# Quadcopter UAVs Extended States/Disturbance Observer-Based Nonlinear Robust Backstepping Control

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## Abstract

**:**

## 1. Introduction

- (i)
- The proposed algorithm overcomes the drawbacks of previous methods in the requirement of full state measurement. The ESDO is able to estimate the velocity state of the vehicle once this parameter cannot be directly measured. Thus, the implementation cost for data acquisition may be reduced and the influence of high measurement noise generated from the velocity sensor is also alleviated.
- (ii)
- The unmeasured velocity states and lumped perturbations are estimated by the presented ESDO integrating with advantages of the recursive structure of the backstepping technique, the convergence of tracking errors is always guaranteed.
- (iii)
- The numerical simulation is fully executed in both attitude and position control mode; these performance results are compared with other existing control methods to confirm the strict stability and efficiency of the presented control scheme in both the convergence error and anti-disturbance capacity.
- (iv)
- Finally, unlike the existing methods, the upper bounds of the uncertainties and/or external disturbances are not demanded during the steps of designing the proposed control scheme.

## 2. Mathematical Model and Problem Description

**Hypothesis**

**1.**

**Hypothesis**

**2.**

## 3. Robust Backstepping Control-Based ESDO

#### 3.1. Extended State/Disturbance Observer (ESDO)

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**1.**

#### 3.2. Robust Backstepping Controller Design

#### 3.3. Stability Analysis of the Proposed Robust Backstepping Controller

**Lemma**

**1**

**.**Let $h(t),W(t):\left[0,\infty \right)\mapsto \mathbb{R}$. Then

**Proof**

**of**

**Lemma**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**P**in Equation (32) into Equation (40) and doing some mathematical conversion steps, the value of $\dot{V}\left({\tilde{\mathit{z}}}_{1},{\tilde{\mathit{z}}}_{2}\right)$ can be obtained as follows,

**Remark**

**2.**

## 4. Simulation Results and Discussions

## 5. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Performance comparison of SMC, ADRC, and RBCESDO in both 3D (

**a**) and 2D (

**b**) trajectory tracking tests on a quadcopter UAV.

**Figure 4.**Disturbance estimation in attitude dynamic (

**a**–

**c**) and position dynamic (

**d**–

**f**) of the proposed ESDO.

**Figure 5.**Comparison of attitude performance between the proposed RBCESDO (

**a**–

**c**), SMC (

**d**–

**f**), and ADRC (

**g**–

**i**).

**Figure 6.**Comparison of position performance between the proposed RBCESDO, SMC, and ADRC: horizontal position x (

**a**), y (

**b**), and vertical position z (

**c**).

**Figure 7.**Comparison of the controller performance between RBCESDO, SMC, and ADRC: vertical force (

**a**), roll torque (

**b**), pitch torque (

**c**) and yaw torque (

**d**).

**Figure 8.**Comparison of attitude tracking error (

**a**–

**c**) and position tracking error (

**d**–

**f**) between the proposed RBCESDO, SMC, and ADRC.

**Figure 9.**Comparison of maximum attitude tracking error (

**a**) and maximum position tracking error (

**b**) of the proposed RBCESDO, SMC, and ADRC.

Symbol | Descriptions | Value and Unit |
---|---|---|

$m$ | Total mass of the vehicle | $1.12\mathrm{kg}$ |

$l$ | Arm length of quadcopter UAV frame | $0.23\mathrm{m}$ |

${J}_{r}$ | Inertial moment of a rotor | $8.5{10}^{-4}\mathrm{kg}.{\mathrm{m}}^{2}$ |

${I}_{x}$ | Inertial moment around x-axis | $0.0019\mathrm{kg}.{\mathrm{m}}^{2}$ |

${I}_{y}$ | Inertial moment around y-axis | $0.0019\mathrm{kg}.{\mathrm{m}}^{2}$ |

${I}_{z}$ | Inertial moment around z-axis | $0.0223\mathrm{kg}.{\mathrm{m}}^{2}$ |

$b$ | Thrust coefficient | $7.73212\left({10}^{-6}\right)\mathrm{N}.{\mathrm{s}}^{2}$ |

$d$ | Drag coefficient | $1.27513\left({10}^{-7}\right)\mathrm{N}.\mathrm{m}.{\mathrm{s}}^{2}$ |

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

$\mathit{\alpha}$ | Observer gain of matrix | $diag\left[2.5,2.5,2,5,5,5,5\right]$ |

$\mathit{\beta}$ | Observer gain of matrix | $diag\left[100,100,100,75,75,75\right]$ |

${\mathit{k}}_{1}$ | Controller gain | $diag\left[25,25,35,12,12,8\right]$ |

${\mathit{k}}_{2}$ | Controller gain | $diag\left[18,18,35,12,12,8\right]$ |

${\mathit{k}}_{3}$ | Controller gain | $diag\left[0.1,0.1,0.1,0.1,0.1,0.1\right]$ |

${\mathit{Z}}_{0}$ | Initial state value | $\left[8,0,0,0,0,0.1\right]$ |

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**MDPI and ACS Style**

Thanh, H.L.N.N.; Huynh, T.T.; Vu, M.T.; Mung, N.X.; Phi, N.N.; Hong, S.K.; Vu, T.N.L.
Quadcopter UAVs Extended States/Disturbance Observer-Based Nonlinear Robust Backstepping Control. *Sensors* **2022**, *22*, 5082.
https://doi.org/10.3390/s22145082

**AMA Style**

Thanh HLNN, Huynh TT, Vu MT, Mung NX, Phi NN, Hong SK, Vu TNL.
Quadcopter UAVs Extended States/Disturbance Observer-Based Nonlinear Robust Backstepping Control. *Sensors*. 2022; 22(14):5082.
https://doi.org/10.3390/s22145082

**Chicago/Turabian Style**

Thanh, Ha Le Nhu Ngoc, Tuan Tu Huynh, Mai The Vu, Nguyen Xuan Mung, Nguyen Ngoc Phi, Sung Kyung Hong, and Truong Nguyen Luan Vu.
2022. "Quadcopter UAVs Extended States/Disturbance Observer-Based Nonlinear Robust Backstepping Control" *Sensors* 22, no. 14: 5082.
https://doi.org/10.3390/s22145082