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Finite-Time Stability of Discrete Switched Singular Positive Systems

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Abstract

The finite-time stability problem of discrete switched singular positive systems (DSSPSs) is investigated in this paper. First, the concept of finite-time stability for DSSPSs is proposed, and a necessary and sufficient condition of finite-time stability for DSSPSs under arbitrary switching is obtained. Second, based on the mode-dependent average dwell time approach, by constructing the quasi-linear Lyapunov function, a sufficient stability criterion of finite-time stability for DSSPSs is derived in terms of a set of linear matrix inequalities. Finally, a numerical example is given to show the effectiveness of the proposed techniques.

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References

  1. B. Cantó, C. Coll, E. Sánchez, Positive solutions of a discrete-time descriptor system. Int. J. Syst. Sci. 39(1), 81–88 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Chen, Y. Yang, Finite-time stability of switched positive linear systems. Int. J. Robust Nonlinear Control 24(1), 179–190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Chen, H.B. Zou, R.Q. Lu, A. Xue, Finite-time stability and dynamic output feedback stabilization of stochastic systems. Circuits Syst. Signal Process. 33(1), 53–69 (2014)

    Article  MathSciNet  Google Scholar 

  4. J. Clote, J. Ferrer, M.D Magret, Switched singular linear systems, in Proceedings of 17th Mediterranean Conference on Control and Automation Thessaloniki (2009), Greece, pp. 1343–1347

  5. L. Dai, Singular Control Systems (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  6. L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)

    Book  MATH  Google Scholar 

  7. A. Herrero, A. Ramírez, N. Thome, An algorithm to check the nonnegativity of singular systems. Appl. Math. Comput. 189(1), 355–365 (2007)

    MathSciNet  MATH  Google Scholar 

  8. A. Herrero, A. Ramírez, N. Thome, Nonnegativity, stability, and regularization of discrete-time descriptor systems. Linear Algebra Appl. 432(4), 837–846 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Ibeas, M. de la Sen, R. Vilanova, J. Herrera, Stability of switched linear discrete-time descriptor systems with explicit calculation of a common quadratic Lyapunov sequence, in Proceedings of American Control Conference (ACC) (2010), Baltimore, pp. 1719–1724

  10. J. Ishihara, M. Terra, On the Lyapunov theorem for singular systems. IEEE Trans. Autom. Control 47(11), 1926–1930 (2002)

    Article  MathSciNet  Google Scholar 

  11. T. Kaczorek, Drazin inverse matrix method for fractional descriptor continuous-time linear systems. Bull. Pol. Acad. Sci. Tech. 62(3), 409–412 (2014)

    Google Scholar 

  12. T. Kaczorek, Positive 1D and 2D Systems (Springer, London, 2002)

    Book  MATH  Google Scholar 

  13. G. Kamenkov, On stability of motion over a finite interval of time. J. Appl. Math. Mech. 17, 529–540 (1953)

    MathSciNet  Google Scholar 

  14. S. Li, Z.R. Xiang, Stability and \(L_{\infty }\)-gain analysis for positive switched systems with time-varying delay under state-dependent switching. Circuits Syst. Signal Process. (2015). doi:10.1007/s00034-015-0099-0

  15. S. Li, Z.R. Xiang, H.R. Karimi, Positive \(L_{1}\) observer design for positive switched systems. Circuits Syst. Signal Process. 33(7), 2085–2106 (2014)

    Article  Google Scholar 

  16. L.L. Liu, J.G. Peng, B.W. Wu, On parameterized Lyapunov-Krasovskii functional techniques for investigating singular time-delay systems. Appl. Math. Lett. 24(3), 703–708 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. L.L. Liu, J.G. Peng, B.W. Wu, \(H_{\infty }\) control of singular time-delay systems via discretized Lyapunov functional. J. Frankl. Inst. 348(4), 749–762 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. T.T. Liu, B.W. Wu, L.L. Liu, Y.E. Wang, Asynchronously finite-time control of discrete impulsive switched positive time-delay systems. J. Frankl. Inst. 352(10), 4503–4514 (2015)

