# Electromagnetic Levitation Control for Bending Flexible Steel Plate: Experimental Consideration on Disturbance Cancellation Control

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## Abstract

**:**

## 1. Introduction

## 2. System for Control Experiment

_{1}–z

_{5}) was measured by five eddy-current gap sensors. Among the five pairs of electromagnets, four pairs at the corners were inclined and a central electromagnet was moved in the vertical direction. In addition, the distance between the surface of the electromagnets and the steel plate was controlled at 5 mm, even when θ was changed. Thus, by moving the five electromagnets, the bending magnetic levitation of the steel plate was possible. Here, θ is defined as an electromagnet angle. The coil current (i

_{1}–i

_{5}) is detected from the external resistance for measurement.

_{1}–w

_{3}). In this system, the sampling frequency was 1000 Hz.

## 3. Equation of Motion

_{z}: mass obtained by dividing the steel plate virtually into five [kg], z: displacement from the equilibrium levitation position of the steel plate [m], and f

_{z}: variation value of the attractive force per electromagnet [N].

_{eff}/Z in inverse proportion to the distance Z and a component L

_{lea}corresponding to the leakage of the magnetic flux are added and approximated to obtain an inductance L per electromagnet as shown in the following equation:

_{z}and the current I

_{z}flowing through the coil is expressed by the following equation when the resistance of the electromagnet coil is R

_{z}.

_{0}, I

_{z}), the fluctuation value v

_{z}from the stationary voltage applied to the electromagnet coil is expressed by the following equation:

_{z}is the inductance of the electromagnetic coil in the equilibrium levitation state, and it is assumed that it can be expressed by the following equation:

_{z}: total value of the static attractive forces generated from both electromagnets pairs in the equilibrium levitation state [N], Z

_{0}: gap between the electromagnet surface and the steel plate surface in the equilibrium levitation state [m], w: displacement of the frame, I

_{z}: steady state current for obtaining static attraction force [A], i

_{z}: variation value of the current flowing through the electromagnet coil [A], L

_{eff}/Z

_{0}: effective inductance per electromagnet [H], L

_{lea}: leakage inductance per electromagnet [H], L

_{z}: inductance of the electromagnetic coil in the equilibrium levitation state [H], R

_{z}: total resistance value of the electromagnet coil of the pair [Ω], v

_{z}: fluctuation value from the stationary voltage applied to the electromagnet of the pair [V].

## 4. State Equation

## 5. Control Theory

#### 5.1. Optimal Control

**Q**is a semi-regular matrix and

**H**is a regular matrix. The feedback control input at this time is as follows:

#### 5.2. Disturbance Cancellation Control

## 6. Levitation Experiment

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**Time histories of displacement and amplitude spectrums of vibrating frames by random disturbance.

**Figure 9.**Bending levitation result when vibrating frames by the random disturbance (θ = 13°). (

**a**) Optimal control; (

**b**) Optimal control with disturbance cancellation control.

Measurement range | 0–15 mm |

Straightness | ±1% of F.S |

Degradability | 0.03% of F.S |

Responsiveness | DC-10 kHz (−3 dB) |

A/D Converter | D/A Converter | ||
---|---|---|---|

Input voltage range | ±10 V | Output voltage range | ±10 V |

Resolution | 12 bit | Resolution | 12 bit |

Conversion time | 3 μs/ch | Conversion time | 5 μs/ch |

Output voltage | ±20 V |

Output current | ±5.5A |

Frequency range | 2–3 kHz |

Maximum excitation force | 10 kgf ± 5% |

Maximum acceleration | 28 G ± 5% |

Frequency range | 3 Hz–13 kHz |

Maximum input current | 5.5 A ± 5% |

Maximum speed | 1.77 m/s |

Maximum amplitude | 10 mm |

Optimal control | 0.129 mm |

Optimal control with disturbance cancellation control | 0.092 mm |

Optimal control | 86% |

Optimal control with disturbance cancellation control | 96% |

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**MDPI and ACS Style**

Ogawa, K.; Tada, M.; Narita, T.; Kato, H. Electromagnetic Levitation Control for Bending Flexible Steel Plate: Experimental Consideration on Disturbance Cancellation Control. *Actuators* **2018**, *7*, 43.
https://doi.org/10.3390/act7030043

**AMA Style**

Ogawa K, Tada M, Narita T, Kato H. Electromagnetic Levitation Control for Bending Flexible Steel Plate: Experimental Consideration on Disturbance Cancellation Control. *Actuators*. 2018; 7(3):43.
https://doi.org/10.3390/act7030043

**Chicago/Turabian Style**

Ogawa, Kazuki, Makoto Tada, Takayoshi Narita, and Hideaki Kato. 2018. "Electromagnetic Levitation Control for Bending Flexible Steel Plate: Experimental Consideration on Disturbance Cancellation Control" *Actuators* 7, no. 3: 43.
https://doi.org/10.3390/act7030043