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Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Fractional Derivatives

## 3. Fractional-Order Models

**Remark**

**1.**

## 4. Control Systems and Bode Diagrams

- If the input is a sinusoidal signal, the output is also sinusoidal of the same frequency. The changes are in amplitude and a shift in time (phase).
- A superposition principle with respect to inputs and outputs according to which, if for inputs ${u}_{1}$ and ${u}_{2}$, we have outputs ${y}_{1}$ and ${y}_{2}$, respectively, then for an input ${u}_{3}=a{u}_{1}+b{u}_{2}$, the output is ${y}_{3}=a{y}_{1}+b{y}_{2}$.

## 5. Oustaloup’s Filter Approximation

## 6. Fractional-Order Modeling and Control Toolboxes

- FOTF (Fractional Order Transfer Function) is a control toolbox for fractional order systems developed by Xue et al. [31,36], which extends many MATLAB built-in functions. The FOTF approximate fractional differential operators by means of a discretization of the Grünwald–Letnikov definition of noninteger derivative, but other approximation methods are possible [36].
- Ninteger (Non-integer) is a toolbox for MATLAB intended to help develop non-integer order controllers for single-input, single-output plants and assess their performance. It was originally developed by Valério and Sá da Costa [37]. It uses integer-order approximations of the fractional-order transfer function, mainly based on Oustaloup’s filter; more generally, the whole toolbox has been inspired by the original CRONE one, from which Ninteger imported several methods.
- CRONE (Commande Robuste d’Ordre Non Entier is a robust command of non-integer order) Toolbox, developed in the nineties by the CRONE team [32]. Many approximation techniques proposed by the CRONE team are considered as foundational in the literature (see e.g., [28,33]). For instance, Oustaloup’s (leader of the CRONE team) method of approximation of transfer functions was one of the cornerstones of the original CRONE toolbox.

## 7. Numerical Results

#### 7.1. Example 1. Experimental Identification of a Furnace

- Time instants in seconds;
- Input u (voltages) in percentage of maximum input voltage;
- Output y (temperatures) in °C.

#### 7.1.1. Integer-Order Identification

#### 7.1.2. Fractional-Order Identification

#### 7.1.3. Validation

#### 7.2. Example 2. Identification of a Simulated Fractional Model

#### 7.2.1. Integer-Order Identification

#### 7.2.2. Oustaloup’s and Matsuda’s Filters

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Phenomena | Model |
---|---|

linear and local | linear ODEs |

linear and nonlocal | linear FDEs |

nonlinear and local | nonlinear ODEs |

nonlinear and nonlocal | nonlinear FDEs |

**Table 2.**Furnace approximation errors. Computed as $\parallel y-\tilde{y}\parallel $, where y are the experimental data, $\tilde{y}$ are the approximated data and $\parallel \xb7\parallel $ is the Euclidean norm. The relative percentage expresses the difference between the errors committed by the two models in a proportion of the highest.

Identification Error | Validation Error | |||||
---|---|---|---|---|---|---|

TF Order | Integer | Fractional | Relative % | Integer | Fractional | Relative % |

1st | $302.26$ | $302.5$ | $0.08$% | $244.26$ | $244.28$ | $0.006$% |

2nd | $302.12$ | $298.68$ | $1.14$% | $235.46$ | $243.51$ | $3.3$% |

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**MDPI and ACS Style**

Conejero, J.A.; Franceschi, J.; Picó-Marco, E. Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide? *Mathematics* **2022**, *10*, 2719.
https://doi.org/10.3390/math10152719

**AMA Style**

Conejero JA, Franceschi J, Picó-Marco E. Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide? *Mathematics*. 2022; 10(15):2719.
https://doi.org/10.3390/math10152719

**Chicago/Turabian Style**

Conejero, J. Alberto, Jonathan Franceschi, and Enric Picó-Marco. 2022. "Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?" *Mathematics* 10, no. 15: 2719.
https://doi.org/10.3390/math10152719