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Limit cycle oscillation suppression controller design and stability analysis of the periodically time-varying flapping flight dynamics in hover

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Abstract

The periodically time-varying forces make the equilibrium state of Beihawk, an X-shaped flapping-wing aircraft, to be a periodic limit cycle oscillation. However, traditional controllers based on averaging theory fail to suppress this oscillation and the derived stability result may be inaccurate. In this study, a period-based method is proposed to design the oscillation suppression controller, locate the corresponding cycle and analyze its stability. A periodically time-varying wing-tail interaction model is built and Discrete Fourier Transform is applied to adapt the model for controller design. The harmonics less than quintuple flapping frequency account for more than 96 percent of the total harmonics and are reserved to present a concise model. Based on this model, active disturbance rejection controller (ADRC) is designed and its Extended State Observer can observe the disturbance to suppress the oscillation. Poincaré map is introduced to convert the stability analysis of the cycle to a fixed point. A multiple shooting method is adopted to locate several points on the cycle and the map is obtained by calculating the submaps between the adjacent points with the Floquet theory. The located points are proved to be accurate compared with the numerical solved cycle and the stability analysis result of the cycle is verified by the dynamic evolution. Compared with the State Feedback Controller, the ADRC performs better in suppressing the limit cycle oscillation and eliminating the attitude control error. The oscillation suppression is meaningful in maintaining a stable flight and capturing high quality images.

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Data availability

All data used to support the findings of this study are available from the corresponding author upon request.

Code availability

This work is done using custom code.

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Acknowledgements

The authors acknowledge the financial support from the National Natural Science Foundation of China under Grants 51905014. The authors would also like to thank the editors and reviewers for their critical review of this manuscript.

Funding

The research is supported by Innovative Research Group Project of the National Natural Science Foundation of China Grant No. 51905014

Author information

Authors and Affiliations

  1. School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, China

    Liang Wang, Wuyao Jiang & Zongxia Jiao

  2. Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing, 100191, China

    Liang Wang, Wuyao Jiang, Zongxia Jiao & Longfei Zhao

  3. Ningbo Institute of Technology, Beihang University, Ningbo, 315800, China

    Zongxia Jiao & Longfei Zhao

  4. Research Institute for Frontier Science, Beihang University, Beijing, 100191, China

    Longfei Zhao

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  1. Liang Wang
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Correspondence to Longfei Zhao.

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Appendices

Appendix

Algorithm description of the LMA: Solving the located points on the circle.

Initialization

Indicative values for the user-defined parameters: \(\tau = 10^{ - 3} ,\varepsilon_{1} = \varepsilon_{2} = \varepsilon_{3} = 10^{ - 15} ,k_{\max } = 100\).

Input: A target error function \(g(\hat{x})\) and an initial state vector \( \hat{x}_{0} \in R^{(m + 1)n}\).

Output: A vector \( \hat{x}^{ + } \in R^{(m + 1)n}\) minimizing \(\chi^{2} (\hat{x})\).

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Wang, L., Jiang, W., Jiao, Z. et al. Limit cycle oscillation suppression controller design and stability analysis of the periodically time-varying flapping flight dynamics in hover. Nonlinear Dyn 107, 3385–3405 (2022). https://doi.org/10.1007/s11071-021-07145-0

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