# Fractional-Order Identification and Control of Heating Processes with Non-Continuous Materials

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## Abstract

**:**

## 1. Introduction

## 2. Fractional Theoretical Model

## 3. Fractional Experimental Model Identification

#### 3.1. Experimental Setup

#### 3.2. Experimental Model

#### 3.3. Experimental Campaign

#### 3.4. Identification and Validation Method

## 4. Modelling Results

#### 4.1. Model Identification at sensor1

#### 4.2. Time Domain Model Validation at sensor1

#### 4.3. Model Validation at sensor2 and sensor3

#### 4.4. Fractional Order and Non-Continuous Materials

#### 4.5. Controllers Implementation and Comparison

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Experimental setup. (

**a**) Schematic, where $SM$ (V/K) is the Seebeck coefficient, $RM$ (Ohm) is the Electrical Resistance, and $KM$ the Thermal Conductance (W/K) of the Peltier cell. Such parameters where calculated using the Ferrotec method (http://www.ferrotec.com) through measurements of the module input current I, the hot side temperature $th$ and the cold side temperature $tc$. NI DAQ: National Instrument

^{©}Data Acquisition Board; (

**b**) Entire acquisition setup; (

**c**) Beam photo with details on the sensors position.

**Figure 2.**Nelder–Mead multi-objective optimization algorithm for heat transfer model identification. (

**a**) Identification of model parameters in the frequency domain; (

**b**) Validation of the identified model in the spatial domain.

**Figure 3.**Metal beam filled with air. Theoretical and experimental model comparison in the frequency domain. (

**a**) Module diagram; (

**b**) Phase diagram.

**Figure 4.**Metal beam filled with Styrofoam. Theoretical and experimental model comparison in the frequency domain. (

**a**) Module diagram; (

**b**) Phase diagram.

**Figure 5.**Metal beam filled with metal buckshots. Theoretical and experimental model comparison in the frequency domain. (

**a**) Module diagram; (

**b**) Phase diagram.

Material | T1 | T2 | T3 | α | ${\mathit{\alpha}}_{1}$ |
---|---|---|---|---|---|

Air | 13.8610 | 289.7543 | 1222344.6 | $N/D$ | $N/D$ |

Styrofoam | 13.7534 | 293.0218 | 1248787.9 | $N/D$ | $N/D$ |

Metal Buckshots | 8.1853 | 282.1817 | 481148.7 | $N/D$ | $N/D$ |

Material | T1 | T2 | T3 | α | ${\mathit{\alpha}}_{1}$ |
---|---|---|---|---|---|

Air | 2.0589 | 117.6201 | 9260.2 | 0.7408 | 0.7077 |

Styrofoam | 2.1589 | 109.8189 | 11252.4 | 0.7382 | 0.7649 |

Metal Buckshots | 2.5013 | 456.3768 | 13193.5 | 0.5771 | 0.6869 |

Material | Model | ${\mathit{J}}_{\mathit{err}}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{d}}_{{\mathit{\lambda}}_{1}}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{d}}_{{\mathit{\lambda}}_{2}}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{d}}_{{\mathit{\lambda}}_{3}}$ |
---|---|---|---|---|---|---|---|---|

Air | Theo. | 10.5569 | 1 | 3.2 | 1.1542 | 3.69 | 1.3496 | 4.32 |

Exp. | 3.5595 | 1 | 3.2 | 1.2031 | 3.85 | 1.50 | 4.8 | |

Styrofoam | Theo. | 9.073 | 1 | 3.2 | 1.1622 | 3.72 | 1.3607 | 4.35 |

Exp. | 3.087 | 1 | 3.2 | 1.2104 | 3.87 | 1.51 | 4.83 | |

Metal Buckshots | Theo. | 6.9831 | 1 | 3.2 | 1.2001 | 3.84 | 1.4096 | 4.51 |

Exp. | 2.3032 | 1 | 3.2 | 1.2929 | 4.14 | 1.6062 | 5.14 |

**Table 4.**Relationship between thermal conductivity of the non-continuous beam material and the experimental fractional order model α.

Material | Thermal Conductivity | α |
---|---|---|

Air | 0.025 | 0.7408 |

Styrofoam | 0.04 | 0.7382 |

Metal Buckshots | 35 | 0.5771 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Caponetto, R.; Sapuppo, F.; Tomasello, V.; Maione, G.; Lino, P.
Fractional-Order Identification and Control of Heating Processes with Non-Continuous Materials. *Entropy* **2016**, *18*, 398.
https://doi.org/10.3390/e18110398

**AMA Style**

Caponetto R, Sapuppo F, Tomasello V, Maione G, Lino P.
Fractional-Order Identification and Control of Heating Processes with Non-Continuous Materials. *Entropy*. 2016; 18(11):398.
https://doi.org/10.3390/e18110398

**Chicago/Turabian Style**

Caponetto, Riccardo, Francesca Sapuppo, Vincenzo Tomasello, Guido Maione, and Paolo Lino.
2016. "Fractional-Order Identification and Control of Heating Processes with Non-Continuous Materials" *Entropy* 18, no. 11: 398.
https://doi.org/10.3390/e18110398