    Article  MathSciNet  Google Scholar 

  19. T.T. Liu, B.W. Wu, L.L. Liu, Y.E. Wang, New stabilization results for discrete-time positive switched systems with forward mode-dependent average dwell time. Trans. Inst. Meas. Control (2015). doi:10.1177/0142331215604894

  20. T.T. Liu, B.W. Wu, Y.X. Tong, Exponential stability of discrete-time linear singular positive time-delay systems, in Proceedings of 27th Chinese Control and Decision Conference (2015), Qingdao, pp. 6069–6073

  21. A. Mustapha, N. Diego, Positivity of discrete singular systems and their stability: an LP-based approach. Automatica 50(1), 84–91 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Phat, N. Sau, On exponential stability of linear singular positive delayed systems. Appl. Math. Lett. 38, 67–72 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. R.Q. Shi, X.M. Tian, X.D. Zhao, X.L. Zheng, Stability and \(L_{1}\)-Gain analysis for switched delay positive systems with stable and unstable subsystems. Circuits Syst. Signal Process. 34(5), 1683–1696 (2015)

    Article  MATH  Google Scholar 

  24. E. Virnik, Stability analysis of positive descriptor systems. Linear Algebra Appl. 429(10), 2640–2659 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. J.Y. Wang, W.H. Qi, X.W. Gao, Finite-time \(L_{1}\) control for positive Markovian jump systems with partly known transition rates. Circuits Syst. Signal Process. (2015). doi:10.1007/s00034-015-0131-4

  26. Y.E. Wang, X.M. Sun, J. Zhao, Asynchronous \(H_{\infty }\) control of switched delay systems with average dwell time. J. Frankl. Inst. 349(10), 3159–3169 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Y.E. Wang, X.M. Sun, J. Zhao, Stabilization of a class of switched Stochastic systems with time delays under asynchronous switching. Circuits Syst. Signal Process. 32(1), 347–360 (2013)

    Article  MathSciNet  Google Scholar 

  28. M. Xiang, Z.R. Xiang, Observer design of switched positive systems with time-varying delays. Circuits Syst. Signal Process. 32(5), 2171–2184 (2013)

    Article  MathSciNet  Google Scholar 

  29. J.F. Zhang, Z.Z. Han, J. Huang, Stabilization of discrete-time positive switched systems. Circuits Syst. Signal Process. 32(3), 1129–1145 (2013)

    Article  MathSciNet  Google Scholar 

  30. J.F. Zhang, Z.Z. Han, H. Wu, Robust finite-time stability and stabilisation of switched positive systems. IET Control Theory A 8(1), 67–75 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Y.L. Zhang, B.W. Wu, Y.E. Wang, X.X. Han, Finite-time stability for switched singular systems. Acta Phys. Sin. 63(17), 170205 (2014). (in Chinese)

    Google Scholar 

  32. Y. Zhang, Q. Zhang, T. Tanaka, M. Cai, Admissibility for positive continuous-time descriptor systems. Int. J. Syst. Sci. 44(11), 2158–2165 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. X.D. Zhao, L. Zhang, P. Shi, M. Liu, 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 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11371233 and 61403241.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, China

    Baowei Wu, Lili Liu & Yue-E Wang

  2. School of Science, Xi’an Polytechnic University, Xi’an, 710048, China

    Tingting Liu

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  1. Tingting Liu
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  2. Baowei Wu
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  3. Lili Liu
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  4. Yue-E Wang
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Correspondence to Baowei Wu.

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Liu, T., Wu, B., Liu, L. et al. Finite-Time Stability of Discrete Switched Singular Positive Systems. Circuits Syst Signal Process 36, 2243–2255 (2017). https://doi.org/10.1007/s00034-016-0423-3

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  • DOI: https://doi.org/10.1007/s00034-016-0423-3

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