#
L_{1} Adaptive Control for a Vertical Rotor Orientation System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

**Assumption**

**1.**

**(Uniform boundedness of**$\mathit{f}(\mathit{t},\mathbf{0})$).

**Assumption**

**2.**

**(Boundedness of partial derivatives).**

## 3. ${\mathit{L}}_{\mathbf{1}}$ Adaptive Control Design of the Rotor Orientation System

#### 3.1. Semi-Global Linearization of the System

#### 3.2. State Predictor

#### 3.3. ${L}_{1}$ Adaptive Control Algorithm

## 4. Adaptive Control System Analysis

#### 4.1. Boundedness of State Error and Parameter Error

**Proposition 1.**

**Proof.**

#### 4.2. Boundedness of State Variables

**Proof.**

#### 4.3. Performance of the Adaptive System

## 5. Simulation Results for the Adaptive Control System

#### 5.1. Simulation of the Adaptive Controller

#### 5.2. Simulation of the Rotor Orientation System

## 6. Experimental Results for the Rotor Orientation System

#### 6.1. Experimental Devices

#### 6.2. Experimental Process

- 1.
- Measurement of the rotor drift angle.

- 2.
- Calculation of the electromagnetic force.

- 3.
- Calculation of the current in every electromagnet.

#### 6.3. Experimental Results

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Prediction error simulation results for: (

**a**) the x-direction component ${e}_{1}$; (

**b**) the y-direction component ${e}_{2}$; (

**c**) the x-direction angular error ${e}_{3}$; and (

**d**) y-direction angular error ${e}_{4}$.

**Figure 5.**Simulation comparison for Γ values: (

**a**) comparison of parameter ${\mathit{K}}_{\mathit{\theta}}$; (

**b**) comparison of error ${e}_{1}$.

**Figure 7.**Experimental platform for the orientation control system: (

**a**) electromagnet; (

**b**) rotor; (

**c**) universal joint; (

**d**) motor.

**Figure 8.**Block diagram of the orientation experiment: (

**a**) gap sensor; (

**b**) electromagnet; (

**c**) rotor; (

**d**) gap calculation; (

**e**) angle calculation; (

**f**) controller; (

**g**) control current calculation; (

**h**) power amplifier.

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

Material | Iron | – |

Outer diameter | 150 | mm |

Inner diameter | 142 | mm |

Height | 150 | mm |

Mass | 1.737 | kg |

© 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**

Liu, S.; Fan, Y.; Di, J.; Ji, M.
*L*_{1} Adaptive Control for a Vertical Rotor Orientation System. *Appl. Sci.* **2016**, *6*, 242.
https://doi.org/10.3390/app6090242

**AMA Style**

Liu S, Fan Y, Di J, Ji M.
*L*_{1} Adaptive Control for a Vertical Rotor Orientation System. *Applied Sciences*. 2016; 6(9):242.
https://doi.org/10.3390/app6090242

**Chicago/Turabian Style**

Liu, Sijia, Yu Fan, Jun Di, and Mingming Ji.
2016. "*L*_{1} Adaptive Control for a Vertical Rotor Orientation System" *Applied Sciences* 6, no. 9: 242.
https://doi.org/10.3390/app6090242