# Offset-Free Model Predictive Control for Active Magnetic Bearing Systems

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## Abstract

**:**

## 1. Introduction

## 2. Single Degree of Freedom Active Magnetic Bearing System

## 3. Modeling

## 4. Offset-Free Model Predictive Control Design

#### 4.1. Control System Architecture

#### 4.2. Target Calculation and MPC Problem Formulation

## 5. Experimental Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Single-degree-of-freedom AMB system. (

**a**) Picture of the test rig. (

**b**) System layout: (1) Pivot; (2) Displacement sensor; (3) Moving mass; (4) Electromagnet.

**Figure 3.**Step excitation through the actuators with a current ${i}_{d}=0.25\mathrm{A}$. (

**a**) Nominal force applied to the system (dotted line); force from model simulation (dashed line); force estimated experimentally by the Kalman filter (solid line). (

**b**) MPC command (dashed line: simulation result; solid line: experimental result). (

**c**) Displacement of the mass (solid line: eddy current sensor direct measurement; dashed line: Kalman filter estimation).

**Figure 4.**Step excitation at different force amplitudes (${i}_{d}=0.3\mathrm{A},0.35\mathrm{A},0.4\mathrm{A}$). (

**a**) Nominal vs. estimated applied force. (

**b**) Displacement of the mass.

**Figure 5.**Effects of the plant-model mismatch on OF-MPC (

**a**) and MPC (

**b**) architectures. (

**c**) is the estimation of the force equivalent to the plant-model mismatch (obtained only with OF-MPC architecture) and (

**d**) is the variation of bias current provided to the plant.

**Figure 6.**Experimental setup for the load variation estimation test. Picture (

**a**) and layout (

**b**) of the setup. $L$: variable load.

**Figure 7.**Load variation estimation test. Dashed line: real applied force. Solid line: estimated force.

Symbol | Name | Value | Unit |
---|---|---|---|

$m$ | Mass | $3.41$ | $\mathrm{kg}$ |

$S$ | Cross-section area at the air gap | $420$ | ${\mathrm{mm}}^{2}$ |

${q}_{0}$ | Nominal airgap | $0.6$ | $\mathrm{mm}$ |

$n$ | Number of turns | $142$ | - |

$R$ | Coil resistance | $0.35$ | $\mathrm{Ohm}$ |

${L}_{0}$ | Coil nominal inductance | $5.8$ | $\mathrm{mH}$ |

${k}_{i}$ | Current-force factor | $13.65$ | $\mathrm{N}/\mathrm{A}$ |

${k}_{q}$ | Electromagnet negative stiffness | $11.4$ | $\mathrm{N}/\mathrm{mm}$ |

${k}_{m}$ | Back-electromotive-force factor | $13.65$ | $\mathrm{Vm}/\mathrm{s}$ |

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

$\mathrm{Variance}\mathrm{of}\left\{{w}_{x}\right\}$ | $\left[\begin{array}{cc}4\times {10}^{-10}& 0\\ 0& 4\times {10}^{-10}\end{array}\right]$ |

$\mathrm{Variance}\mathrm{of}\left\{{w}_{d}\right\}$ | $\left[1\right]$ |

$\mathrm{Variance}\mathrm{of}\left\{{w}_{n}\right\}$ | $\left[4\times {10}^{-10}\right]$ |

${L}_{x}$ | $\left[\begin{array}{c}1\\ 119\end{array}\right]$ |

${L}_{d}$ | $\left[28325\right]$ |

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

$N$ | 12 |

$Q$ | $\left[\begin{array}{cc}5\times {10}^{6}& 0\\ 0& 0.1\end{array}\right]$ |

$R$ | 1 |

$P$ | $\left[\begin{array}{cc}98255756& 804198\\ 804198& 10020\end{array}\right]$ |

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**MDPI and ACS Style**

Bonfitto, A.; Castellanos Molina, L.M.; Tonoli, A.; Amati, N.
Offset-Free Model Predictive Control for Active Magnetic Bearing Systems. *Actuators* **2018**, *7*, 46.
https://doi.org/10.3390/act7030046

**AMA Style**

Bonfitto A, Castellanos Molina LM, Tonoli A, Amati N.
Offset-Free Model Predictive Control for Active Magnetic Bearing Systems. *Actuators*. 2018; 7(3):46.
https://doi.org/10.3390/act7030046

**Chicago/Turabian Style**

Bonfitto, Angelo, Luis Miguel Castellanos Molina, Andrea Tonoli, and Nicola Amati.
2018. "Offset-Free Model Predictive Control for Active Magnetic Bearing Systems" *Actuators* 7, no. 3: 46.
https://doi.org/10.3390/act7030046