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Exponential Stability and Asynchronous Stabilization of Nonlinear Impulsive Switched Systems via Switching Fuzzy Lyapunov Function Approach

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Abstract

In this paper, the Takagi–Sugeno (T-S) fuzzy model is first used to deal with the exponential stability and asynchronous stabilization problem of a class of continuous-time nonlinear impulsive switched systems with asynchronous behaviors. In order to reduce the conservativeness resulting from the quadratic Lyapunov functions (QLFs) and nonlinearity, the switching fuzzy Lyapunov functions (FLFs) are proposed using the switching information and structural information of membership function in the rule base. Using the switching FLFs approach and the mode-dependent average dwell time (MDADT) technique, we obtain stability conditions for the open-loop nonlinear impulsive switched systems and stabilization conditions for the closed-loop nonlinear impulsive switched systems. Moreover, the stability and stabilization results are formulated in the form of LMIs. Finally, a numerical example and a chemical process example are given to demonstrate the advantage and applicability of the proposed method.

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References

  1. Liberzon, D.: Switching in Systems and Control, Berlin. Birkhauser, Germany (2003)

    Book  MATH  Google Scholar 

  2. Ye, H., Michel, A.N., Hou, L.: Stability theory for hybrid dynamical systems. IEEE Trans. Autom. Control 43(4), 461–474 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

  4. Lee, S.H., Kim, T.H., Lim, J.T.: A new stability analysis of switched systems. Automatica 36(6), 917–922 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng, D.Z., Guo, L., Lin, Y.D., Wang, Y.: Stabilization of switched linear systems. IEEE Trans. Autom. Control 50(5), 661–666 (2005)

    Article  MathSciNet  Google Scholar 

  6. Dayawansa, W.P., Martin, C.F.: A converse Lyapunov theorem for a class of dynamical systems which undergo switching. IEEE Trans. Autom. Control 44(4), 751–760 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Daafouz, J., Riedinger, P., Iung, C.: Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach. IEEE Trans. Autom. Control 47(11), 1883–1887 (2002)

    Article  MathSciNet  Google Scholar 

  9. Chiou, J.S., Wang, C.J., Cheng, C.M., Wang, C.C.: Stability analysis and controller design of the nonlinear switched systems via T-S discrete-time fuzzy model. Int. J. Fuzzy Syst. 11(4), 213–224 (2009)

    MathSciNet  Google Scholar 

  10. Wu, L., Qi, T., Feng, Z.: Average dwell time approach to \({L_2} - {L_\infty }\) control of switched delay systems via dynamic output feedback. IET Control Theory Appl. 3(10), 1425–1436 (2009)

    Article  MathSciNet  Google Scholar 

  11. Zhang, L., Gao, H.: Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, L., Shi, P.: Stability, \({L_2}\)-gain and asynchronous \({H_\infty }\) control of discrete-time switched systems with average dwell time. IEEE Trans. Autom. Control 54(9), 2193–2200 (2009)

    MathSciNet  Google Scholar 

  13. Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012)

    Article  MathSciNet  Google Scholar 

  14. Zhao, X., Liu, H., Wang, Z.: Weighted \(H_\infty\) performance analysis of switched linear systems with mode-dependent average dwell time. Int. J. Syst. Sci. 44(11), 2130–2139 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lu, Q., Zhang, L., Karimi, H.R., Shi, Y.: \(H_\infty\) control for asynchronously switched linear parameter-varying systems with mode-dependent average dwell time. IET Control Theory Appl. 7(5), 673–683 (2013)

    Article  MathSciNet  Google Scholar 

  16. Guan, Z., Hill, D.J., Shen, X.: On hybrid impulsive and switching systems and application to nonlinear control. IEEE Trans. Autom. Control 50(7), 1058–1062 (2005)

    Article  MathSciNet  Google Scholar 

  17. Li, C., Ma, F., Feng, G.: Hybrid impulsive and switching time-delay systems. IET Control Theory Appl. 3(11), 1487–1498 (2009)

    Article  MathSciNet  Google Scholar 

  18. Xu, H., Teo, K.L.: Exponential stability with \({L_2}\)-gain condition of nonlinear impulsive switched systems. IEEE Trans. Autom. Control 55(10), 2429–2433 (2010)

    Article  MathSciNet  Google Scholar 

  19. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its application to modeling and control. IEEE Trans. Syst. Man Cybern. 15(1), 116–132 (1958)

    MATH  Google Scholar 

  20. Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, New York (2001)

