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Stabilization for a Class of Discrete-Time Switched \(\Phi \)-Systems

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Abstract

This paper investigates the stability for a class of discrete-time switched \(\Phi \)-systems without assuming the stability for any subsystem. Based on the single Lyapunov function method and the multiple Lyapunov functions method, respectively, switching laws depending on the system state are designed, which can guarantee the stability of the switched system. In particular, when the proposed single Lyapunov function method is used, the stability of the convex combination of the system matrices is not required, which extends the classic single Lyapunov method. Two examples illustrate the effectiveness of the results.

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References

  1. L.I. Allerhand, U. Shaked, Robust stability and stabilization of linear switched systems with dwell time. IEEE Trans. Autom. Control 56, 381–386 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. N. Barabanov, D. Prokhorov, Stability analysis of discrete time recurrent neural networks. IEEE Trans. Neural Netw. 13, 292–303 (2002)

    Article  Google Scholar 

  3. G. Chesi, P. Colaneri, J.C. Geromel, R. Middleton, R. Shorten, A nonconservative LMI condition for stability of switched systems with guaranteed dwell time. IEEE Trans. Autom. Control 57, 1297–1320 (2010)

    Article  MathSciNet  Google Scholar 

  4. Y. Chu, K. Glover, Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities. IEEE Trans. Autom. Control 44, 471–483 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilization of hybrid systems. Proc. IEEE 88, 1069–1082 (2000)

    Article  Google Scholar 

  7. J.C. Geromel, P. Colaneri, Stability and stabilization of discrete time switched systems. Int. J. Control 79, 719–728 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. J.C. Geromel, P. Colaneri, Stability and stabilization of continuous-time switched linear systems. SIAM J. Control Optim. 45, 1915–1930 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems (Springer, Berlin, 2003)

    Book  MATH  Google Scholar 

  10. Q. Han, A. Xue, S. Liu, X. Yu, Robust absolute stability criteria for uncertain Lur’e systems of neutral type. Int. J. Robust Nonlinear Control 18, 278–295 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Kaszkurewicz, A. Bhaya, On a class of globally stable neural circuits. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 171–174 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Z.G. Li, C.Y. Wen, Y.C. Soh, Stabilization of a class of switched systems via designing switching laws. IEEE Trans. Autom. Control 46, 665–670 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Liberzon, Switching in Systems and Control (Birkhauser, Basel, 2003)

    Book  MATH  Google Scholar 

  14. D. Liu, A.N. Michel, Dynamical systems with saturation nonlinearities: analysis and design. Lecture Notes in Control and Information Sciences, vol. 195 (London, UK, 1994)

  15. M. Margaliot, G. Langholz, Necessary and sufficient conditions for absolute stability: the case of second-order systems. IEEE Trans. Circuit Syst. I Fundam. Theory Appl. 50, 227–234 (2003)

    Article  MathSciNet  Google Scholar 

  16. S. Pettersson, B. Lennartson, Stabilization of hybrid systems using a min-projection strategy, in Proceedings of the American Control Conference, Arlington, pp. 223–228 (2001)

  17. H.R. Shaker, J.P. How, Stability analysis for a class of switched nonlinear systems. Automatica 47, 2286–2291 (2011)

    Article  MathSciNet  Google Scholar 

  18. E.D. Sontag, M. Karny, K. Warwick, V. Kurkova, Recurrent Neural Networks: Some Systems-Theoretic Aspects in Dealing with Complexity: A Neural Network Approach (Springer, London, 1997)

    Google Scholar 

  19. Z. Sun, S. Ge, Switched linear systems: control and design. IEEE Trans. Autom. Control 51, 1585–1586 (2006)

    Article  Google Scholar 

  20. M. Wicks, P. Peleties, R. DeCarlo, Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems. Eur. J. Control 4, 140–147 (1998)

    Article  MATH  Google Scholar 

  21. W. Xiang, J. Xiao, M.N. Iqbal, Asymptotic stability, \(L_2\) gain, boundness analysis and control synthesis for switched systems: a switching frequency approach. Int. J. Adapt. Control Signal Process. 26, 350–373 (2012)

    Article  MATH  Google Scholar 

  22. W. Xiang, J. Xiao, Stability analysis and control synthesis of switched impulsive systems. Int. J. Robust Nonlinear Control 22, 1440–1459 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Z. Xiang, W. Xiang, Stability analysis of switched systems under dynamical dwell time control approach. Int. J. Syst. Sci. 40, 347–355 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach, in Proceedings of the American control conference, Chicago, pp. 200–204 (2000)

  25. G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Disturbance attenuation properties of time-controlled switched systems. J. Frankl. Inst. 338, 765–779 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Qualitative analysis of discrete-time switched systems, in Proceedings of the American control conference, Anchorage, pp. 1880–1885 (2002)

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

    Article  MathSciNet  Google Scholar 

  28. L. Zhang, P. Shi, Asynchronously switched control of switched linear systems with average dwell time. Automatica 46, 953–958 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. X. Zhao, S. Yin, H. Li, B. Niu, Switching stabilization for a class of slowly switched systems. IEEE Trans. Autom. Control 60, 221–226 (2015)

    Article  MathSciNet  Google Scholar 

  30. X. Zhao, X. Zheng, B. Niu, L. Liu, Adaptive tracking control for a class of uncertain switched nonlinear systems. Automatica 52, 185–191 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61233002 and 61503117, and IAPI Fundamental Research Funds under Grant 2013ZCX03-01.

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Authors and Affiliations

  1. State Key Laboratory of Synthetical Automation for Process Industries (Northeastern University), College of Information Science and Engineering, Northeastern University, Shenyang, 110819, China

    Haibiao Sun, Jiao Li & Jun Zhao

  2. College of Science, Dalian Nationalities University, Dalian, 116600, China

    Jiao Li

Authors
  1. Haibiao Sun
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  2. Jiao Li
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  3. Jun Zhao
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Correspondence to Jun Zhao.

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Sun, H., Li, J. & Zhao, J. Stabilization for a Class of Discrete-Time Switched \(\Phi \)-Systems. Circuits Syst Signal Process 36, 834–844 (2017). https://doi.org/10.1007/s00034-016-0325-4

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