# Fixed-Time Extended Observer-Based Adaptive Sliding Mode Control for a Quadrotor UAV under Severe Turbulent Wind

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

**Lemma**

**1**

**.**System

**Lemma**

**2**

**.**We assume that vector field ${\nu}_{f}$ is homogeneous of degree k with respect to vector field ${\nu}_{g}$. Then, the origin is a finite-time-stable equilibrium under ${\nu}_{f}$ if and only if the origin is an asymptotically stable equilibrium under ${\nu}_{f}$ and $k<0$.

**Lemma**

**3**

**.**We assume that there is a continuous unbounded function V. We set ${\Theta}_{s}\subset {\mathbb{R}}^{n}$; it is globally fixed-time attractive if $V\left(q\right)=0\Rightarrow q\in {\Theta}_{s}$ and any $q\left(t\right)$ satisfies $\dot{V}\left(q\right)\le -{(\alpha {V}^{a}\left(q\left(t\right)\right)+\beta {V}^{b}\left(q\left(t\right)\right))}^{c}$ for $\alpha ,\beta ,a,b,c>0$, with $ac<1$, $bc>1$, and settling time function

**Lemma**

**4**

**.**We consider system (2) and suppose that there is a continuous function $\mathcal{V}\left(x\right)$ that fulfills $\mathcal{V}\left(0\right)=0$ and $\mathcal{V}\left(x\right)\in {\mathbb{R}}^{+}$, $\forall x\ne 0$. Then, the origin of (2) is practical finite-time stable if

## 3. UAV Dynamics and the Von Karman Wind Model

## 4. FxtESO-Based ASMC Design

#### 4.1. Fixed-Time Extended State Observer

#### 4.2. Attitude Controller

#### 4.3. Positioning Controller

#### 4.4. Stability Analysis

**Theorem**

**1.**

**Proof.**

- Matrix $\mathbf{P}$ is the solution of Lyapunov equation ${\mathbf{K}}^{T}\mathbf{P}+\mathbf{P}\mathbf{K}=-\mathbf{I}$ with identity matrix $\mathbf{I}\in {\mathbb{R}}^{\mathbf{3}\times \mathbf{3}}$.
- $$\begin{array}{cc}\hfill \dot{V}\left({\tilde{\mathbf{x}}}_{\alpha}\right)& ={\dot{\tilde{\xi}}}_{\alpha}^{T}\mathbf{P}{\tilde{\xi}}_{\alpha}+{\tilde{\xi}}_{\alpha}^{T}\mathbf{P}{\dot{\tilde{\xi}}}_{\alpha}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\tilde{\xi}}_{\alpha}^{T}{\mathbf{K}}^{T}\mathbf{P}+{\tilde{\xi}}_{\alpha}^{T}\mathbf{P}\mathbf{K}{\tilde{\xi}}_{\alpha}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\tilde{\xi}}_{\alpha}^{T}({\mathbf{K}}^{T}\mathbf{P}+\mathbf{P}\mathbf{K}){\tilde{\xi}}_{\alpha}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-{\tilde{\xi}}_{\alpha}^{T}\mathbf{I}{\tilde{\xi}}_{\alpha}<0.\hfill \end{array}$$

## 5. Results

#### 5.1. Disturbance Compensation

#### 5.2. Comparison Study

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ASMC | Adaptive sliding mode control |

ESO | Extended state observer |

FxtESO | Fixed-time extended state observer |

NESO | Nonlinear extended state observer |

UAV | Unamnned Aerial Vehicle |

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**Figure 5.**Disturbance compensation analysis. Trajectory tracking performance of (

