# Fourier Series Learning Control for Torque Ripple Minimization in Permanent Magnet Synchronous Motors

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## Abstract

**:**

## 1. Introduction

## 2. Model Description and Problem Formulation

## 3. Learning Controller Design Using Fourier Series

## 4. Error Convergence Conditions of the Closed-Loop System

**Proposition**

**1.**

**Proof.**

## 5. Experimental Implementation

## 6. Results and Discussion

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FSLC | Fourier Series Learning Controller |

PMSM | Permanent Magnet Synchronous Motor |

FOC | Field Oriented Control |

DTC | Direct Torque Control |

ILC | Iterative Learning Control |

SRM | Switched Reluctance Machine |

NN | Neural Network |

FSNN | Fourier Series Neural Network |

AFNN | Adaptive Fourier Neural Network |

FNN | Fourier Neural Network |

AUV | Autonomous Underwater Vehicles |

FA | Function Approximation |

DFT | Discrete Fourier Transform |

IDFT | Inverse Discrete Fourier Transform |

ANN | Artificial Neural Network |

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**Figure 1.**Block diagram of the FOC (Field Oriented Control) for the velocity control of a PMSM (Permanent Magnet Synchronous Motor) with three different master controllers, the FSLC (Fourier Series Learning Controller), a traditional PI (Proportional Integral) and an ANN (Artificial Neural Network).

**Figure 3.**Angular positions when the velocity is controlled to $0.3142$ rad/s. (

**a**) FSLC as the velocity controller; (

**b**) PI as the velocity controller; and (

**c**) ANN as the velocity controller.

**Figure 4.**Estimation of the angular velocity when it is controlled to $0.3142$ rad/s. (

**a**) FSLC as the velocity controller; (

**b**) PI as the velocity controller; and (

**c**) ANN as the velocity controller.

**Figure 5.**${I}_{q}$ currents of the closed-loop systems when the velocity is controlled to $0.3142$ rad/s. (

**a**) FSLC as the velocity controller; (

**b**) PI as the velocity controller and (

**c**) ANN as the velocity controller.

**Figure 6.**Errors of the control systems when the velocity is controlled to $0.3142$ rad/s. (

**a**) FSLC as the velocity controller; (

**b**) PI as the velocity controller; and (

**c**) ANN as the velocity controller.

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

Proportional gain of the slave PI controllers for currents ${I}_{q}$ and ${I}_{d}$ | ${K}_{ps}$ | $1.0$ |

Integral gain of the slave PI controllers for currents ${I}_{q}$ and ${I}_{d}$ | ${K}_{is}$ | $10.0$ |

Proportional gain of the FSLC (Fourier Series Learning Controller) | ${\mathsf{\alpha}}_{n}$ | $0.037$ |

Learning gain of the FSLC | ${\mathsf{\gamma}}_{n}$ | $0.03$ |

Number of terms of the FSLC | N | 4 |

Proportional gain of the master PI (Proportional Integral) controller | ${K}_{pm}$ | $0.0477$ |

Integral gain of the master PI controller | ${K}_{im}$ | $2.38$ |

Learning coefficient of the ANN (Artificial Neural Network) | $\mathsf{\eta}$ | $4.9$ |

Number of neurons in the intermediate layer of the ANN | ${N}_{n}$ | 3 |

Gain of the position filter | A | $5.0$ |

Sample period (s) | ${T}_{s}$ | $0.005$ |

Velocity reference (rad/s) | ${\mathsf{\omega}}_{Ref}$ | $0.3142$ |

© 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**

Espíndola-López, E.; Gómez-Espinosa, A.; Carrillo-Serrano, R.V.; Jáuregui-Correa, J.C.
Fourier Series Learning Control for Torque Ripple Minimization in Permanent Magnet Synchronous Motors. *Appl. Sci.* **2016**, *6*, 254.
https://doi.org/10.3390/app6090254

**AMA Style**

Espíndola-López E, Gómez-Espinosa A, Carrillo-Serrano RV, Jáuregui-Correa JC.
Fourier Series Learning Control for Torque Ripple Minimization in Permanent Magnet Synchronous Motors. *Applied Sciences*. 2016; 6(9):254.
https://doi.org/10.3390/app6090254

**Chicago/Turabian Style**

Espíndola-López, Eduardo, Alfonso Gómez-Espinosa, Roberto V. Carrillo-Serrano, and Juan C. Jáuregui-Correa.
2016. "Fourier Series Learning Control for Torque Ripple Minimization in Permanent Magnet Synchronous Motors" *Applied Sciences* 6, no. 9: 254.
https://doi.org/10.3390/app6090254