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Switched Singularly Perturbed Systems with Reliable Controllers

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Abstract

This paper addresses the problem of exponential stability for a class of switched control singularly perturbed systems (SCSPS) not only when all the control actuators are operational, but also when some of them experience failures. Multiple Lyapunov functions and average dwell-time switching signal approach are used to establish the stability criteria for the proposed systems. In this paper, we assume that a full access to all the system modes is available, though the mode-dependent, slow-state feedback controllers experience faulty actuators of an outage type. In the stability analysis, the system under study is viewed as an interconnected system that has been decomposed into isolated, lower order, slow and fast subsystems, and the interconnection between them. It has been observed that if the degree of stability of each isolated mode is greater than the interconnection between them, the interconnected mode is exponentially stable, and, then, the full order SCSPS is also exponentially stable for all admissible switching signals with average dwell-time. A numerical example with simulations is introduced to illustrate the validity of the proposed theoretical results.

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References

  1. Alwan, M.S., Liu, X.Z.: Stability of singularly perturbed switched systems with time delay and impulsive effects. Nonlinear Anal. 17, 4297–4308 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alwan, M.S., Liu, X.Z., Ingalls, B.: Exponential stability of singularly perturbed switched systems with time delay. Nonlinear Anal.: Hybrid Syst. 2 (3), 913–921 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched hybrid systems. IEEE Trans. Autom. Control 43 (4), 475–482 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheng, X.M, Gui, W.H., Gan, Z.J.: Robust reliable control for a class of time-varying uncertain impulsive systems. J. Cent. S. Univ. Technol. 12 (1), 199–202 (2005)

    Article  Google Scholar 

  5. Hespanha, J.P., Morse, A.S.: Stability of switched systems with average dwell-time. In: Proceedings of the 38th IEEE Conference on Decision and Control, vol. 3, Phoenix, pp. 2655–2660 (1999)

    Google Scholar 

  6. Kang, K.-I., Park, K.-S., Lim, J.-T.: Exponential stability of singularly perturbed systems with time delay and uncertainties. Int. J. Syst. Sci. 46 (1), 170–178 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River (2002)

    MATH  Google Scholar 

  8. Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic, London (1986)

    MATH  Google Scholar 

  9. Liberzon, D.: Switching in Systems and Control. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

  10. Liberzon, D., Morse, A.S.: Basic Problems is Stability and Design of Switched Systems. IEEE Control Syst. Mag. 19 (5), 59–70 (1999)

    Article  Google Scholar 

  11. Liu, X., Shen, X., Zhang, Y.: Exponential stability of singularly perturbed systems with time delay. Appl. Anal. 82 (2), 117–130 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Morse, S.: Supervisory control of families of linear set-point controllers, part 1: exact matching. IEEE Trans. Autom. Control 41 (10), 1413–1431 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Seo, C.J., Kim, B.K.: Robust and reliable H ∞ control for linear systems with parameter uncertainty and actuator failure. Automatica 32 (3), 465–467 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shorten, R., Wirth, F., Mason, O., Wulff, K., King, C.: Stability criteria for switched and hybrid systems. SIAM Rev. 49 (4), 545–592 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Veillette, R.J.: Reliable state feedback and reliable observers. In: Proceedings of the 31st Conference on Decision and Control, Tucson, pp. 2898–2903 (1992)

    Google Scholar 

  16. Veillette, R.J., Medanic, J.V., Perkins, W.R.: Design of reliable control systems. IEEE Trans. Autom. Control 37 (3), 290–304 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhai, G., Hu, B., Yasuda, K., Michel, A.N.: Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Proc. Am. Control Conf. 1 (6), 200–204 (2000)

    MATH  Google Scholar 

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Acknowledgements

This work was partially supported by NSERC Canada. The third author acknowledges the sponsorship of King Abdulaziz University, Saudi Arabia.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Mohamad S. Alwan, Xinzhi Liu & Taghreed G. Sugati

Authors
  1. Mohamad S. Alwan
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  2. Xinzhi Liu
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  3. Taghreed G. Sugati
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Corresponding author

Correspondence to Mohamad S. Alwan .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Montreal, Montreal, Québec, Canada

    Jacques Bélair

  2. Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada

    Ian A. Frigaard

  3. Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada

    Herb Kunze

  4. Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roman Makarov

  5. MS2Discovery Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick Melnik

  6. Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    Raymond J. Spiteri

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© 2016 Springer International Publishing Switzerland

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Alwan, M.S., Liu, X., Sugati, T.G. (2016). Switched Singularly Perturbed Systems with Reliable Controllers. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (eds) Mathematical and Computational Approaches in Advancing Modern Science and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30379-6_35

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