Skip to main content
SpringerLink
Log in

Fixed-time stabilization of underactuated cart-pendulum system based on hierarchical sliding mode control method

  • Research
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A cart-pendulum system is a typical underactuated mechanical system with two degrees of freedom and one control input. The fixed-time stabilization control problem for this system is studied in this paper. A fixed-time hierarchical sliding mode control method is developed to solve this problem. Firstly, the system is divided into two subsystems. And two first-level sliding mode surfaces are designed for these two subsystems respectively. Secondly, a second-level sliding mode surface is constructed by linearly combining two first-level sliding mode surfaces. After that, a hierarchical sliding mode controller is designed based on the designed sliding mode surfaces. It enables the state variables of closed-loop control system to be stabilized at the origin within a fixed-time. This means that the settling time can be limited within a range and does not depend on the initial state value of the system. Finally, a numerical example is given to show the effectiveness of the developed control method.

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

Similar content being viewed by others

Data availability

The readers can access the data reported in this paper from the corresponding author.

References

  1. Yang, T., Sun, N., Fang, Y.: Neuroadaptive control for complicated underactuated systems with simultaneous output and velocity constraints exerted on both actuated and unactuated states. IEEE Trans. Neural Netw. Learn. Syst. 34(8), 4488–4498 (2021)

    Article  MathSciNet  Google Scholar 

  2. Yang, T., Chen, H., Sun, N., Fang, Y.: Adaptive neural network output feedback control of uncertain underactuated systems with actuated and unactuated state constraints. IEEE Trans. Syst. Man Cybern. Syst. 52(11), 7027–7043 (2021)

    Article  Google Scholar 

  3. Ye, H.: Stabilization of uncertain feedforward nonlinear systems with application to underactuated systems. IEEE Trans. Autom. Control 64(8), 3484–3491 (2019)

    Article  MathSciNet  Google Scholar 

  4. Zhang, A., She, J., Qiu, J., Yang, C., Alsaadi, F.: Design of motion trajectory and tracking control for underactuated cart-pendulum system. Int. J. Robust Nonlinear Control 29(8), 2458–2470 (2019)

    Article  MathSciNet  Google Scholar 

  5. Lee, J., Mukherjee, R., Khalil, H.: Output feedback stabilization of inverted pendulum on a cart in the presence of uncertainties. Automatica 54, 146–157 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bettayeb, M., Boussalem, C., Mansouri, R., Al-Saggaf, U.: Stabilization of an inverted pendulum-cart system by fractional PI-state feedback. ISA Trans. 53(2), 508–516 (2014)

    Article  Google Scholar 

  7. She, J., Zhang, A., Lai, X.: Global stabilization of 2-DOF underactuated mechanical systems - an equivalent-input-disturbance approach. Nonlinear Dyn. 69(1–2), 495–509 (2012)

    Article  Google Scholar 

  8. Kumar, E., Jerome, J.: Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum. Procedia Engineering 100(64), 69–178 (2013)

    Google Scholar 

  9. Yang, T., Sun, N., Fang, Y.: Adaptive fuzzy control for a class of MIMO underactuated systems with plant uncertainties and actuator deadzones: design and experiments. IEEE Trans. Cybern. 52(8), 8213–8226 (2021)

    Article  Google Scholar 

  10. Park, M., Chwa, D.: Orbital stabilization of inverted-pendulum systems via coupled sliding-mode control. IEEE Trans. Industr. Electron. 56(9), 3556–3570 (2009)

    Article  Google Scholar 

  11. Jiang, B., Wu, Z., Karimi, H.: A distributed dynamic event-triggered mechanism to HMM-based observer design for \(H\infty \) sliding mode control of Markov jump systems. Automatica 142, 110357 (2022)

  12. Moulay, E., Lechappe, V., Bernuau, E., Plestan, F.: Robust fixed-time stability: application to sliding-mode control. IEEE Trans. Autom. Control 67(2), 1061–1066 (2022)

    Article  MathSciNet  Google Scholar 

  13. Pan, Y., Yang, C., Pan, L., Yu, H.: Integral sliding mode control: performance, modification, and improvement. IEEE Trans. Industr. Inf. 14(7), 3087–3096 (2017)

    Article  Google Scholar 

  14. Tang, Y.: Terminal sliding mode control for rigid robots. Automatica 34(1), 51–56 (1998)

    Article  MathSciNet  Google Scholar 

  15. Mobayen, S.: Fast terminal sliding mode controller design for nonlinear second-order systems with time-varying uncertainties. Complexity 21(2), 239–244 (2015)

    Article  MathSciNet  Google Scholar 

  16. Su, Y., Zheng, C.: A new nonsingular integral terminal sliding mode control for robot manipulators. Int. J. Syst. Sci. 51(8), 1418–1428 (2020)

    Article  MathSciNet  Google Scholar 

  17. Wang, W., Yi, J., Yi, D., Liu, D.: Design of a stable sliding-mode controller for a class of second-order underactuated systems. IET Control Theory Appl. 151(6), 683–690 (2004)

    Article  Google Scholar 

  18. Wang, W., Liu, X., Yi, J.: Structure design of two types of sliding-mode controllers for a class of under-actuated mechanical systems. IET Control Theory Appl. 1(1), 163–172 (2007)

    Article  MathSciNet  Google Scholar 

  19. Qian, D., Yi, J., Zhao, D.: Hierarchical sliding mode control for a class of SIMO under-actuated systems. Control. Cybern. 37(1), 159–175 (2008)

    MathSciNet  Google Scholar 

  20. Singh, A., Ha, Q.: Fast terminal sliding control application for second-order underactuated systems. Int. J. Control Autom. Syst. 17, 1884–1898 (2019)

    Article  Google Scholar 

  21. Pham, D., Lee, S.: Hierarchical sliding mode control for a two-dimensional ball segway that is a class of a second-order underactuated system. J. Vib. Control 25(1), 72–83 (2019)

    Article  MathSciNet  Google Scholar 

  22. Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57(8), 2106–2110 (2011)

    Article  MathSciNet  Google Scholar 

  23. Bhat, S., Bernstein, D.: Finite-time stability of continuous autonomous systems. SIAM J. Control. Optim. 38(3), 751–766 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Taishan Scholar Program of Shandong Province under Grant No. tsqn202211240, the Natural Science Foundation Program of Shandong Province under Grant No. ZR2019YQ28, the Development Plan of Youth Innovation Team of University in Shandong Province under Grant No. 2019KJN007, the Postgraduate Education Reform Project of Shandong Province,Grant/Award Number: SDYJG21045.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. School of Automation and Electrical Engineering, Linyi University, Linyi, 276000, People’s Republic of China

    Lu Fan, Ancai Zhang, Xiao Liang, Xinghui Zhang & Jianlong Qiu

Authors
  1. Lu Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ancai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xinghui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jianlong Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ancai Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no Conflict of interest

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, L., Zhang, A., Liang, X. et al. Fixed-time stabilization of underactuated cart-pendulum system based on hierarchical sliding mode control method. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09650-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11071-024-09650-4

Keywords

Search

Navigation