm{F}}=5$.

Frequency | $\mathit{\delta}\left|\mathit{G}\right|$ | $\mathbf{\Delta}\mathit{\phi}$ | $\mathit{\delta}\left|\mathit{G}\right|$ | $\mathbf{\Delta}\mathit{\phi}$ | $\mathit{\delta}\left|\mathit{G}\right|$ | $\mathbf{\Delta}\mathit{\phi}$ | |||
---|---|---|---|---|---|---|---|---|---|

Hz | % | ${}^{\circ}$ | % | ${}^{\circ}$ | % | ${}^{\circ}$ | |||

Model | Oustaloup | Ref. Oust. | Xue’s Oust. | ||||||

0.00100 | −1.38 | 0.203 | 0.118 | 0.516 | 0.0180 | −0.00590 | |||

0.00158 | −1.28 | 0.229 | 0.300 | 0.765 | 0.0104 | −0.00532 | |||

0.00251 | −1.16 | 0.268 | 0.643 | 1.11 | 0.0149 | −0.000791 | |||

0.00398 | −1.01 | 0.300 | 1.26 | 1.53 | 0.0147 | −0.00860 | |||

0.00631 | −0.831 | 0.335 | 2.28 | 1.98 | −0.00200 | −0.00126 | |||

0.0100 | −0.584 | 0.343 | 3.82 | 2.30 | 0.0219 | −0.00143 | |||

0.0158 | −0.384 | 0.298 | 5.58 | 2.30 | −0.00670 | −0.0129 | |||

0.0251 | −0.214 | 0.259 | 7.19 | 2.04 | −0.000391 | 0.00989 | |||

0.0398 | −0.0883 | 0.178 | 8.37 | 1.58 | 0.0195 | −0.0112 | |||

0.0631 | −0.0749 | 0.135 | 9.04 | 1.15 | −0.0247 | −0.00247 | |||

0.100 | −0.0134 | 0.118 | 9.47 | 0.815 | 0.0171 | 0.0101 | |||

0.158 | −0.0141 | 0.0860 | 9.67 | 0.537 | 0.00328 | −0.0161 | |||

0.251 | −0.0313 | 0.119 | 9.76 | 0.382 | −0.0216 | 0.00749 | |||

0.398 | 0.0126 | 0.149 | 9.88 | 0.258 | 0.0260 | 0.00192 | |||

0.631 | −0.0229 | 0.202 | 9.87 | 0.169 | −0.0122 | −0.0161 | |||

1.00 | −0.0149 | 0.340 | 9.90 | 0.149 | −0.0102 | 0.0105 |

**Table 3.**Magnitude errors dependent on filter order for refined Oustaloup filter with $\alpha =0.9$. The errors are relative and expressed in %.

Frequency | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

0.00100 | 16.8 | 9.8 | 1.2 | 1.3 | 1.7 | 1.6 | 1.5 | 1.5 |

0.0100 | 103.2 | −8.5 | 5.8 | 4.7 | 4.2 | 4.4 | 4.4 | 4.4 |

0.100 | 36.2 | 21.2 | 10.8 | 7.2 | 6.2 | 6.0 | 6.0 | 6.0 |

1.00 | −47.8 | −7.7 | 11.3 | 7.0 | 6.1 | 6.4 | 6.4 | 6.4 |

**Table 4.**Errors of FO approximation models calculated for FO inertia with $\alpha =0.6$ and ${N}_{\mathrm{F}}=5$.

Frequency | $\mathit{\delta}\left|\mathit{G}\right|$ | $\mathbf{\Delta}\mathit{\phi}$ | $\mathit{\delta}\left|\mathit{G}\right|$ | $\mathbf{\Delta}\mathit{\phi}$ | $\mathit{\delta}\left|\mathit{G}\right|$ | $\mathbf{\Delta}\mathit{\phi}$ | |||
---|---|---|---|---|---|---|---|---|---|

Hz | % | ${}^{\circ}$ | % | ${}^{\circ}$ | % | ${}^{\circ}$ | |||

Model | Oustaloup | Ref. Oust. | Xue’s Oust. | ||||||

0.00100 | −2.01 | 1.42 | 1.59 | 0.811 | 0.0530 | −0.0954 | |||

0.00158 | −1.44 | 1.16 | 2.08 | 0.907 | −0.00114 | −0.0506 | |||

0.00251 | −0.914 | 0.925 | 2.71 | 0.973 | 0.0616 | −0.0275 | |||

0.00398 | −0.595 | 0.666 | 3.29 | 0.937 | 0.00696 | −0.0559 | |||

0.00631 | −0.397 | 0.527 | 3.83 | 0.939 | −0.0182 | 0.0057 | |||

0.0100 | −0.157 | 0.364 | 4.45 | 0.846 | 0.0661 | −0.0208 | |||

0.0158 | −0.157 | 0.230 | 4.81 | 0.721 | −0.0484 | −0.0318 | |||

0.0251 | −0.0649 | 0.209 | 5.23 | 0.670 | 0.0106 | 0.0323 | |||

0.0398 | 0.0117 | 0.0950 | 5.59 | 0.503 | 0.0480 | −0.0360 | |||

0.0631 | −0.0854 | 0.0843 | 5.71 | 0.427 | −0.0737 | −0.00225 | |||

0.100 | 0.0330 | 0.0966 | 6.01 | 0.372 | 0.0527 | 0.0293 | |||

0.158 | −0.00155 | 0.0225 | 6.11 | 0.232 | 0.00784 | −0.0468 | |||

0.251 | −0.0629 | 0.0941 | 6.15 | 0.238 | −0.0642 | 0.0236 | |||

0.398 | 0.0629 | 0.0993 | 6.36 | 0.175 | 0.0779 | 0.00726 | |||

0.631 | −0.0466 | 0.101 | 6.30 | 0.099 | −0.0364 | −0.0465 | |||

1.00 | −0.0269 | 0.248 | 6.37 | 0.149 | −0.0314 | 0.0348 |

**Table 5.**Tuning parameters of the PI${}^{\lambda}$D${}^{\mu}$ controllers used in simulations—taken from [24].

k | ${\mathit{T}}_{\mathbf{i}}$ | ${\mathit{T}}_{\mathbf{d}}$ | $\mathit{\lambda}$ | $\mathit{\mu}$ |
---|---|---|---|---|

$0.133$ | $2.5$ | $0.76$ | $0.89$ | $0.44$ |

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**MDPI and ACS Style**

Wiora, J.; Wiora, A.
Influence of Methods Approximating Fractional-Order Differentiation on the Output Signal Illustrated by Three Variants of Oustaloup Filter. *Symmetry* **2020**, *12*, 1898.
https://doi.org/10.3390/sym12111898

**AMA Style**

Wiora J, Wiora A.
Influence of Methods Approximating Fractional-Order Differentiation on the Output Signal Illustrated by Three Variants of Oustaloup Filter. *Symmetry*. 2020; 12(11):1898.
https://doi.org/10.3390/sym12111898

**Chicago/Turabian Style**

Wiora, Józef, and Alicja Wiora.
2020. "Influence of Methods Approximating Fractional-Order Differentiation on the Output Signal Illustrated by Three Variants of Oustaloup Filter" *Symmetry* 12, no. 11: 1898.
https://doi.org/10.3390/sym12111898