Skip to main content
SpringerLink
Log in

On Robust, Multi-Input Sliding-Mode Based Control with a State-Dependent Boundary Layer

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper proposes a nonlinear multi-input control law using sliding mode concepts for continuous-time, uncertain, linear systems. The control law introduces a state-dependent layer around the sliding mode plane to remove chattering. This layer combines two types of boundary layers: a constant layer and a sector-shaped layer. The states will always enter the state-dependent boundary layer and the choice of the sliding mode will be seen to determine the ultimate system performance. A proof of stability shows ultimate boundedness. The controller is applied to a nonlinear simulation model of a cart-pendulum and exhibits a high degree of robustness. The new boundary layer in connection with a novel dynamically changing, state-dependent gain can be used to obtain a narrow boundary-layer shape in the operating region of interest. This permits rejection of disturbances without chattering of the control and improves on the performance expected of a sliding-mode control with constant boundary layer.

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.

Similar content being viewed by others

References

  1. 1. Ambrosino, G., Celantano, G., and Garofalo, F., Variable Structure Model Reference Adaptive Control Systems, International Journal of Control, Vol. 39, No. 6, pp. 1339–1349, 1984.

    Article  MATH  Google Scholar 

  2. 2. Ryan, E., and Corless, M., Ultimate Boundedness and Asymptotic Stability of a Class of Uncertain Dynamical Systems via Continuous and Discontinuous Feedback Control, IMA Journal of Mathematical Control and Information, Vol. 1, No. 3, pp. 223–242, 1984.

    Article  MATH  Google Scholar 

  3. 3. Burton, J. A., and Zinober, A. S. I., Continuous Approximation of Variable Structure Control, International Journal of Systems Science, Vol. 17, No. 6, pp. 875–885, 1986.

    Article  MATH  Google Scholar 

  4. 4. Slotine, J. J., and Li, W., Applied Nonlinear Control, Prentice-Hall International Editions, London, UK, 1991.

    MATH  Google Scholar 

  5. 5. Spurgeon, S. K., and Davies, R., A Nonlinear Control Strategy for Robust Sliding Mode Performance in the Presence of Unmatched Uncertainty, International Journal of Control, Vol. 57, No. 5, pp. 1107–1123, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  6. 6. Utkin, V., Sliding Modes in Control Optimization. Springer Verlag, New York, NY, 1992.

    MATH  Google Scholar 

  7. 7. Levant, A., Higher-Order Sliding: Collection of Design Tools, Proceedings of the European Control Conference, Brussels, Belgium, File ECC766, 1997.

  8. 8. Bartolini, G., Ferrara, A., Usai, E., and Utkin, V., On Multi-Input Chattering-Free Second-Order Sliding-Mode Control, IEEE Transactions on Automatic Control, Vol. 45, No. 9, pp. 1711–1717, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  9. 9. Van De Wal, M., De Jager, B., and Veldpaus, F., The Slippery Road to Sliding Control: Conventional versus Dynamical Sliding-Mode Control, International Journal of Robust and Nonlinear Control, Vol. 8, No. 6, pp. 535–549, 1998.

    Article  MATH  Google Scholar 

  10. 10. Dejager, B., Comparison of Methods to Eliminate Chattering and Avoid Steady State Errors in Sliding-Mode Control, Proceedings of the IEEE International Workshop on Variable Structure and Lyapunov Control of Uncertain Dynamical Systems, Sheffield, UK, pp. 37–42, 1992.

  11. 11. Furuta, K., and Morisada, M., Sliding-Mode Control of Discrete Systems, Transactions of the Society of Instrument and Control Engineers, Vol. 25, No. 5, pp. 574–578, 1989.

    Google Scholar 

  12. 12. Wang, W. J., Lee, R. C., and Yang, D. C., Sliding-Mode Control Design in Multi-Input Perturbed Discrete-Time Systems, Journal of Dynamic Systems, Measurements, and Control, Vol. 118, No. 2, pp. 322–327, 1996.

    Article  MATH  Google Scholar 

  13. 13. Furuta, K., and Pan, Y., Variable Structure Control with Sliding Sector, Automatica, Vol. 36, No. 2, pp. 211–228, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  14. 14. Herrmann, G., Spurgeon, S. K., and Edwards, C., A New Nonlinear, Continuous-Time Control Law Using Sliding-Mode Control Approaches, Proceedings of the 5th International Workshop on Variable Structure Systems, Longboat Key, Florida, pp. 50–56, 1998.

  15. 15. Herrmann, G., Spurgeon, S. K., and Edwards, C., On Sliding-Mode Based Control via Cone-Shaped Boundary Layers, Proceedings of the 15th IFAC World Congress on Automatic Control, Barcelona, Spain, File 2014, 2002.

  16. 16. Chen, M. S., Hwang, Y. R., and Tomizuka, M., A State-Dependent Boundary Layer Design for Sliding-Mode Control, IEEE Transactions on Automatic Control, Vol. 47, No. 10, pp. 1677–1681, 2002.

    Article  MathSciNet  Google Scholar 

  17. 17. Boyd, S., EL Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, Pennsylvania, 1994.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester, UK

    G. Herrmann (Research Associate), S. K. Spurgeon (Professor) & C. Edwards (Senior Lecturer)

Authors
  1. G. Herrmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. K. Spurgeon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Edwards
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Simaan

The first author would like to acknowledge the support from the European Commission TMR Grant, Project FMBICT983463.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herrmann, G., Spurgeon, S.K. & Edwards, C. On Robust, Multi-Input Sliding-Mode Based Control with a State-Dependent Boundary Layer. J Optim Theory Appl 129, 89–107 (2006). https://doi.org/10.1007/s10957-006-9045-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-006-9045-4

Keywords

Search

Navigation