# Nonlinear Robust Control on Yaw Motion of a Variable-Speed Unmanned Aerial Helicopter under Multi-Source Disturbances

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## Abstract

**:**

## 1. Introduction

- A fourth-order yaw error dynamic model considering actuator output fault, matched and unmatched disturbances is constructed. Based on the adaptive command filtered method, we built a simple control framework with no need for using DOB and solving complex HJI inequality to attenuate system multi-source perturbation.
- A rigorous stability analysis of the closed-loop yaw system was completed, which achieved the predefined performance of ${\mathcal{L}}_{2}$-gain disturbance suppression. Four groups of comparative simulation results show the effectiveness and superiority of this control law.

## 2. Yaw Error Dynamic Model

**Lemma**

**1.**

**Lemma**

**2**

**Lemma**

**3**

**Assumption**

**1.**

**Assumption**

**2.**

## 3. Controller Design

**Definition**

**1.**

- (1)
- When $w=\mathbf{0}$, all signals are uniform-ultimately bounded stable.
- (2)
- When $w\ne \mathbf{0}$, the evaluation signal vector $z$ for initial state $x\left(0\right)=\mathbf{0}$, $T>0$ satisfies$${\int}_{0}^{T}{\parallel z\parallel}^{2}dt\le {\gamma}^{2}{\int}_{0}^{T}{\parallel w\parallel}^{2}dt+\epsilon ,\forall w\in {\mathcal{L}}_{2}[0,T]$$ε is a sufficiently small constant [27], the $z=h\left(x\right)$ is a state-dependent vector and $h\left(\mathit{x}\right)$ is designed later.

