Skip to main content
SpringerLink
Log in

Multi-variable adaptive high-order sliding mode quasi-optimal control with adjustable convergence rate for unmanned helicopters subject to parametric and external uncertainties

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper presents a novel multi-variable high-order sliding mode quasi-optimal control method for unmanned helicopters. In order to facilitate the theoretical design and engineering implementation, the control system is divided into attitude and position subsystem, and the latter is further subdivided into two parts, horizontal and vertical position control. Then the multi-variable adaptive high-order continuous sliding mode controllers are designed for attitude and position, respectively, based on integral sliding mode surface. A new quadratic performance index is proposed in the design process, which enables the control system to converge in finite time, and the convergence rate can be adjusted by control parameters. Finally, the effectiveness and robustness of the proposed method are verified and compared with existing literature by simulation and practical experiments. The comparison results show that the proposed method has higher tracking accuracy and better robustness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Data Availibility Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Ahmed, B., Pota, H.R., Garratt, M.: Flight control of a rotary wing uav–a practical approach. In: IEEE Conference on Decision and Control (2008)

  2. Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 17(2), 101–127 (2005)

    Article  MathSciNet  Google Scholar 

  3. Bin, X., Jianchuan, G., Yao, Z.: Adaptive backstepping tracking control of a 6-dof unmanned helicopter. IEEE/CAA J. Autom. Sinica 2(1), 19–24 (2015)

    Article  MathSciNet  Google Scholar 

  4. Cisneros, P.S.G., Hoffmann, C., Bartels, M., Werner, H.: Linear parameter-varying controller design for a nonlinear quad-rotor helicopter model for high speed trajectory tracking. In: 2016 American Control Conference (ACC), pp. 486–491 (2016)

  5. Ding, S., Mei, K., Yu, X.: Adaptive second-order sliding mode control: a lyapunov approach. IEEE Trans. Autom. Control (2021). https://doi.org/10.1109/TAC.2021.3115447

    Article  Google Scholar 

  6. El-Ferik, S., Syed, A.H., Omar, H.M., Deriche, M.A.: Nonlinear forward path tracking controller for helicopter with slung load. Aerosp. Sci. Technol. 69, 602–608 (2017)

    Article  Google Scholar 

  7. Fang, X., Wu, A., Shang, Y., Dong, N.: A novel sliding mode controller for small-scale unmanned helicopters with mismatched disturbance. Nonlinear Dyn 83(1), 1053–1068 (2016)

    Article  MathSciNet  Google Scholar 

  8. Fang, X., Wu, A., Shang, Y., Dong, N.: Robust control of small-scale unmanned helicopter with matched and mismatched disturbances. J. Franklin Inst. 353(18), 4803–4820 (2016)

    Article  MathSciNet  Google Scholar 

  9. Fang, Z., Tian, H., Li, P.: Probabilistic robust linear parameter-varying control of a small helicopter using iterative scenario approach. IEEE/CAA J. Autom. Sinica 2(1), 85–93 (2015)

    Article  MathSciNet  Google Scholar 

  10. Filippov, A.F.: Differential Equations with Discontinuous Right-hand Side. Kluwer, Dordrecht, The Netherlands (1998)

    MATH  Google Scholar 

  11. Frost, W., Turner, R.E.: A discrete gust model for use in the design of wind energy conversion systems. J. Appl. Meteorol. 21(6), 770–776 (1982)

    Article  Google Scholar 

  12. Godbolt, B., Vitzilaios, N.I., Lynch, A.F.: Experimental validation of a helicopter autopilot design using model-based pid control. Journal of Intelligent and Robotic Systems 70(1), 385–399 (2013)

    Article  Google Scholar 

  13. Halbe, O., Hajek, M.: Robust helicopter sliding mode control for enhanced handling and trajectory following. Journal of Guidance, Control, and Dynamics 43(10), 1805–1821 (2020)

    Article  Google Scholar 

  14. He, Y., Pei, H., Sun, T.: Robust tracking control of helicopters using backstepping with disturbance observers. Asian Journal of Control 16(5), 1387–1402 (2015)

    Article  Google Scholar 

  15. Hua, C., Chen, J., Guan, X.: Fractional-order sliding mode control of uncertain quavs with time-varying state constraints. Nonlinear Dynamics 95(2), 1347–1360 (2019)

    Article  Google Scholar 

  16. Huang, Y., Zhu, M., Zheng, Z., Feroskhan, M.: Fixed-time autonomous shipboard landing control of a helicopter with external disturbances. Aerospace Science and Technology 84, 18–30 (2019)

    Article  Google Scholar 

  17. Jafar, A., Bhatti, A.I., Ahmad, S., Ahmed, N.: Robust gain-scheduled linear parameter-varying control algorithm for a lab helicopter: A linear matrix inequality-based approach. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 232(5), 558–571 (2018)

