# Load Torque Observer for BLDC Motors Based on a HOSM Differentiator

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## Abstract

**:**

## 1. Introduction

- The proposed method can estimate time-varying load torque, provided that this torque and its k-th derivatives are bounded;
- A position estimation algorithm (PEA) is included to use with Hall effect sensor signals when needed;
- The proposed scheme can guarantee a bounded error in the disturbance estimation, the size of which depends on the measurement noise and sampling time.

## 2. Preliminaries

#### 2.1. Brushless DC Motor Model

#### 2.2. Observability Definitions

**Definition 1**

**([20]).**

**Definition 2**

**([21]).**

#### 2.3. Mechanical System

**Assumption 1.**

**Assumption 2.**

**Fact 1.**

## 3. Main Result

#### 3.1. Luenberger Observer

#### 3.2. HOSM Differentiator

**Assumption 3**

**([18]).**

**Lemma 2**

**([17]).**

#### 3.3. Estimated States and Load Torque Reconstruction

## 4. Simulations

## 5. Experimental Results

#### 5.1. Experimental Test Bench

#### 5.2. Position Estimation Algorithm-PEA

#### 5.3. Disturbance Estimation

#### 5.4. Results Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Simulation results, test 1: estimated velocity $\widehat{\omega}$ ($--$) vs. velocity $\omega $ (−) and the velocity estimation error ${e}_{\omega}$ (−).

**Figure 3.**Simulation results, test 1: estimated load torque ${\widehat{\tau}}_{L}$ ($--$) vs. load toque ${\tau}_{L}$ (−) and the load torque estimation error ${e}_{\tau}$ (−).

**Figure 4.**Simulation results, test 2: estimated velocity $\widehat{\omega}$ ($--$) vs. velocity $\omega $ (−) and the velocity estimation error ${e}_{\omega}$ (−).

**Figure 5.**Simulation results, test 2: estimated load torque ${\widehat{\tau}}_{L}$ ($--$) vs. load toque ${\tau}_{L}$ (−) and the load torque estimation error ${e}_{\tau}$ (−).

**Figure 7.**Relationship between Hall sensor signals and wrapped position ${\theta}_{elec}$ (−). With Hall effect sensor signal A (−), Hall effect sensor signal B (−) and Hall effect sensor signal C (−).

**Figure 9.**Experimental results with flywheel: estimated electrical position with PEA ${\widehat{\theta}}_{e}$($--$) vs. encoder position ${\theta}_{enc}$ (−).

**Figure 10.**Experimental results, test 1: estimated velocity $\widehat{\omega}$ ($--$) vs. measured velocity $\omega $ (−) and the velocity estimation error ${e}_{\omega}$ (−).

**Figure 11.**Experimental results, test 1: estimated load torque ${\widehat{\tau}}_{L}$ ($--$) vs. measured load toque ${\tau}_{L}$ (−) and the load torque estimation error ${e}_{\tau}$ (−).

**Figure 12.**Experimental results, test 2: estimated velocity $\widehat{\omega}$ ($--$) vs. measured velocity $\omega $ (−) and the velocity estimation error ${e}_{\omega}$ (−).

**Figure 13.**Experimental results, test 2: estimated load torque ${\widehat{\tau}}_{L}$ ($--$) vs. measured load toque ${\tau}_{L}$ (−) and the load torque estimation error ${e}_{\tau}$ (−).

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

Rated Voltage | 240 [$\mathrm{V}$] |

Rated Torque | 2.1 [$\mathrm{N}\mathrm{m}$] |

Rated Power | 600 [$\mathrm{W}$] |

Resistance | 1.2 [$\Omega $] |

Inductance | 0.00205 [$\mathrm{m}\mathrm{H}$] |

Electric constant | 0.40355 [$\mathrm{V}/\mathrm{rad}/\mathrm{s}$] |

Mechanical constant | 0.65997 [$\mathrm{N}\mathrm{m}/\mathrm{A}$] |

Inertia | 0.00027948 [$\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}$] |

Viscous friction coefficient | 0.0006738 [$\mathrm{Nms}$] |

Test 1 | ${e}_{\omega}=0.046329$ |

${e}_{\tau}=0.0012986$ | |

Test 2 | ${e}_{\omega}=0.0411790$ |

${e}_{\tau}=0.0018641$ |

Test 1 | ${e}_{\omega}=0.3256$ |

${e}_{\tau}=0.0030$ | |

Test 2 | ${e}_{\omega}=0.3946$ |

${e}_{\tau}=0.0043$ |

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**MDPI and ACS Style**

Coronado-Andrade, A.; de la Guerra, A.; Alvarez-Icaza, L.
Load Torque Observer for BLDC Motors Based on a HOSM Differentiator. *Machines* **2023**, *11*, 1065.
https://doi.org/10.3390/machines11121065

**AMA Style**

Coronado-Andrade A, de la Guerra A, Alvarez-Icaza L.
Load Torque Observer for BLDC Motors Based on a HOSM Differentiator. *Machines*. 2023; 11(12):1065.
https://doi.org/10.3390/machines11121065

**Chicago/Turabian Style**

Coronado-Andrade, Axel, Alejandra de la Guerra, and Luis Alvarez-Icaza.
2023. "Load Torque Observer for BLDC Motors Based on a HOSM Differentiator" *Machines* 11, no. 12: 1065.
https://doi.org/10.3390/machines11121065