**Theorem**

**1.**

**Proof**

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

UAH | Unmanned Aerial Helicopter |

AYDS | Artificial Yaw Damping System |

ADRC | Active Disturbance Rejection Control |

SMC | Sliding Mode Control |

DOB | Disturbance-Observer-Based |

UUB | Uniform-Ultimately bounded |

HJI | Hamilton–Jacobi-Issacs |

Bs | Backstepping |

ESO | Extended State Observer |

RBF | Radial Basis Function |

## References

- La Civita, M.; Papaueorgiou, G.; Messner, W.C.; Kanade, T. Design and flight testing of an H(infinity) controller for a robotic helicopter. J. Guid. Control Dyn.
**2006**, 29, 485–494. [Google Scholar] [CrossRef] - Nedjati, A.; Vizvari, B.; Izbirak, G. Post-earthquake response by small UAV helicopters. Nat. Hazards
**2016**, 80, 1669–1688. [Google Scholar] [CrossRef] - Ding, L.; Ma, R.; Wu, H.; Feng, C.; Li, Q. Yaw control of an unmanned aerial vehicle helicopter using linear active disturbance rejection control. Proc. Inst. Mech. Eng. Part I-J. Syst. Control Eng.
**2017**, 231, 427–435. [Google Scholar] [CrossRef] - Ding, L.; He, Q.; Wang, C.; Qi, R. Disturbance Rejection Attitude Control for a Quadrotor: Theory and Experiment. Int. J. Aerosp. Eng.
**2021**, 2021, 8850071. [Google Scholar] [CrossRef] - Xu, D.Z.; Jiang, B.; Shi, P. Global Robust Tracking Control of Non-affine Nonlinear Systems with Application to Yaw Control of UAV Helicopter. Int. J. Control Autom. Syst.
**2013**, 11, 957–965. [Google Scholar] [CrossRef] - Jiang, T.; Lin, D.; Song, T. Novel integral sliding mode control for small-scale unmanned helicopters. J. Frankl. Inst.
**2019**, 356, 2668–2689. [Google Scholar] [CrossRef] - Krishna, A.B.; Sen, A.; Kothari, M. Super Twisting Algorithm for Robust Geometric Control of a Helicopter. J. Intell. Robot. Syst. Theory Appl.
**2021**, 102, 61. [Google Scholar] [CrossRef] - Ullah, I.; Pei, H.L. Fixed Time Disturbance Observer Based Sliding Mode Control for a Miniature Unmanned Helicopter Hover Operations in Presence of External Disturbances. IEEE Access
**2020**, 8, 73173–73181. [Google Scholar] [CrossRef] - Liu, L.; Chen, M.; Li, T.; Wang, H. Composite Anti-Disturbance Reference Model L
_{2}–L_{∞}Control for Helicopter Slung Load System. J. Intell. Robot. Syst. Theory Appl.**2021**, 102, 15. [Google Scholar] [CrossRef] - Tang, S.; Mao, L.; Liu, G.; Wang, W. Active Disturbance Rejection Control Design of the Yaw Channel for a Small-scale Helicopter based on Backstepping. In Proceedings of the 38th 2019 Chinese Control Conference (CCC), Guangzhou, China, 27–30 July 2019; pp. 8073–8078. [Google Scholar]
- Zhao, W.; Meng, Z.; Wang, K.; Zhang, H. Backstepping Control of an Unmanned Helicopter Subjected to External Disturbance and Model Uncertainty. Appl. Sci.
**2021**, 11, 5331. [Google Scholar] [CrossRef] - Chen, X.; Zhang, Q.; Duan, Y. Finite-Time Backstepping Sliding Mode Control Applied to the Yaw Control of UAV Helicopters with Actuator Faults. In Proceedings of the 39th Chinese Control Conference, Shenyang, China, 27–29 July 2020; pp. 6911–6916. [Google Scholar] [CrossRef]
- Li, Y.; Chen, M.; Ge, S.S. Anti-disturbance control for attitude and altitude systems of the helicopter under random disturbances. Aerosp. Sci. Technol.
**2020**, 96, 105561. [Google Scholar] [CrossRef] - Sheng, S.; Sun, C. Yaw Control of an Unmanned Helicopter Using Adaptive Model Feedback and Error Compensation. J. Aerosp. Eng.
**2016**, 29, 06015002. [Google Scholar] [CrossRef] - Soltanpour, M.R.; Hasanvand, F.; Hooshmand, R. Robust linear parameter varying attitude control of a quadrotor unmanned aerial vehicle with state constraints and input saturation subject to wind disturbance. Trans. Inst. Meas. Control
**2020**, 42, 1083–1096. [Google Scholar] [CrossRef] - Fan, X.; Yi, Y.; Zhang, T. Disturbance rejection control of yaw channel of a small-scale unmanned helicopter via Takagi-Sugeno disturbance modeling approach. Int. J. Adv. Robot. Syst.
**2016**, 13, 1729881416671113. [Google Scholar] [CrossRef] - Yan, W.X.; Huang, J.; Xu, D.Z. Adaptive fuzzy tracking control for non-affine nonlinear yaw channel of unmanned aerial vehicle helicopter. Int. J. Adv. Robot. Syst.
**2017**, 14, 1729881416678137. [Google Scholar] [CrossRef] [Green Version] - Le, T.Q.; Lai, Y.C.; Yeh, C.L. Adaptive tracking control based on neural approximation for the yaw motion of a small-scale unmanned helicopter. Int. J. Adv. Robot. Syst.
**2019**, 16, 1729881419828277. [Google Scholar] [CrossRef] [Green Version] - Lai, Y.C.; Le, T.Q. Adaptive Learning-Based Observer With Dynamic Inversion for the Autonomous Flight of an Unmanned Helicopter. IEEE Trans. Aerosp. Electron. Syst.
**2021**, 57, 1803–1814. [Google Scholar] [CrossRef] - Shen, S.; Xu, J. Adaptive neural network-based active disturbance rejection flight control of an unmanned helicopter. Aerosp. Sci. Technol.
**2021**, 119, 107062. [Google Scholar] [CrossRef] - Zhang, Q.; Chen, X.; Xu, D. Adaptive Neural Fault-Tolerant Control for the Yaw Control of UAV Helicopters with Input Saturation and Full-State Constraints. Appl. Sci.
**2020**, 10, 1404. [Google Scholar] [CrossRef] [Green Version] - Ezekiel, D.M.; Samikannu, R.; Matsebe, O. Pitch and Yaw Angular Motions (Rotations) Control of the 1-DOF and 2-DOF TRMS: A Survey. Arch. Comput. Methods Eng.
**2021**, 28, 1449–1458. [Google Scholar] [CrossRef] - Singh, R.; Bhushan, B. Data-Driven Technique-Based Fault-Tolerant Control for Pitch and Yaw Motion in Unmanned Helicopters. IEEE Trans. Instrum. Meas.
**2021**, 70, 3502711. [Google Scholar] [CrossRef] - Dong, W.; Farrell, J.A.; Polycarpou, M.M.; Djapic, V.; Sharma, M. Command filtered adaptive backstepping. IEEE Trans. Control Syst. Technol.
**2011**, 20, 566–580. [Google Scholar] [CrossRef] - Sun, Z.Y.; Yun, M.M.; Li, T. A new approach to fast global finite-time stabilization of high-order nonlinear system. Automatica
**2017**, 81, 455–463. [Google Scholar] [CrossRef] - Yu, J.; Shi, P.; Zhao, L. Finite-time command filtered backstepping control for a class of nonlinear systems. Automatica
**2018**, 92, 173–180. [Google Scholar] [CrossRef] - Ishii, C.; Shen, T.; Qu, Z. Lyapunov recursive design of robust adaptive tracking control with L2-gain performance for electrically-driven robot manipulators. Int. J. Control
**2001**, 74, 811–828. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Tang, P.; Dai, Y.; Chen, J.
Nonlinear Robust Control on Yaw Motion of a Variable-Speed Unmanned Aerial Helicopter under Multi-Source Disturbances. *Aerospace* **2022**, *9*, 42.
https://doi.org/10.3390/aerospace9010042

**AMA Style**

Tang P, Dai Y, Chen J.
Nonlinear Robust Control on Yaw Motion of a Variable-Speed Unmanned Aerial Helicopter under Multi-Source Disturbances. *Aerospace*. 2022; 9(1):42.
https://doi.org/10.3390/aerospace9010042

**Chicago/Turabian Style**

Tang, Peng, Yuehong Dai, and Junfeng Chen.
2022. "Nonlinear Robust Control on Yaw Motion of a Variable-Speed Unmanned Aerial Helicopter under Multi-Source Disturbances" *Aerospace* 9, no. 1: 42.
https://doi.org/10.3390/aerospace9010042