    Google Scholar 

  18. Jiang, T., Lin, D., Song, T.: Finite-time control for small-scale unmanned helicopter with disturbances. Nonlinear Dynamics 96(3), 1747–1763 (2019)

    Article  Google Scholar 

  19. Kumar, M.V., Omkar, S.N., Suresh, S., Sampath, P., Ganguli, R.: Design of a stability augmentation system for a helicopter using lqr control and ads-33 handling qualities specifications. Aircraft Engineering and Aerospace Technology 80(2), 111–123 (2008)

    Article  Google Scholar 

  20. Liu, H., Derawi, D., Kim, J., Zhong, Y.: Robust optimal attitude control of hexarotor robotic vehicles. Nonlinear Dynamics 74(4), 1155–1168 (2013)

    Article  Google Scholar 

  21. Lu, H., Liu, C., Guo, L., Chen, W.H.: Flight control design for small-scale helicopter using disturbance-observer-based backstepping. Journal of Guidance, Control, and Dynamics 38(11), 2235–2240 (2015)

    Article  Google Scholar 

  22. Lv, M., De Schutter, B., Shi, C., Baldi, S.: Logic-based distributed switching control for agents in power-chained form with multiple unknown control directions. Automatica 137, 110,143 (2022). https://doi.org/10.1016/j.automatica.2021.110143

  23. Lv, M., Yu, W., Cao, J., Baldi, S.: Consensus in high-power multiagent systems with mixed unknown control directions via hybrid nussbaum-based control. IEEE Transactions on Cybernetics (2020). https://doi.org/10.1109/TCYB.2020.3028171

    Article  Google Scholar 

  24. Lv, M., Yu, W., Cao, J., Baldi, S.: A separation-based methodology to consensus tracking of switched high-order nonlinear multiagent systems. IEEE transactions on neural networks and learning systems (2021). https://doi.org/10.1109/TNNLS.2021.3070824

    Article  Google Scholar 

  25. Mahmoud, M.S., Koesdwiady, A.B.: Improved digital tracking controller design for pilot-scale unmanned helicopter. Journal of the Franklin Institute 349(1), 42–58 (2012)

    Article  MathSciNet  Google Scholar 

  26. Maqsood, H., Qu, Y.: Nonlinear disturbance observer based sliding mode control of quadrotor helicopter. Journal of Electrical Engineering & Technology 15(3), 1453–1461 (2020)

    Article  Google Scholar 

  27. Masajedi, P., Ghanbarzadeh, A.: Optimal controller designing based on linear quadratic regulator technique for an unmanned helicopter at hover with the presence of wind disturbance. International Journal of Dynamics and Control 1(3), 214–222 (2013)

    Article  Google Scholar 

  28. Raptis, I.A., Valavanis, K.P., Moreno, W.A.: A novel nonlinear backstepping controller design for helicopters using the rotation matrix. IEEE Transactions on Control Systems Technology 19(2), 465–473 (2011)

    Article  Google Scholar 

  29. Sandino, L.A., Bejar, M., Kondak, K., Ollero, A.: Advances in modeling and control of tethered unmanned helicopters to enhance hovering performance. Journal of Intelligent & Robotic Systems 73(1–4), 3–18 (2014)

    Article  Google Scholar 

  30. Thomas, F., Thottungal, A.V., Johnson, M.S.: Composite control of a hovering helicopter based on optimized sliding mode control. Journal of Optimization Theory and Applications pp. 1–20 (2021)

  31. Tijani, I.B., Akmeliawati, R., Legowo, A., Budiyono, A.: robust controller for autonomous helicopter hovering control. Aircraft Engineering & Aerospace Technology 87(4), 330–344 (2015)

    Article  Google Scholar 

  32. Tijani, I.B., Akmeliawati, R., Legowo, A., Budiyono, A., Muthalif, A.A.: robust controller for autonomous helicopter hovering control. Aircraft Engineering & Aerospace Technology 83(6), 363–374 (2011)

    Article  Google Scholar 

  33. Wang, B., Shen, Y., Zhang, Y.: Active fault-tolerant control for a quadrotor helicopter against actuator faults and model uncertainties. Aerospace Science and Technology 99, 105,745 (2020)

  34. Wang, D., Zong, Q., Tian, B., Lu, H., Wang, J.: Adaptive finite-time reconfiguration control of unmanned aerial vehicles with a moving leader. Nonlinear Dynamics 95(2), 1099–1116 (2019)

    Article  Google Scholar 

  35. Wang, H.Q., Mian, A.A., Wang, D.B., Duan, H.B.: Robust multi-mode flight control design for an unmanned helicopter based on multi-loop structure. International Journal of Control Automation & Systems 7(5), 723–730 (2009)

    Article  Google Scholar 

  36. Wang, T., Yang, C., Liang, J., Wu, Y., Wang, C., Zhang, Y.: Chaos-genetic algorithm for the system identification of a small unmanned helicopter. Journal of Intelligent & Robotic Systems 67(3–4), 323–338 (2012)