    Book  Google Scholar 

  21. Li, H., Gao, Y., Wu, L., Lam, H.K.: Fault detection for T-S fuzzy time-delay systems: delta operator and input-output methods. IEEE Trans. Cybern. 45(2), 229–241 (2015)

    Article  Google Scholar 

  22. Li, H., Pan, Y., Zhou, Q.: Filter design for interval type-2 fuzzy systems with D stability constraints under a unified frame. IEEE Trans. Fuzzy Syst. 23(3), 719–725 (2015)

    Article  Google Scholar 

  23. Cao, K., Gao, X.Z., Lam, H.K., Vasilakos, A.V., Pedrycz, W.: A new relaxed stability condition for Takagi-Sugeno fuzzy control systems using quadratic fuzzy Lyapunov functions and staircase membership functions. Int. J. Fuzzy Syst. 16(3), 327–337 (2014)

    MathSciNet  Google Scholar 

  24. Lee, C., Chen, H.: Identification and fuzzy controller design for nonlinear uncertain systems with input time-delay. Int. J. Fuzzy Syst. 11(2), 73–86 (2009)

    MathSciNet  Google Scholar 

  25. Wang, L., Liu, X.: Robust \(H_{\infty }\) fuzzy output feedback control for uncertain discrete-time nonlinear systems. Int. J. Fuzzy Syst. 12(3), 218–226 (2010)

    MathSciNet  Google Scholar 

  26. Sun, C.H., Wang, W.J., Lin, W.W.: Linear control and parallel distributed fuzzy control design for T-S fuzzy time-delay system. Int. J. Fuzzy Syst. 9(4), 229–235 (2007)

    MathSciNet  Google Scholar 

  27. Chiou, J.S., Wang, C.J., Cheng, C.M., Wang, C.C.: Analysis and synthesis of switched nonlinear systems using the T-S fuzzy model. Appl. Math. Model. 34(6), 1467–1481 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mao, Y., Zhang, H., Xu, S.: Exponential stability and asynchronous stabilization of a class of switched nonlinear system via T-S fuzzy model. IEEE Trans. Fuzzy Syst. 22(4), 817–828 (2014)

    Article  Google Scholar 

  29. Tanaka, K., Hori, T., Wang, H.O.: A fuzzy Lyapunov approach to fuzzy control system design. Proc. Am. Control Conf. 2001, 4790–4795 (2001)

    Article  Google Scholar 

  30. Tanaka, K., Hori, T., Wang, H.O.: A multiple Lyapunov function approach to stabilization of fuzzy control systems. IEEE Trans. Fuzzy Syst. 11(4), 582–589 (2003)

    Article  Google Scholar 

  31. Lee, D.H., Park, J.B., Joo, Y.H.: A new fuzzy Lyapunov function for relaxed stability condition of continuous-time Takagi-Sugeno fuzzy systems. IEEE Trans. Fuzzy Syst. 19(4), 785–791 (2011)

    Article  Google Scholar 

  32. Wang, B., Zhang, H., Xie, D., Wang, G., Dang, C.: Asynchronous stabilisation of impulsive switched systems. IET Control Theory Appl. 7(16), 2021–2027 (2013)

    Article  MathSciNet  Google Scholar 

  33. Huang, L.: Linear Algebria in Systems and Control Theory. Science Press, Beijing (1984)

    Google Scholar 

  34. Shi, G., Zou, Y., Wang, C.: An algebraic approach to robust \({H_\infty }\) control via state feedback. Syst. Control Lett. 18(5), 365–370 (1992)

    Article  MathSciNet  Google Scholar 

  35. de Oliveira, M.C., Bernussou, J., Geromel, J.C.: A new discrete-time robust stability condition. Syst. Control Lett. 37(4), 261–265 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yazdi, M.B., Jahed-Motlagh, M.R., Attia, S.A., Raisch, J.: Modal exact linearization of a class of second-order switched nonlinear systems. Nonlinear Analysis: Real World Applications 11(4), 2243–2252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Qunxian Zheng & Hongbin Zhang

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  1. Qunxian Zheng
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  2. Hongbin Zhang
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Correspondence to Hongbin Zhang.

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Zheng, Q., Zhang, H. Exponential Stability and Asynchronous Stabilization of Nonlinear Impulsive Switched Systems via Switching Fuzzy Lyapunov Function Approach. Int. J. Fuzzy Syst. 19, 257–271 (2017). https://doi.org/10.1007/s40815-015-0086-4

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  • DOI: https://doi.org/10.1007/s40815-015-0086-4

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