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Reliable Finite-Time H∞ Control for T–S Fuzzy Semi-Markovian Jump Delay Systems

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Abstract

This paper addresses the observer-based finite-time \(H_{\infty }\) control problem for Takagi–Sugeno (T–S) fuzzy semi-Markovian jump systems with mode-dependent fast-varying delays, actuator faults, parameter uncertainties and partly uncertain transition rates. Firstly, a more general actuator fault model is considered and the fuzzy reliable observer-based control scheme is designed. By applying the new Lyapunov functional candidates, free-weighting matrix method and slack matrix variables, some negative integral terms can be adopted and the nonlinear terms caused by time delay and partly unknown transition rates can be dealt. Furthermore, novel sufficient conditions on guaranteeing the closed-loop system is stochastically finite-time bounded (SFTB) with the prescribed \(H_{\infty }\) level are proposed. Particularly, different from traditional slowly varying delays, the case of mode-dependent fast-varying delays which are more difficult to be dealt is considered in this paper. Lastly, a practical example is exploited to affirm the validity of the obtained results.

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References

  1. Martinelli, F.: Optimality of a two-threshold feedback control for a manufacturing system with a production dependent failure rate. IEEE Trans. Autom. Control 52(10), 1937–1942 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Kim, S.H., Park, P.: Networked-based robust \(H_{\infty }\) control design using multiple levels of network traffic. Automatica 45(3), 764–770 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Schwartz, C.: Control of semi-Markov jump linear systems with application to the bunch-train cavity interaction. PhD Thesis, Northwestern University (2003)

  4. Barbu, V.S., Limnios, N.: Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications: Their Use in Reliability and DNA Analysis. Springer, New York (2008)

    MATH  Google Scholar 

  5. Mariton, M.: On systems with non-Markovian regime changes. IEEE Trans. Autom. Control 34(3), 346–349 (1989)

    MathSciNet  MATH  Google Scholar 

  6. Huang, J., Shi, Y.: Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int. J. Robust Nonlinear Control 23, 2028–2043 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Shen, H., Wu, Z.G., Park, J.H.: Reliable mixed passive and \(H_{\infty }\) filtering for semi-Markov jump systems with randomly occurring uncertainties and sensor failures. Int. J. Robust Nonlinear Control 25, 3231–3251 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Wei, Y.L., Park, J.H., Qiu, J.B., Wu, L.G.: Sliding mode control for semi-Markovian jump systems via output feedback. Automatica 81, 133–141 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Wu, X.H., Mu, X.W.: Event-triggered control for networked nonlinear semi-Markovian jump systems with randomly occurring uncertainties and transmission delay. Inf. Sci. 487, 84–96 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Wu, X.H., Mu, X.W.: \(H_{\infty }\) stabilization for networked semi-Markovian jump systems with randomly occurring uncertainties via improved dynamic event-triggered scheme. Int. J. Robust Nonlinear Control 29, 4609–4626 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Hu, Z.H., Mu, X.W.: Stabilization for switched stochastic systems with semi-Markovian switching signals and actuator saturation. Inf. Sci. 483, 419–431 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Tian, Y.X., Yan, H.C., Zhang, H., Zhan, X.S., Peng, Y.: Dynamic output-feedback control of linear semi-Markov jump systems with incomplete semi-Markov kernel. Automatica (2020). https://doi.org/10.1016/j.automatica.2020.108997

    Article  MATH  Google Scholar 

  13. Mu, X.W., Hu, Z.H.: Stability analysis for semi-Markovian switched singular stochastic systems. Automatica (2020). https://doi.org/10.1016/j.automatica.2020.109014

    Article  MATH  Google Scholar 

  14. Zhang, L.X., Boukas, E.K.: Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica 45, 463–468 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, Y., He, Y., Wu, M., Zhang, J.: Stabilization for Markovian jump linear systems with partial information on transition probability based on free-connection weighting matrices. Automatica 47(1), 79–84 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Jiang, B.P., Kao, Y.G., Karimi, H.R., Chen, C.C.: Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates. IEEE Trans. Autom. Control 63(11), 3919–3926 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Qi, W.H., Park, J.H., Cheng, J., Kao, Y.G.: Robust stabilisation for non-linear time-delay semi-Markovian jump systems via sliding mode control. IET Control Theory Appl. 11(10), 1504–1513 (2017)

