Advertisement

Guaranteed Cost Sliding Mode Control of Switched Systems with Known Sojourn Probabilities

  • Haijuan Zhao
  • Yugang Niu
Regular Papers Control Theory and Applications
  • 28 Downloads

Abstract

This paper investigates the problem of the optimal guaranteed cost sliding mode control (SMC) for a class of uncertain switched systems, in which the sojourn probabilities staying in each subsystem are available. By introducing a set of stochastic variables, a new type of switched system model with known sojourn probabilities is constructed. And then, the SMC law is synthesized such that the reachability of the specified common sliding surface can be ensured. Moreover, some sufficient conditions are derived to ensure the mean-square stability of the closed-loop system with quadratic guaranteed cost function. The optimal solution of the guaranteed SMC scheme is established for the closed-loop systems. Finally, a numerical example is provided to illustrate the proposed method.

Keywords

Guaranteed cost control sliding mode control sojourn probability switched systems 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Liberzon, Switching in Systems and Control, Springer Science & Business Media, 2012.zbMATHGoogle Scholar
  2. [2]
    L. Gurvits, R. Shorten, and O. Mason, “On the stability of switched positive linear systems,” IEEE Trans. on Automatic Control, vol. 52, no. 6, pp. 1099–1103, June 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Y. Li, Y. Sun, and F. Meng, “New criteria for exponential stability of switched time-varying systems with delays and nonlinear disturbances,” Nonlinear Analysis: Hybrid Systems, vol.26, pp. 284–291, November 2017.Google Scholar
  4. [4]
    H. Ren, G. Zong, and T. Li. “Event-triggered finite-time control for networked switched linear systems with asynchronous switching,” IEEE Trans. on Systems, Man, and Cybernetics: Systems, vol. PP, no. 99, pp. 1–11, January 2018.Google Scholar
  5. [5]
    X. Zhao, L. Zhang, P. Shi, and M. Liu, “Stability of switched positive linear systems with average dwell time switching,” Automatica, vol. 48, no. 6, pp. 1132–1137, June 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    H. Zhao, Y. Niu, and J. Song, “Finite-time output feedback control of uncertain switched systems via sliding mode design,” International Journal of Systems Science, vol. 49, no. 5, pp. 984–996, February 2018.MathSciNetCrossRefGoogle Scholar
  7. [7]
    L. Rodrigues and S. Boyd, “Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization,” Systems & Control Letters, vol. 54, no. 9, pp. 835–853, September 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Yuno and T. Ohtsuka, “A sufficient condition for the stability of discrete-time systems with state-dependent coefficient matrices,” IEEE Trans. on Automatic Control, vol. 59, no. 1, pp. 243–248, June 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    W. Qi, X. Kao, and X. Gao, “Further results on finite-time stabilisation for stochastic Markovian jump systems with time-varying delay,” International Journal of Systems Science, vol. 48, no. 14, pp. 2967–2975, August 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    W. Qi, X. Kao, and X. Gao, “Passivity and passification for stochastic systems with Markovian switching and generally uncertain transition rates,” International Journal of Control, Automation and Systems, vol. 15, no. 5, pp. 2174–2181, October 2017.CrossRefGoogle Scholar
  11. [11]
    E. Tian, W K. Wong, and D. Yue, “Robust H¥ control for switched systems with input delays: A sojourn-probabilitydependent method,” Information Sciences, vol. 283, pp. 22–35, November 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    E. Tian, W K. Wong, D. Yue, and T. C. Yang, “H¥ filtering for discrete-time switched systems with known sojourn probabilities measurements,” IEEE Trans. on Automatic Control, vol. 60, no. 9, pp. 2446–2451, March 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. Song, Y. Niu, J. Lam, and H. K. Lam, “Fuzzy remote tracking control for randomly varying local nonlinear models under fading and missing measurements,” IEEE Trans. on Fuzzy Systems, vol. 26, no. 3, pp. 1125–1137, May 2017.CrossRefGoogle Scholar
  14. [14]
    Y. Niu and D.W. Ho, “Design of sliding mode control subject to packet losses,” IEEE Trans. on Automatic Control, vol. 55, no. 11, pp. 2623–2628, August 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    H. Dong, Z. Wang, S X. Ding, and H. Gao, “Finite-horizon reliable control with randomly occurring uncertainties and nonlinearities subject to output quantization,” Automatica, vol. 52, pp. 355–362, February 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Kalidass, H. Su, Y. Q. Wu, and S. Rathinasamy, “H¥ filtering for impulsive networked control systems with random packet dropouts and randomly occurring nonlinearities,” International Journal of Robust and Nonlinear Control, vol. 25, no. 12, pp. 1767–1782, August 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A. Elahi and A. Alfi, “Finite-time H¥ control of uncertain networked control systems with randomly varying communication delays,” ISA Transactions, vol. 69, pp. 65–88, July 2017.CrossRefGoogle Scholar
  18. [18]
    B. Zheng and J. H. Park, “Sliding mode control design for linear systems subject to quantization parameter mismatch,” Journal of the Franklin Institute, vol. 353, no. 1, pp. 37–53, January 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    E. Tian and D. Yue, “A new state feedback H¥ control of networked control systems with time-varying network conditions,” Journal of the Franklin Institute, vol. 349, no. 3, pp. 891–914, April 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    R. Galván-Guerra, L. Fridman, J. E. Velázquez-Velázquez, S. Kamal, and B. Bandyopadhyay, “Continuous output integral sliding mode control for switched linear systems,” Nonlinear Analysis: Hybrid Systems, vol. 22, pp. 284–305, November 2016.MathSciNetzbMATHGoogle Scholar
  21. [21]
    X. Su, X. Liu, P. Shi, and R. Yang, “Sliding mode control of discrete-time switched systems with repeated scalar nonlinearities,” IEEE Trans. on Automatic Control, vol. 62, no. 9, pp. 4604–4610, November 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Y. Liu, Y. Niu, Y. Zou, and H. R. Karimi, “Adaptive sliding mode reliable control for switched systems with actuator degradation,” IET Control Theory & Applications, vol. 9, no. 8, pp. 1197–1204, May 2015.MathSciNetCrossRefGoogle Scholar
  23. [23]
    E. Hernandez-Vargas, P. Colaneri, R. Middleton, and F. Blanchini, “Discrete-time control for switched positive systems with application to mitigating viral escape,” International Journal of Robust and Nonlinear Control, vol. 21, no. 10, pp. 1093–1111, July 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    H. Trinh, “Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser model,” Nonlinear Analysis: Hybrid Systems, vol. 24, pp. 45–57, May 2017.MathSciNetzbMATHGoogle Scholar
  25. [25]
    G. Zong, X. Wang, and H. Zhao, “Robust finite-time guaranteed cost control for impulsive switched systems with time-varying delay,” International Journal of Control, Automation and Systems, vol. 15, no. 1, pp. 113–121, February 2017.CrossRefGoogle Scholar
  26. [26]
    J. Song, Y. Niu, and Y. Zou, “Asynchronous sliding mode control of Markovian jump systems with time-varying delays and partly accessible mode detection probabilities,” Automatica, vol. 93, pp. 33–41, July 2018.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Advanced Control and Optimization for Chemical Process (East China University of Science and Technology)Ministry of EducationShanghaiChina

Personalised recommendations