# Design and High-Order Precision Numerical Implementation of Fractional-Order PI Controller for PMSM Speed System Based on FPGA

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## Abstract

**:**

## 1. Introduction

## 2. The Plant Model and Controller Design

#### 2.1. The Plant Model of PMSM Servo System

#### 2.2. The Controller Design

#### 2.2.1. Fractional Order PI Controller Design

#### 2.2.2. Integer-Order PI Controller Design

## 3. Numerical Implementation of Fractional Order Operators

#### 3.1. Impulse Response Invariance Method

#### 3.2. Oustaloup Method

#### 3.3. GL Method

## 4. Simulation Analysis

#### 4.1. Comparison of Three Discretization Methods

#### 4.2. The Influence of Discretization Order of Impulse Response Invariant Method

## 5. FPGA Design Experimental Verification

#### 5.1. FOC Algorithm Implementation Based on FPGA

#### 5.2. FPGA Implementation of Fractional-Order Operator

#### 5.3. Experimental Verification

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**7th−order discretization approximate Bode graph of ${s}^{0.5}$ in the frequency band [0.01, 1000] Hz.

**Figure 5.**24th−order discretization approximate Bode graph of ${s}^{0.5}$ in the frequency band [0.01, 1000] Hz.

**Figure 6.**24th−order discretization approximate Bode graph of ${s}^{0.5}$ in the frequency band [0.01, 10,000] Hz.

**Figure 7.**24th−order discretization approximate Bode graph of ${s}^{0.1}$ in the frequency band [0.01, 1000] Hz.

**Figure 8.**24th−order discretization approximate Bode graph of ${s}^{0.9}$ in the frequency band [0.01, 1000] Hz.

**Figure 22.**The speed responses of the PMSM system with different IOPI controller gains (Experiment).

**Figure 23.**The speed responses of the PMSM system with different FOPI controller gains (Experiment).

Operator | Considered Frequency | The Discretization | |
---|---|---|---|

Band (Hz) | Order N | ||

Case 1 | ${s}^{0.5}$ | [0.01,1000] | 7 |

Case 2 | ${s}^{0.5}$ | [0.01,1000] | 24 |

Case 3 | ${s}^{0.5}$ | [0.01,10,000] | 24 |

Case 4 | ${s}^{0.1}$ | [0.01,1000] | 24 |

Case 5 | ${s}^{0.9}$ | [0.01,1000] | 24 |

Motor Parameters | Value | Unit |
---|---|---|

Rated power | 2.0 | kW |

Rated speed | 2000 | r/min |

Rated voltage | 220 | V |

Rated current | 9.1 | A |

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**MDPI and ACS Style**

Wang, B.; Wang, S.; Peng, Y.; Pi, Y.; Luo, Y.
Design and High-Order Precision Numerical Implementation of Fractional-Order PI Controller for PMSM Speed System Based on FPGA. *Fractal Fract.* **2022**, *6*, 218.
https://doi.org/10.3390/fractalfract6040218

**AMA Style**

Wang B, Wang S, Peng Y, Pi Y, Luo Y.
Design and High-Order Precision Numerical Implementation of Fractional-Order PI Controller for PMSM Speed System Based on FPGA. *Fractal and Fractional*. 2022; 6(4):218.
https://doi.org/10.3390/fractalfract6040218

**Chicago/Turabian Style**

Wang, Baokun, Shaohua Wang, Yibing Peng, Youguo Pi, and Ying Luo.
2022. "Design and High-Order Precision Numerical Implementation of Fractional-Order PI Controller for PMSM Speed System Based on FPGA" *Fractal and Fractional* 6, no. 4: 218.
https://doi.org/10.3390/fractalfract6040218