    MathSciNet  Google Scholar 

  18. Shen, Y., Wu, Z.G., Shi, P., Shu, Z., Karimi, H.R.: \(H_{\infty }\) control of Markov jump time-delay systems under asynchronous controller and quantizer. Automatica 99, 352–360 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Gao, H.J., Fei, Z.Y., Lam, J., Du, B.Z.: Further results on exponential estimates of Markovian jump systems with mode-dependent time-varying delays. IEEE Trans. Autom. Control 56(1), 223–229 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Li, F.B., Wu, L.G., Shi, P.: Stochastic stability of semi-Markovian jump systems with mode-dependent delays. Int. J. Robust Nonlinear Control 24, 3317–3330 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Zhang, H.G., Wang, J.Y., Wang, Z.S., Liang, H.J.: Mode-dependent stochastic synchronization for Markovian coupled neural networks with time-varying mode-delays. IEEE Trans. Neural Netw. Learn. Syst. 26(11), 2621–2634 (2015)

    MathSciNet  Google Scholar 

  22. Ma, H., Li, H.Y., Lu, R.Q., Huang, T.W.: Adaptive event-triggered control for a class of nonlinear systems with periodic disturbances. Sci. China Inf. Sci. 63(5), 150212 (2020)

    MathSciNet  Google Scholar 

  23. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. Syst. 15(1), 116–132 (1985)

    MATH  Google Scholar 

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

    Google Scholar 

  25. Chen, J., Xu, S.Y., Li, Y.M., Chu, Y.M., Zou, Y.: Further studies on stability and stabilization conditions for discrete-time T–S systems with the order relation information of membership functions. J. Frankl. Inst. 352(12), 5796–5809 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Shi, L., Kao, Y.G.: A fuzzy control approach to stabilization of Markovian jump systems with general unknown transition probabilities. Int. J. Fuzzy Syst. 18, 1–11 (2016)

    MathSciNet  Google Scholar 

  27. Tong, S.C., Min, X., Li, Y.X.: Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions. IEEE Trans. Cybern. 50(9), 3903–3913 (2020)

    Google Scholar 

  28. Li, Y.M., Yang, T.T., Tong, S.C.: Adaptive neural networks finite-time optimal control for a class of nonlinear systems. IEEE Trans. Neural Netw. Learn. Syst. 31(11), 4451–4460 (2020)

    MathSciNet  Google Scholar 

  29. Jiang, B.P., Karimi, H.R., Kao, Y.G., Gao, C.C.: Takagi–Sugeno model based event-triggered fuzzy sliding-mode control of networked control systems with semi-Markovian switchings. IEEE Trans. Fuzzy Syst. 28(4), 673–683 (2020)

    Google Scholar 

  30. Wang, W.J., Wu, H.N., Guo, L., Luo, Y.S.: Robust \(H_{\infty }\) fuzzy control for uncertain nonlinear Markovian jump systems with time-varying delay. Fuzzy Sets Syst. 212, 41–46 (2013)

    MATH  Google Scholar 

  31. Li, L., Zhang, Q.L., Zhu, B.Y.: \(H_{\infty }\) fuzzy control for nonlinear time-delay singular Markovian jump systems with partly unknown transition rates. Fuzzy Sets Syst. 254, 106–125 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Shi, P., Zhang, Y.Q., Chadli, M., Agaewal, R.K.: Mixed \(H_{\infty }\) and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays. IEEE Trans. Neural Netw. Learn. Syst. 27(4), 903–909 (2016)

    MathSciNet  Google Scholar 

  33. Zhang, Y.Q., Shi, P., Agarwal, R.K., Shi, Y.: Dissipativity analysis for discrete time-delay fuzzy neural networks with Markovian jumps. IEEE Trans. Fuzzy Syst. 24(2), 432–443 (2016)

    Google Scholar 

  34. Zhuang, G.M., Su, S.F., Xia, J.W., Sun, W.: HMM-based asynchronous \(H_{\infty }\) filtering for fuzzy singular Markovian switching systems with retarded time-varying delays. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2020.2977127

    Article  Google Scholar 

  35. Lin, G.H., Li, H.Y., Ma, H., Yao, D.Y., Lu, R.Q.: Human-in-the-loop consensus control for nonlinear multi-agent systems with actuator faults. IEEE/CAA J. Autom. Sin. (2020). https://doi.org/10.1109/JAS.2020.1003596

    Article  Google Scholar 

  36. Li, H.Y., Wu, Y., Chen, M.: Adaptive fault-tolerant tracking control for discrete-time multi-agent systems via reinforcement learning algorithm. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2020.2982168