    Article  Google Scholar 

  37. Yang, J.H., Hsu, W.C.: Adaptive backstepping control for electrically driven unmanned helicopter. Control Engineering Practice 17(8), 903–913 (2009)

    Article  Google Scholar 

  38. Yu, X., Yang, J., Li, S.: Disturbance observer-based autonomous landing control of unmanned helicopters on moving shipboard. Nonlinear Dynamics 102(1), 131–150 (2020)

  39. Zhao, Z., Cao, D., Yang, J., Wang, H.: High-order sliding mode observer-based trajectory tracking control for a quadrotor uav with uncertain dynamics. Nonlinear Dynamics 102(4), 2583–2596 (2020)

    Article  Google Scholar 

  40. Zhong, Y.: Optimal Control. Tsinghua University Press, Beijing (2015)

    Google Scholar 

  41. Zhou, B., Li, Z., Zheng, Z., Tang, S.: Nonlinear adaptive tracking control for a small-scale unmanned helicopter using a learning algorithm with the least parameters. Nonlinear Dynamics 89(2), 1289–1308 (2017)

    Article  Google Scholar 

  42. Zhou, B., Lu, X., Tang, S., Zheng, Z.: Nonlinear system identification and trajectory tracking control for a flybarless unmanned helicopter: theory and experiment. Nonlinear Dynamics 96(4), 2307–2326 (2019)

    Article  Google Scholar 

  43. Zhu, B., Huo, W.: 3-d path-following control for a model-scaled autonomous helicopter. IEEE Transactions on Control Systems Technology 22(5), 1927–1934 (2014)

    Article  Google Scholar 

  44. Zou, Y., Zheng, Z.: A robust adaptive rbfnn augmenting backstepping control approach for a model-scaled helicopter. IEEE Transactions on Control Systems Technology 23(6), 2344–2352 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

The author of this paper owes great thanks to Mr. Xuanying Li for his disinterested assistance in outdoor flight experiments. Moreover, the author would like to express sincere gratitude to the Editor and the anonymous reviewers whose insightful comments have helped to improve the quality of this paper considerably.

Funding

The authors have no relevant financial or non-financial interests to disclose. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.The authors have no financial or proprietary interests in any material discussed in this article. The authors have no financial or proprietary interests in any material discussed in this article.

Author information

Authors and Affiliations

  1. Unmanned Systems Technology Research Center, National Innovation Institute Defense Technology, Beijing, 100081, People’s Republic of China

    Bin Zhou

Authors
  1. Bin Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Zhou.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Lemma 4

([40]). Assuming that \(\left( \varvec{A}_\mathrm {n} , \varvec{B}_\mathrm {n}\right) \) is stabilized and \(\left( \varvec{A}_\mathrm {n} , \varvec{D}\right) \) is detectable, then the linear time-invariant infinite time LQR optimal control is asymptotically stable. If \(\left( \varvec{A}_\mathrm {n} , \varvec{D}\right) \) is observable, then the solution \( \bar{\varvec{P}} \) of the Riccati matrix algebraic equation (117) is positive definite.

$$\begin{aligned} \bar{\varvec{P}} \varvec{A}_\mathrm {n} + \varvec{A}_\mathrm {n} ^\mathrm {T} \bar{\varvec{P}} + \varvec{Q}- \bar{\varvec{P}}\varvec{B}_\mathrm {n}{\varvec{R}^{ - 1}}{\varvec{B}_\mathrm {n}^\mathrm {T}} \bar{\varvec{P}} = \varvec{0} \end{aligned}$$
(117)

\(\square \)

Proof of Lemma 2

Since matrix \(\left( \varvec{A}_\mathrm {n}+ \delta {{{\varvec{I}}_{ln \times ln}}}, \varvec{B}_\mathrm {n}\right) \) is stabilized and \(\left( \varvec{A}_\mathrm {n}+ \delta {{{\varvec{I}}_{ln \times ln}}}, \varvec{D} \right) \) is detectable, according to Lemma 4, for the linear time-invariant infinite time LQR problem described by system (36) and performance index (37), the optimal control is given by equation (38), then the system (36) is asymptotically stable. In other words, \({\bar{\varvec{z}}}(t)\) is bounded and asymptotically approaches zero. Thus,

$$\begin{aligned} \varvec{z}(t)= e^{-\delta t} {\bar{\varvec{z}}}(t) \rightarrow 0, \quad t \rightarrow \infty \end{aligned}$$

That’s to say, the state \( {{\varvec{z}}}(t) \) converges to the origin at a rate no less than the exponential function \(e^{-\delta t} \). \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, B. Multi-variable adaptive high-order sliding mode quasi-optimal control with adjustable convergence rate for unmanned helicopters subject to parametric and external uncertainties. Nonlinear Dyn 108, 3671–3692 (2022). https://doi.org/10.1007/s11071-022-07433-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07433-3

Keywords

Search

Navigation