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Asynchronous Control for Positive Discrete-Time Markovian Jump Systems

  • Hui Shang
  • Wenhai Qi
  • Guangdeng ZongEmail author
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)

Abstract

This paper is concerned with the asynchronous behaviours of discrete-time positive Markovian jump systems (PMJSs). In previous literatures about PMJSs, asynchronous behaviors which mean that the system modes and controller modes are not synchronous are always overlooked when designing controller. A sufficient condition for stochastic stability is first given by using Lyapunov–Krasovskii functional. The asynchronous controller is constructed in terms of linear matrix inequality forms to ensure the closed-loop system stochastic stability. Finally, a numerical example is stated to show the effectiveness of the proposed design.

Keywords

Asynchronous controller Lyapunov–Krasovskii functional Positive Markovian jump systems Stochastic stability 

Notes

Acknowledgements

This work is supported by National Natural Science Foundation of China (61703231) and (61773235), Natural Science Foundation of Shandong (ZR2017QF001) and (ZR2017MF063), Postdoctoral Science Foundation of China (2017M612235), Taishan Scholar Project of Shandong Province (TSQN20161033), and Excellent Experiment Project of Qufu Normal University (jp201728).

References

  1. 1.
    Boukas, E.K.: Stabilization of stochastic nonlinear hybrid systems. Int. J. Innov. Comput. Inf. Control 1, 131–141 (2005)MathSciNetGoogle Scholar
  2. 2.
    Boukas, E.K.: Stochastic Switching Systems: Analysis and Design. Springer Science & Business Media, Berlin (2007)Google Scholar
  3. 3.
    Bolzern, P., Colaneri, P., Nicolao, G.: Stochastic stability of positive Markov jump linear systems. Automatica 50, 1181–1187 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Costa, O.L.V., Fragoso, M.D., Marques, R.P.: Discrete-Time Markov Jump Linear Systems, pp. 280. Springer, London (2005)Google Scholar
  5. 5.
    Caswell, H.: Matrix population models: construction, analysis and interpretation. Nat. Resour. Model. 14, 593–595 (2001)CrossRefGoogle Scholar
  6. 6.
    Chen, X.M., Lama, J., Li, P., Shu, Z.: \(l_1\)-induced norm and controller synthesis of positive systems. Automatica 49, 1377–1385 (2013)CrossRefGoogle Scholar
  7. 7.
    Ding, Y.C., Liu, H., Shi, K.: \(H_\infty \) state-feedback controller design for continuous-time nonhomogeneous Markov jump systems. Optim. Control Appl. Methods 38, 133–144 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications, pp. 305. Wiley, New York (2000)CrossRefGoogle Scholar
  9. 9.
    Hien, L.V., Trinh, H.: Observer-based control of \(2\)-\(D\) Markov jump systems. IEEE Trans. Circuits Syst. 64, 1322–1326 (2017)CrossRefGoogle Scholar
  10. 10.
    Jiang, B.P., Gao, C.C., Gui, Y.G.: Stochastic admissibility and stabilization of singular Markovian jump systems with multiple time-varying delays. Int. J. Control. Autom. Syst. 14, 1280–1288 (2016)CrossRefGoogle Scholar
  11. 11.
    Kaczorek, T.: Positive 1D and 2D Systems, vol. 431. Springer, London (2002)CrossRefGoogle Scholar
  12. 12.
    Li, L.C., Shen, M.Q., Zhang, G.M., Yan, S.: \(H_\infty \) control of Markov jump systems with time-varying delay and incomplete transition probabilities. Appl. Math. Comput. 301, 95–106 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, S., Xiang, Z.R., Lin, H., Karimi, H.R.: State estimation on positive Markovian jump systems with time-varying delay and uncertain transition probabilities. Inf. Sci. 369, 251–266 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Qi, W.H., Gao, X.W.: \(H_\infty \) observer design for stochastic time-delayed systems with Markovian switching under partly known transition rates and actuator saturation. Appl. Math. Comput. 289, 80–97 (2016)MathSciNetGoogle Scholar
  15. 15.
    Qi, W.H., Kao, Y.G., Gao, X.W.: Passivity and passification for stochastic systems with Markovian switching and generally uncertain transition rates. Int. J. Control. Autom. Syst. 15, 2174–2181 (2017)CrossRefGoogle Scholar
  16. 16.