    Article  Google Scholar 

  37. Shen, H., Su, L., Wu, Z.G., Park, J.H.: Reliable dissipative control for Markov jump systems using an event-triggered sampling information scheme. Nonlinear Anal Hybrid Syst. 25, 41–59 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Shen, H., Su, L., Park, J.H.: Reliable mixed \(H_{\infty }\)/passive control for T–S fuzzy delayed systems based on a semi-Markov jump model approach. Fuzzy Sets Syst. 314, 79–98 (2017)

    MATH  Google Scholar 

  39. Shen, H., Chen, M.S., Wu, Z.G., Cao, J.D., Park, J.H.: Reliable event-triggered asynchronous extended passive control for semi-Markov jump fuzzy systems and its application. IEEE Trans. Fuzzy Syst. 28(8), 1708–1722 (2020)

    Google Scholar 

  40. Li, H.Y., Shi, P., Yao, D.Y., Wu, L.G.: Observer-based adaptive sliding mode control for nonlinear Markovian jump systems. Automatica 64, 132–142 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Xia, W.F., Li, Y.M., Chu, Y.M., Xu, S.Y., Chen, W.M., Zhang, Z.Q.: Observer-based mixed passive and \(H_{\infty }\) control for uncertain Markovian jump systems with time delays using quantized measurements. Nonlinear Anal Hybrid Systems 31, 233–246 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Amato, F., Ariola, M., Dorato, P.: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37, 1459–1463 (2001)

    MATH  Google Scholar 

  43. He, S.P., Liu, F.: Finite-time \(H_{\infty }\) fuzzy control of nonlinear jump systems with time delays via dynamic observer-based state feedback. IEEE Trans. Fuzzy Syst. 2(4), 605–614 (2012)

    Google Scholar 

  44. Zhang, Y.Q., Shi, P., Nguang, S.K., Karimi, H.R., Agarwal, R.K.: Robust finite-time fuzzy \(H_{\infty }\) control for uncertain time-delay systems with stochastic jumps. J. Frankl. Inst. 351, 4211–4229 (2014)

    MathSciNet  MATH  Google Scholar 

  45. Cheng, J., Park, J.H., Liu, Y.J., Liu, Z.J., Tang, L.M.: Finite-time \(H_{\infty }\) fuzzy control of nonlinear Markovian jump delayed systems with partly uncertain transition descriptions. Fuzzy Sets Syst. 314, 99–115 (2017)

    MATH  Google Scholar 

  46. Shen, H., Li, F., Yan, H.C., Karimi, H.R., Lam, H.K.: Finite-time event-triggered \(H_{\infty }\) control for T–S fuzzy Markov jump systems. IEEE Trans. Fuzzy Syst. 26(5), 3122–3135 (2018)

    Google Scholar 

  47. Xie, L.H.: Output feedback \(H_{\infty }\) control of systems with parameter uncertainties. Int. J. Control 63(4), 741–750 (1996)

    MATH  Google Scholar 

  48. Xie, L.H., De Souza, C.E.: Robust \(H_{\infty }\) control for linear systems with norm-bounded time-varying uncertainty. IEEE Trans. Autom. Control 37, 1188–1191 (1992)

    MathSciNet  MATH  Google Scholar 

  49. Boukas, E.K.: Control of Singular Systems with Random Abrupt Changes. Springer, Berlin (2008)

    MATH  Google Scholar 

  50. Tuan, H.D., Apkarian, P., Narikiyo, T., Yamamoto, Y.: Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Trans. Fuzzy Syst. 9, 324–332 (2001)

    Google Scholar 

  51. Seuret, A., Gouaisbaut, F.: Wirtinger-based integral inequality: application to time-delay systems. Automatica 49(9), 2860–2866 (2013)

    MathSciNet  MATH  Google Scholar 

  52. Chen, Y.G., Fei, S.M., Li, Y.M.: Robust stabilization for uncertain saturated time-delay systems: a distributed-delay-dependent polytopic approach. IEEE Trans. Autom. Control 62(7), 3455–3460 (2017)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (11571322, 11971444).

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  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Xin Li, Zenghui Hu & Xiaowu Mu

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  1. Xin Li
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  2. Zenghui Hu
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  3. Xiaowu Mu
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Correspondence to Xiaowu Mu.

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Li, X., Hu, Z. & Mu, X. Reliable Finite-Time H∞ Control for T–S Fuzzy Semi-Markovian Jump Delay Systems. Int. J. Fuzzy Syst. 23, 1524–1538 (2021). https://doi.org/10.1007/s40815-021-01052-7

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