Advertisement

Sliding Modes for Switched Uncertain Linear Time Invariant Systems: An Output Integral Sliding Mode Approach

  • Leonid Fridman
  • Rosalba Galván-GuerraEmail author
  • Juan-Eduardo Velázquez-Velázquez
  • Rafael Iriarte
Chapter

Abstract

A robustifying methodology for switched uncertain linear time invariant systems with matched uncertainties/perturbations and state-dependent location transitions using only output information is presented. An output integral sliding mode control technique, based on an algebraic hierarchical observer is proposed. This approach allows the theoretically exact compensation of the matched uncertainties/perturbations right after the initial time; but it requires the use of filters to reconstruct the state vector and produces a high level of chattering. To eliminate the necessity of filtering and to diminish the chattering, a continuous output integral sliding mode controller is designed. This controller is based on the super-twisting algorithm and it compensates the matched uncertainties/perturbations after a finite transient. For this case, sufficient conditions to ensure the convergence of the controller and the observer before every switching are given. The proposed approach is illustrated via numerical simulations.

Notes

Acknowledgements

My co-authors and myself are happy to prepare this chapter for this book reflecting some new results developing the concept of output integral sliding mode that we proposed with you, Professor Alexander Poznyak, 10 years ago. Thank you for your creativity, kindness, and constant support. Happy birthday and long life.

Leonid Fridman would like to acknowledge the financial support of the project 113216 of PAPIIT-UNAM.

Leonid Fridman

References

  1. 1.
    Alessandri, A., Coletta, P.: Switching observers for continuous-time and discrete-time linear systems. In: American Control Conference, 2001. Proceedings of the 2001, vol. 3, pp. 2516–2521. IEEE, New York (2001)Google Scholar
  2. 2.
    Bainov, D., Simeonov, P.: Integral inequalities and applications. In: Mathematics and its Applications, vol. 57, 1 edn. Springer, Netherlands (1992)Google Scholar
  3. 3.
    Bejarano, F.J., Fridman, L.: State exact reconstruction for switched linear systems via a super-twisting algorithm. Int. J. Syst. Sci. 42(5), 717–724 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bejarano, F.J., Fridman, L., Poznyak, A.S.: Output integral sliding mode control based on algebraic hierarchical observer. Int. J. Control 80, 443–453 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bejarano, F.J., Pisano, A.: Switched observers for switched linear systems with unknown inputs. IEEE Trans. Autom. Control 56(3), 681–686 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bejarano, F.J., Pisano, A., Usai, E.: Finite-time converging jump observer for switched linear systems with unknown inputs. Nonlinear Anal. Hybrid Syst. 5(2), 174–188 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Branicky, M.S.: Introduction to hybrid systems. In: Handbook of Networked and Embedded Control Systems, pp. 91–116. Birkhäuser Boston (2005)Google Scholar
  8. 8.
    Caravani, P., De Santis, E.: Observer-based stabilization of linear switching systems. Int. J. Robust Nonlinear Control 19(14), 1541–1563 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chalanga, A., Kamal, S., Fridman, L., Bandyopadhyay, B., Moreno, J.A.: How to implement super-twisting controller based on sliding mode observer? In: 2014 13th International Workshop on Variable Structure Systems (VSS), pp. 1–6 (2014)Google Scholar
  10. 10.
    Dávila, J., Ríos, H., Fridman, L.: State observation for nonlinear switched systems using nonhomogeneous high-order sliding mode observers. Asian J. Control 14(4), 911–923 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    El-Farra, N.H., Mhaskar, P., Christofides, P.D.: Output feedback control of switched nonlinear systems using multiple lyapunov functions. Syst. Control Lett. 54(12), 1163–1182 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Floquet, T., Barbot, J.P.: Super twisting algorithm-based step-by-step sliding mode observers for nonlinear systems with unknown inputs. Int. J. Syst. Sci. 38(10), 803–815 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fridman, L., Poznyak, A., Bejarano, F.J.: Robust Output LQ Optimal Control via Integral Sliding Modes. Birkhäuser (2014)Google Scholar
  14. 14.
    Galván-Guerra, R., Fridman, L., Velázquez-Velázquez, J., Kamal, S., Bandyopadhyay, B.: Continuous output integral sliding mode control for switched linear systems. Nonlinear Anal. Hybrid Syst. 22, 284–305 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hautus, M.: Strong detectability and observers. Linear Algebra Appl. 50, 353–368 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hespanha, J.P., Morse, A.: Switching between stabilizing controllers. Automatica 38(11), 1905–1917 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Levant, A.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 58(6), 1247–1263 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, Z., Wen, C., Soh, Y.: Observer-based stabilization of switching linear systems. Automatica 39(3), 517–524 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lian, J., Zhao, J., Dimirovski, G.M.: Integral sliding mode control for a class of uncertain switched nonlinear systems. Eur. J. Control 16(1), 16–22 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mohrenschildt, M.V.: Hybrid systems: solutions, stability, control. In: American Control Conference (ACC), vol. 1, pp. 692 –698 (2000)Google Scholar
  21. 21.
    Polyakov, A., Poznyak, A.: Reaching time estimation for super-twisting second order sliding mode controller via lyapunov function designing. IEEE Trans. Autom. Control 54(8), 1951–1955 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ríos, H., Kamal, S., Fridman, L., Zolghadri, A.: Fault tolerant control allocation via continuous integral sliding-modes: a hosm-observer approach. Automatica 51, 318–325 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rugh, W.J.: Linear System Theory. Prentice Hall, Upper Saddle River (1993)Google Scholar
  24. 24.
    Saadaoui, H., Manamanni, N., Djemaï, M., Barbot, J., Floquet, T.: Exact differentiation and sliding mode observers for switched lagrangian systems. Nonlinear Anal. Theory, Methods Appl. 65(5), 1050–1069 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Springer, Berlin (2000)Google Scholar
  26. 26.
    Shaikh, M.S., Caines, P.E.: On the hybrid optimal control problem: theory and algorithms. IEEE Trans. Autom. Control 52(9), 1587–1603 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, Berlin (1992)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Leonid Fridman
    • 1
  • Rosalba Galván-Guerra
    • 2
    Email author
  • Juan-Eduardo Velázquez-Velázquez
    • 2
  • Rafael Iriarte
    • 1
  1. 1.Facultad de IngenieríaUniversidad Nacional Autonoma de MéxicoMexico CityMexico
  2. 2.Unidad Profesional Interdisciplinaria de Ingeniería Campus HidalgoInstituto Politécnico NacionalHidalgoMexico

Personalised recommendations