Skip to main content
SpringerLink
Log in

Min-Max Sliding-Mode Control

  • Chapter
The Robust Maximum Principle

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

Abstract

This chapter deals with the Min-Max Sliding-Mode Control design where the original linear time-varying system with unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances. Such a trade-off between an original uncertain linear time-varying dynamic system and a corresponding higher order multimodel system with complete knowledge leads to a linear multimodel system with known bounded disturbances. Each model from a given finite set is characterized by a quadratic performance index. The developed Min-Max Sliding-Mode Control strategy gives an optimal robust sliding-surface design algorithm, which is reduced to a solution of the equivalent LQ Problem that corresponds to the weighted performance indices with weights from a finite-dimensional simplex. An illustrative numerical example is presented.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aubin, J. (1979), Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Boltyanski, V., & Poznyak, A. (1999b), ‘Robust maximum principle in minimax control’, Int. J. Control 72(4), 305–314.

    Article  MathSciNet  Google Scholar 

  • Boltyanski, V.G., & Poznyak, A.S. (2002a), ‘Linear multi-model time optimization’, Optim. Control Appl. Methods 23, 141–161.

    Article  MathSciNet  Google Scholar 

  • Boltyanski, V.G., & Poznyak, A.S. (2002b), ‘Robust maximum principle for a measured space as uncertainty set’, Dyn. Syst. Appl. 11, 277–292.

    MathSciNet  MATH  Google Scholar 

  • Dorling, C., & Zinober, A. (1986), ‘Two approaches to hyperplane design in multivariable variable structure control systems’, Int. J. Control 44, 65–82.

    Article  Google Scholar 

  • Edwards, C., & Spurgeon, S. (1998), Sliding Mode Control: Theory and Applications, Taylor and Francis, London.

    MATH  Google Scholar 

  • Fridman, L. (2002), ‘Sliding mode control for systems with fast actuators: Singularity perturbed approach’, in Variable Structure Systems: Towards the 21st Century (X. Yu & J.X. Xu, eds.), Vol. 274 of Lecture Notes in Control and Information Science, Springer, London, pp. 391–415.

    Chapter  Google Scholar 

  • Fridman, L., & Levant, A. (2002), ‘Higher order sliding modes’, in Sliding Mode Control in Engineering (W. Perruquetti & J.P. Barbot, eds.), Vol. 11 of Control Engineering Series, Dekker, New York, pp. 53–102.

    Chapter  Google Scholar 

  • Khalil, H. (1980), ‘Approximation of Nash strategies’, IEEE Trans. Autom. Control AC-25(2), 247–250.

    Article  MathSciNet  Google Scholar 

  • Poznyak, A., Duncan, T., Pasik-Duncan, B., & Boltyanski, V. (2002a), ‘Robust maximum principle for multi-model LQ-problem’, Int. J. Control 75(15), 1770–1777.

    Article  MathSciNet  Google Scholar 

  • Poznyak, A., Shtessel, Y., & Jimenez, C. (2003), ‘Min-max sliding-mode control for multimodel linear time varying systems’, IEEE Trans. Autom. Control 48(12), 2141–2150.

    Article  MathSciNet  Google Scholar 

  • Shtessel, Y. (1996), ‘Principle of proportional damages in multiple criteria LQR problem’, IEEE Trans. Autom. Control 41(3), 461–464.

    Article  MathSciNet  Google Scholar 

  • Shtessel, Y., Hall, C., & Jackson, M. (2000), ‘Reusable launch vehicle control in multiple time-scale sliding modes’, J. Guid. Control Dyn. 23(6), 1013–1020.

    Article  Google Scholar 

  • Tam, H.K., Ho, D.W.C., & Lam, J. (2002), ‘Robust hyperplane synthesis for sliding mode control systems via sensitivity minimization’, Optim. Control Appl. Methods 23, 125–139.

    Article  MathSciNet  Google Scholar 

  • Utkin, V. (1991), Slides Modes in Control and Optimization, Springer, Berlin.

    Google Scholar 

  • Utkin, V., Guldner, J., & Shi, J. (1999), Sliding Mode Control in Electromechanical Systems, Taylor and Francis, London.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CIMAT, Jalisco S/N, Col. Valenciana, 36240, Guanajuato, GTO, Mexico

    Vladimir G. Boltyanski

  2. Automatic Control Department, CINVESTAV-IPN, AP-14-740 Av. IPN-2508, Col. San Pedro Zacatenco, 07000, México, D.F., Mexico

    Alexander S. Poznyak

Authors
  1. Vladimir G. Boltyanski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander S. Poznyak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir G. Boltyanski .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Boltyanski, V.G., Poznyak, A.S. (2012). Min-Max Sliding-Mode Control. In: The Robust Maximum Principle. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8152-4_13

Download citation

Publish with us

Policies and ethics

Search

Navigation