# Fractional Order Sliding Mode Control of a Class of Second Order Perturbed Nonlinear Systems: Application to the Trajectory Tracking of a Quadrotor

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## Abstract

**:**

## 1. Introduction

## 2. FOSMC of an Integer Second Order Perturbed Nonlinear System

**Assumption**

**A1.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Application to the Trajectory Tracking of a Quadrotor

#### 3.1. Quadrotor’s Dynamic Model

#### 3.2. Trajectory Tracking of a Quadrotor

## 4. Simulation Results

#### 4.1. Simulations Varying the Parameters ${\sigma}_{\chi}$

#### 4.2. Comparison with the IOSMC

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Projection in the x-y plane of the trajectory.Orange-Desired trajectory, Else-Trajectory being tracked.

**Figure 16.**(

**a**) Control signal ${u}_{1}$ with $\beta =0.5$; (

**b**) Control signal ${u}_{1}$ with $\beta =0.9$.

**Figure 23.**(

**a**) Vertical Thrust with distinct ${\sigma}_{\chi}$’s; (

**b**) Vertical Thrust with ${\sigma}_{\chi}=0.05$.

**Figure 24.**(

**a**) Roll Torque with distinct ${\sigma}_{\chi}$’s; (

**b**) Roll Torque with ${\sigma}_{\chi}=0.05$.

**Figure 25.**(

**a**) Pitch Torque with distinct ${\sigma}_{\chi}$’s; (

**b**) Pitch Torque with ${\sigma}_{\chi}=0.05$.

**Figure 26.**(

**a**) Yaw Torque with distinct ${\sigma}_{\chi}$’s; (

**b**) Yaw Torque with ${\sigma}_{\chi}=0.05$.

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

Mass m | 1.4 | kg |

Gravity g | 9.81 | $\frac{m}{{s}^{2}}$ |

Ixx | 0.02 | kg·m${}^{2}$ |

Iyy | 0.02 | kg·m${}^{2}$ |

Izz | 0.04 | kg·m${}^{2}$ |

Name | Value |
---|---|

$\u03f5$ | 0.005 |

${\sigma}_{z}$ | 0.055 |

${\lambda}_{z}$ | 6.0 |

${\mu}_{z}$ | 6.0 |

${\sigma}_{y}$ | 0.055 |

${\lambda}_{y}$ | 6.0 |

${\mu}_{y}$ | 6.0 |

${\sigma}_{x}$ | 0.055 |

${\lambda}_{x}$ | 6.0 |

${\mu}_{x}$ | 6.0 |

$K{d}_{\varphi}$ | 1.5 |

$K{p}_{\varphi}$ | 25 |

$K{d}_{\theta}$ | 1.5 |

$K{p}_{\theta}$ | 25 |

$K{d}_{\psi}$ | 1.3 |

$K{p}_{\psi}$ | 9 |

Fractional Order $\mathit{\beta}$ | Color |
---|---|

0.1 | Blue |

0.2 | Green |

0.3 | Red |

0.4 | Cyan |

0.5 | Purple |

0.6 | Yellow |

0.7 | Brown |

0.8 | Dark Blue |

0.9 | Light Green |

Signal | Maximal Value | Units |
---|---|---|

Thrust | 35 | N |

Roll Torque | 4 | Nm |

Pitch Torque | 4 | Nm |

Yaw Torque | 2 | Nm |

Name | Value |
---|---|

$\u03f5$ | 0.005 |

${\sigma}_{z}$ | 0.01 |

${\lambda}_{z}$ | 6.0 |

${\mu}_{z}$ | 6.0 |

${\sigma}_{y}$ | 0.01 |

${\lambda}_{y}$ | 6.0 |

${\mu}_{y}$ | 6.0 |

${\sigma}_{x}$ | 0.01 |

${\lambda}_{x}$ | 6.0 |

${\mu}_{x}$ | 6.0 |

$K{d}_{\varphi}$ | 1.5 |

$K{p}_{\varphi}$ | 25 |

$K{d}_{\theta}$ | 1.5 |

$K{p}_{\theta}$ | 25 |

$K{d}_{\psi}$ | 1.3 |

$K{p}_{\psi}$ | 9 |

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## Share and Cite

**MDPI and ACS Style**

Govea-Vargas, A.; Castro-Linares, R.; Duarte-Mermoud, M.A.; Aguila-Camacho, N.; Ceballos-Benavides, G.E.
Fractional Order Sliding Mode Control of a Class of Second Order Perturbed Nonlinear Systems: Application to the Trajectory Tracking of a Quadrotor. *Algorithms* **2018**, *11*, 168.
https://doi.org/10.3390/a11110168

**AMA Style**

Govea-Vargas A, Castro-Linares R, Duarte-Mermoud MA, Aguila-Camacho N, Ceballos-Benavides GE.
Fractional Order Sliding Mode Control of a Class of Second Order Perturbed Nonlinear Systems: Application to the Trajectory Tracking of a Quadrotor. *Algorithms*. 2018; 11(11):168.
https://doi.org/10.3390/a11110168

**Chicago/Turabian Style**

Govea-Vargas, Arturo, Rafael Castro-Linares, Manuel A. Duarte-Mermoud, Norelys Aguila-Camacho, and Gustavo E. Ceballos-Benavides.
2018. "Fractional Order Sliding Mode Control of a Class of Second Order Perturbed Nonlinear Systems: Application to the Trajectory Tracking of a Quadrotor" *Algorithms* 11, no. 11: 168.
https://doi.org/10.3390/a11110168