**a**) Top view, (

**b**) y response, (

**c**) x coordinate and (

**d**) z axis evolution.

**Figure 6.**Disturbance compensation analysis. Attitude performance and control input evolution. (

**a**) Yaw angle (

**b**) Roll and pitch responses, (

**c**) Adaptive gains, (

**d**) Thrust.

**Figure 8.**FxtESO and NESO–based adaptive controller comparison. Trajectory tracking: (

**a**) top view, (

**b**) z axis, (

**c**) x response, (

**d**) y behavior, (

**e**) Roll and pitch estimation, (

**f**) yaw estimation.

**Figure 9.**Quadrotor UAV trajectory. FxtESO and NESO–based adaptive controller comparison. (

**a**) Yaw convergence, (

**b**) Adaptive gains, (

**c**) Required thrust, (

**d**) Generated torques.

Disturbance Estimation | |||||
---|---|---|---|---|---|

MSE | RMSE | IAE | ISE | ITAE | |

x-axis | |||||

FxtESO-ASMC | 0.0469 | 0.2166 | 9.9390 | 2.8160 | 302.7 |

NESO-ASMC | 0.0586 | 0.2421 | 11.00 | 3.5190 | 305.2 |

y-axis | |||||

FxtESO-ASMC | 0.0875 | 0.2958 | 13.51 | 5.25 | 377.2 |

NESO-ASMC | 0.1059 | 0.3254 | 14.34 | 6.3550 | 364.6 |

z-axis | |||||

FxtESO-ASMC | 0.4698 | 0.6854 | 29.25 | 28.19 | 729.3 |

NESO-ASMC | 1.0988 | 1.0482 | 33.37 | 65.94 | 699.5 |

Position Estimation | |||||

MSE | RMSE | IAE | ISE | ITAE | |

x-axis | |||||

FxtESO-ASMC | 1.41 × 10 ${}^{-6}$ | 1.19 × 10 ${}^{-3}$ | 0.0569 | 8.50 × 10 ${}^{-5}$ | 1.7110 |

NESO-ASMC | 1.38 × 10 ${}^{-\mathbf{6}}$ | 1.17 × 10 ${}^{-\mathbf{3}}$ | 0.0562 | 8.30 × 10 ${}^{-\mathbf{5}}$ | 1.6484 |

y-axis | |||||

FxtESO-ASMC | 1.43 × 10 ${}^{-6}$ | 1.19 × 10 ${}^{-3}$ | 0.0572 | 8.62 × 10 ${}^{-5}$ | 1.7180 |

NESO-ASMC | 1.40 × 10 ${}^{-\mathbf{6}}$ | 1.18 × 10 ${}^{-\mathbf{3}}$ | 0.0565 | 8.43 × 10 ${}^{-\mathbf{5}}$ | 1.69 |

z-axis | |||||

FxtESO-ASMC | 5.43 × 10 ${}^{-\mathbf{6}}$ | 2.33 × 10 ${}^{-\mathbf{3}}$ | 0.0655 | 3.26 × 10 ${}^{-\mathbf{4}}$ | 1.799 |

NESO-ASMC | 6.95 × 10 ${}^{-6}$ | 2.63 × 10 ${}^{-3}$ | 0.0697 | 4.17 × 10 ${}^{-4}$ | 1.768 |

MSE | RMSE | IAE | ISE | ITAE | |

x-axis | |||||

FxtESO-ASMC | 0.0075 | 0.0866 | 4.1990 | 0.4472 | 104.3 |

NESO-ASMC | 0.0566 | 0.2379 | 6.0870 | 3.3990 | 114.7 |

FxtESO-ASMC w/o DC | 0.0413 | 0.2032 | 9.186 | 2.481 | 234.9 |

y-axis | |||||

FxtESO-ASMC | 0.0111 | 0.1054 | 3.5910 | 0.6640 | 77.02 |

NESO-ASMC | 0.2925 | 0.5408 | 11.65 | 17.55 | 117.30 |

FxtESO-ASMC w/o DC | 0.0536 | 0.2315 | 8.7970 | 3.214 | 153.8 |

z-axis | |||||

FxtESO-ASMC | 0.0465 | 0.2156 | 3.1020 | 2.7910 | 34.13 |

NESO-ASMC | 0.1499 | 0.3872 | 6.0870 | 8.9940 | 39.97 |

FxtESO-ASMC w/o DC | 0.0917 | 0.3028 | 8.0420 | 5.50 | 77.95 |

Quadrotor UAV. Norms of Control | |||||

$\parallel {T}_{h}\parallel $ | $\parallel \mathit{\tau}\parallel $ | ||||

FxtESO-ASMC | 1717.70 | 395.9815 | |||

NESO-ASMC | 1721.80 | 1041.7 | |||

FxtESO-ASMC w/o DC | 1716.80 | 785.2556 |

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**MDPI and ACS Style**

Miranda-Moya, A.; Castañeda, H.; Wang, H.
Fixed-Time Extended Observer-Based Adaptive Sliding Mode Control for a Quadrotor UAV under Severe Turbulent Wind. *Drones* **2023**, *7*, 700.
https://doi.org/10.3390/drones7120700

**AMA Style**

Miranda-Moya A, Castañeda H, Wang H.
Fixed-Time Extended Observer-Based Adaptive Sliding Mode Control for a Quadrotor UAV under Severe Turbulent Wind. *Drones*. 2023; 7(12):700.
https://doi.org/10.3390/drones7120700

**Chicago/Turabian Style**

Miranda-Moya, Armando, Herman Castañeda, and Hesheng Wang.
2023. "Fixed-Time Extended Observer-Based Adaptive Sliding Mode Control for a Quadrotor UAV under Severe Turbulent Wind" *Drones* 7, no. 12: 700.
https://doi.org/10.3390/drones7120700