# Incremental Nonlinear Dynamic Inversion Attitude Control for Helicopter with Actuator Delay and Saturation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

## 3. Control Law Design

**Theorem**

**1.**

**Proof.**

## 4. Attitude Controller Design for Helicopter

#### 4.1. Helicopter Model

#### 4.2. Anti-Windup Design

#### 4.3. Rate Loop

#### 4.4. Attitude Loop

#### 4.5. Control Law for Collective Pitch of Main Rotor

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Padfield, G.D. Helicopter Flight Dynamics: The Theory and Application of Flying Qualities and Simulation Modelling, 2nd ed.; American Institute of Aeronautics and Astronautics: Reston, VA, USA, 2007; Volume 113. [Google Scholar]
- Prouty, R.W.; Curtiss, H.C. Helicopter Control Systems: A History. J. Guid. Control. Dyn.
**2003**, 26, 12–18. [Google Scholar] [CrossRef] - Hu, J.; Gu, H. Survey on Flight Control Technology for Large-Scale Helicopter. Int. J. Aerosp. Eng.
**2017**, 2017, 5309403. [Google Scholar] [CrossRef] - Kim, N.; Calise, A.J.; Hovakimyan, N.; Prasad, J.V.R.; Corban, E. Adaptive Output Feedback for High-Bandwidth Flight Control. J. Guid. Control. Dyn.
**2002**, 25, 993–1002. [Google Scholar] [CrossRef] - Lee, S.; Ha, C.; Kim, B. Adaptive nonlinear control system design for helicopter robust command augmentation. Aerosp. Sci. Technol.
**2005**, 9, 241–251. [Google Scholar] [CrossRef] - Amaral, T.G.; Crisóstomo, M.M.; Pires, V.F. Helicopter motion control using adaptive neuro-fuzzy inference controller. In Proceedings of the IEEE 2002 28th Annual Conference of the Industrial Electronics Society, Seville, Spain, 5–8 November 2002; Volume 3, pp. 2090–2095. [Google Scholar]
- Leitner, J.; Calise, A.; Prasad, J.V.R. Analysis of Adaptive Neural Networks for Helicopter Flight Control. J. Guid. Control Dyn.
**1997**, 20, 972–979. [Google Scholar] [CrossRef] - Moelans, P. Adaptive Helicopter Control Using Feedback Linearization and Neural Networks. Master’s Thesis, Delft University of Technology, Delft, The Netherlands, 2008. [Google Scholar]
- Kutay, A.T.; Calise, A.J.; Idan, M.; Hovakimyan, N. Experimental results on adaptive output feedback control using a laboratory model helicopter. IEEE Trans. Control Syst. Technol.
**2005**, 13, 196–202. [Google Scholar] [CrossRef] - Johnson, E.N.; Kannan, S.K. Adaptive Trajectory Control for Autonomous Helicopters. J. Guid. Control Dyn.
**2005**, 28, 524–538. [Google Scholar] [CrossRef] - Johnson, E.N.; Calise, A.J. Pseudo-control hedging: A new method for adaptive control. In Proceedings of the Advances in Navigation Guidance and Control Technology Workshop, Alabama, AL, USA, 1–2 November 2000. [Google Scholar]
- Acquatella, B.P.; Chu, Q.P. Agile Spacecraft Attitude Control: An Incremental Nonlinear Dynamic Inversion Approach. IFAC-PapersOnLine
**2020**, 53, 5709–5716. [Google Scholar] [CrossRef] - Wang, X.; van Kampen, E.-J.; Chu, Q.P.; De Breuker, R. Flexible Aircraft Gust Load Alleviation with Incremental Nonlinear Dynamic Inversion. J. Guid. Control. Dyn.
**2019**, 42, 1519–1536. [Google Scholar] [CrossRef] - Smeur, E.; de Croon, G.; Chu, Q. Cascaded incremental nonlinear dynamic inversion for MAV disturbance rejection. Control Eng. Pr.
**2018**, 73, 79–90. [Google Scholar] [CrossRef] - Chen, G.; Liu, A.; Hu, J.; Feng, J.; Ma, Z. Attitude and Altitude Control of Unmanned Aerial-Underwater Vehicle Based on Incremental Nonlinear Dynamic Inversion. IEEE Access
**2020**, 8, 156129–156138. [Google Scholar] [CrossRef] - Cervantes, T.J.L.; Choi, S.H.; Kim, B.S. Flight Control Design using Incremental Nonlinear Dynamic Inversion with Fixed-lag Smoothing Estimation. Int. J. Aeronaut. Space Sci.
**2020**, 21, 1047–1058. [Google Scholar] [CrossRef] - Wang, X.; van Kampen, E.-J.; Chu, Q.; Lu, P. Stability Analysis for Incremental Nonlinear Dynamic Inversion Control. J. Guid. Control Dyn.
**2019**, 42, 1116–1129. [Google Scholar] [CrossRef] - Simplício, P.; Pavel, M.; van Kampen, E.; Chu, Q. An acceleration measurements-based approach for helicopter nonlinear flight control using Incremental Nonlinear Dynamic Inversion. Control. Eng. Pr.
**2013**, 21, 1065–1077. [Google Scholar] [CrossRef] - Pavel, M.D.; Perumal, S.; Olaf, S.; Mike, W.; Chu, Q.P.; Harm, C. Incremental Nonlinear Dynamic Inversion for the Apache AH-64 Helicopter Control. J. Am. Helicopter Soc.
**2020**, 65, 1–16. [Google Scholar] [CrossRef] - Richard, J.-P. Time-delay systems: An overview of some recent advances and open problems. Automatica
**2003**, 39, 1667–1694. [Google Scholar] [CrossRef] - Diagne, M.; Bekiaris-Liberis, N.; Krstic, M. Compensation of input delay that depends on delayed input. Automatica
**2017**, 85, 362–373. [Google Scholar] [CrossRef] - Krstic, M. Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems. IEEE Trans. Autom. Control
**2009**, 55, 287–303. [Google Scholar] [CrossRef] - Krstic, M. Lyapunov Stability of Linear Predictor Feedback for Time-Varying Input Delay. IEEE Trans. Autom. Control
**2010**, 55, 554–559. [Google Scholar] [CrossRef] - Yue, D.; Han, Q.-L. Delayed feedback control of uncertain systems with time-varying input delay. Automatica
**2005**, 41, 233–240. [Google Scholar] [CrossRef] - Sharma, N.; Bhasin, S.; Wang, Q.; Dixon, W.E. Predictor-based control for an uncertain Euler–Lagrange system with input delay. Automatica
**2011**, 47, 2332–2342. [Google Scholar] [CrossRef] - Artstein, Z. Linear systems with delayed controls: A reduction. IEEE Trans. Autom. Control
**1982**, 27, 869–879. [Google Scholar] [CrossRef] - Deng, W.; Yao, J.; Ma, D. Time-varying input delay compensation for nonlinear systems with additive disturbance: An output feedback approach. Int. J. Robust Nonlinear Control
**2017**, 28, 31–52. [Google Scholar] [CrossRef] - Rohollah, M.; Sarangapani, J. Optimal Adaptive Control of Uncertain Nonlinear Continuous-Time Systems with Input and State Delays. IEEE Trans. Neural Netw. Learn. Syst.
**2021**, 2021, 1–10. [Google Scholar] - Rohollah, M.; Sarangapani, J. Optimal control of linear continuous-time systems in the presence of state and input delays with application to a chemical reactor. In Proceedings of the 2020 American Control Conference (ACC), Denver, CO, USA, 1–3 July 2020; pp. 999–1004. [Google Scholar]
- Tarbouriech, S.; Turner, M. Anti-windup design: An overview of some recent advances and open problems. IET Control Theory Appl.
**2009**, 3, 1–19. [Google Scholar] [CrossRef] - Ma, J.; Ge, S.S.; Zheng, Z.; Hu, D. Adaptive NN Control of a Class of Nonlinear Systems with Asymmetric Saturation Actuators. IEEE Trans. Neural Networks Learn. Syst.
**2014**, 26, 1532–1538. [Google Scholar] [CrossRef] [PubMed] - Zhang, S.; Meng, Q. An anti-windup INDI fault-tolerant control scheme for flying wing aircraft with actuator faults. ISA Trans.
**2019**, 93, 172–179. [Google Scholar] [CrossRef] [PubMed] - Quang, L.; Richard, H.; William, S.; Brett, R. Investigation and Preliminary Development of a Modified Pseudo Control Hedging for Missile Performance Enhancement. In Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, CA, USA, 15–18 August 2005. [Google Scholar] [CrossRef]
- Edwards, C.; Lombaerts, T.; Smaili, H. Fault Tolerant Flight Control: A Benchmark Challenge; Springer: Berlin/Heidelberg, Germany, 2010; pp. 1–560. [Google Scholar]
- Hartjes, S. An Optimal Control Approach to Helicopter Noise and Emissions Abatement Terminal Procedures. Master’s Thesis, Delft University of Technology, Delft, The Netherlands, 2015. [Google Scholar]