    Qi, W.H., Gao, X.W.: \(L_1\) control for positive Markovian jump systems with partly known transition rates. Int. J. Control. Autom. Syst. 15, 274–280 (2017)CrossRefGoogle Scholar
  17. 17.
    Qi, W.H., Gao, X.W.: Positive \(L_1\)-gain filter design for positive continuous-time Markovian jump systems with partly known transition rates. Int. J. Control. Autom. Syst. 14, 1413–1420 (2016)CrossRefGoogle Scholar
  18. 18.
    Shen, M., Park, J.H., Ye, D.: A separated approach to control of Markov jump nonlinear systems with general transition probabilities. IEEE Trans. Cybern. 46, 2010–2018 (2016)CrossRefGoogle Scholar
  19. 19.
    Shen, M.Q., Ye, D., Wang, Q.G.: Event-triggered \(H_\infty \) filtering of Markov jump systems with general transition probabilities. Inf. Sci. 418–419, 635–651 (2017)CrossRefGoogle Scholar
  20. 20.
    Song, X.N., Men, Y.Z., Zhou, J.P., Zhao, J.J., Shen, H.: Event-triggered \(H_\infty \) control for networked discrete-time Markov jump systems with repeated scalar nonlinearities control. Appl. Math. Comput. 298, 123–132 (2017)MathSciNetGoogle Scholar
  21. 21.
    Shorten, R., Wirth, F., Leith, D.: A positive systems model of TCP-like congestion control: asymptotic results. IEEE/ACM Trans. Netw. 14, 616–629 (2006)CrossRefGoogle Scholar
  22. 22.
    Song, Y., Xie, J.X., Fei, M.R., Hou, W.Y.: Mean square exponential stabilization of networked control systems with Markovian packet dropouts. Trans. Inst. Meas. Control 35, 75–82 (2013)CrossRefGoogle Scholar
  23. 23.
    Song, J., Niu, Y.G., Zou, Y.Y.: Asynchronous output feedback control of time-varying Markovian jump systems within a finite-time interval. J. Franklin Inst. 354, 6747–6765 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vargas, A.N., Pujol, G., Acho, L.: Stability of Markov jump systems with quadratic terms and its application to RLC circuits. J. Franklin Inst. 354, 332–344 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, G.L., Zhang, Q.L., Yang, C.Y., Su, C.L.: Stability and stabilization of continuous-time stochastic Markovian jump systems with random switching signals. J. Frankin Inst. 353, 1339–1357 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang, H.J., Zhang, D., Lu, R.Q.: Event-triggered \(H_\infty \) filter design for Markovian jump systems with quantization. Nonlinear Anal. Hybrid Syst. 28, 23–41 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, J.Y., Qi, W.H., Gao, X.W., Kao, Y.G.: Positive observer design for positive Markovian jump systems with mode-dependent time-varying delays and incomplete transition rates. Int. J. Control. Autom. Syst. 15, 640–646 (2017)CrossRefGoogle Scholar
  28. 28.
    Wu, Z.G., Shi, P., Shu, Z., Su, H.Y., Lu, R.Q.: Passivity-based asynchronous control for Markov jump systems. IEEE Trans. Autom. Control 62, 2020–2025 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wu Z.G., Dong S.L., Su H.Y., Li C.D.: Asynchronous dissipative control for fuzzy Markov jump systems. IEEE Trans. Cybern. 1–11 (2017)Google Scholar
  30. 30.
    Wu, Z.G., Shi, P., Su, H.Y., Chu, J.: Asynchronous \(l_2\)-\(l_\infty \) filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities. Automatica 50, 180–186 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Xie, J., Kao, Y.G., Zhang, C.H., Karimi, H.R.: Quantized control for uncertain singular Markovian jump linear systems with general incomplete transition rates. Int. J. Control. Autom. Syst. 15, 1107–1116 (2017)CrossRefGoogle Scholar
  32. 32.
    Zhang, L.H., Qi, W.H., Kao, Y.G., Gao, X.W., Zhao, L.J.: New results on finite-time stabilization for stochastic systems with time-varying delay. Int. J. Control. Autom. Syst. 16, 1–10 (2018)CrossRefGoogle Scholar
  33. 33.
    Zhang, J., Han, Z., Zhu, F.: Stochastic stability and stabilization of positive systems with Markovian jump. Nonlinear Anal. Hybrid Syst. 12, 147–155 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhu, S.Q., Wang, B., Zhang, C.H.: Delay-dependent stochastic finite-time \(l_\infty \)-gain filtering for discrete-time positive Markov jump linear systems with time-delay. J. Franklin Inst. 354, 6894–6913 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of EngineeringQufu Normal UniversityRizhaoChina

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