**Figure 10.**Actuator input corresponding to the responses in Figure 8.

Different Control Channel | Maximum Rates of Attitude Angles | Maximum Rates of Attitude Rates |
---|---|---|

Lateral cyclic pitch | $\frac{d\varphi}{dt}=5.7\mathrm{deg}/\mathrm{s}$ | $\frac{dp}{dt}=145.8\mathrm{deg}/{\mathrm{s}}^{2}$ |

Longitudinal cyclic pitch | $\frac{d\theta}{dt}=-2.5\mathrm{deg}/\mathrm{s}$ | $\frac{dq}{dt}=-60.1\mathrm{deg}/{\mathrm{s}}^{2}$ |

Collective of the tail rotor | $\frac{d\psi}{dt}=-0.7\mathrm{deg}/\mathrm{s}$ | $\frac{dr}{dt}=-16.5\mathrm{deg}/{\mathrm{s}}^{2}$ |

Actuator Name | Variable Name | Maximum Rate Limit |
---|---|---|

Main rotor | ${\theta}_{0}$ | $16.0\mathrm{deg}/\mathrm{s}$ |

Longitudinal cyclic | ${\theta}_{1s}$ | $28.8\mathrm{deg}/\mathrm{s}$ |

Lateral cyclic | ${\theta}_{1c}$ | $16.0\mathrm{deg}/\mathrm{s}$ |

Tail rotor | ${\theta}_{0tr}$ | $32.0\mathrm{deg}/\mathrm{s}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, S.; Zhang, H.; Ji, K.
Incremental Nonlinear Dynamic Inversion Attitude Control for Helicopter with Actuator Delay and Saturation. *Aerospace* **2023**, *10*, 521.
https://doi.org/10.3390/aerospace10060521

**AMA Style**

Zhang S, Zhang H, Ji K.
Incremental Nonlinear Dynamic Inversion Attitude Control for Helicopter with Actuator Delay and Saturation. *Aerospace*. 2023; 10(6):521.
https://doi.org/10.3390/aerospace10060521

**Chicago/Turabian Style**

Zhang, Shaojie, Han Zhang, and Kun Ji.
2023. "Incremental Nonlinear Dynamic Inversion Attitude Control for Helicopter with Actuator Delay and Saturation" *Aerospace* 10, no. 6: 521.
https://doi.org/10.3390/aerospace10060521