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Stabilization of Discrete-time Fuzzy Systems Via Delta Operators and its Application to Truck–Trailer Model

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Abstract

This paper addresses the problem of robust \(L_2{-}L_\infty \) control in delta domain for a class of Takagi–Sugeno (TS) fuzzy systems with interval time-varying delays and disturbance input. In particular, the system under study involves state time delay, uncertainties and fast sampling period \(\mathcal {T}\). The main aim of this work was to design a \(L_2{-}L_\infty \) controller such that the proposed TS fuzzy system is robustly asymptotically stable with a \(L_2{-}L_\infty \) prescribed performance level \(\gamma >0\). Based on the proper Lyapunov–Krasovskii functional (LKF) involving lower and upper bound of time delay and free-weighting technique, a new set of delay-dependent sufficient conditions in terms of linear matrix inequalities (LMIs) are established for obtaining the required result. The result reveals that the asymptotic stability is achieved quickly when the sampling frequency is high. Finally, a numerical example based on the truck–trailer model is given to demonstrate the effectiveness and potential of the proposed design technique.

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References

  1. C.K. Ahn, M.K. Song, New sets of criteria for exponential \(L_2\)-\( L_\infty \) stability of Takagi-Sugeno fuzzy systems combined with hopfield neural networks. Int. J. Innov. Comput. Inf. Control 9(7), 2979–2986 (2013)

    Google Scholar 

  2. P. Balasubramaniam, V.M. Revathi, J.H. Park, \(L_2-L_\infty \) filtering for neutral Markovian switching system with mode-dependent time-varying delays and partially unknown transition probabilities. Appl. Math. Comput. 219(17), 9524–9542 (2013)

    MathSciNet  MATH  Google Scholar 

  3. S. Boyd, L.E.L. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994)

    Book  MATH  Google Scholar 

  4. Y.Y. Cao, P.M. Frank, Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy model. Fuzzy Sets Syst. 124(2), 213–229 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Chadli, H.R. Karimi, P. Shi, On stability and stabilization of singular uncertain Takagi-Sugeno fuzzy systems. J. Franklin Inst. 351(3), 1453–1463 (2014)

    Article  MathSciNet  Google Scholar 

  6. H. Chen, L. Wang, Delay-dependent robust \(L_2-L_\infty \) filter design for uncertain neutral stochastic systems with mixed delays. Digit. Signal Process. 30, 184–194 (2014)

    Article  Google Scholar 

  7. Y. Cui, \(L_2-L_\infty \) consensus control for high-order multi-agent systems with nonuniform time-varying delays. Asian J. Control 16(6), 1751–1759 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. S.F. Derakhshan, A. Fatehi, Non-monotonic Lyapunov functions for stability analysis and stabilization of discrete time Takagi-Sugeno fuzzy systems. Int. J. Innov. Comput. Inf. Control 10(4), 1567–1586 (2014)

    Google Scholar 

  9. X.Z. Dong, M. Xiao, Admissible control of linear singular delta operator systems. Circuits Syst. Signal Process. 33(7), 2043–2064 (2014)

    Article  MathSciNet  Google Scholar 

  10. J. Dong, G.H. Yang, Reliable state feedback control of T-S fuzzy systems with sensor faults. IEEE Trans. Fuzzy Syst. 23(2), 421–433 (2014)

    Article  Google Scholar 

  11. S. He, F. Liu, \(L_2-L_\infty \) fuzzy control for Markov jump systems with neutral time-delays. Math. Comput. Simul. 92, 1–13 (2013)

    Article  Google Scholar 

  12. J. Huang, Y. Shi, H. Huang, Z. Li, \(L_2-L_\infty \) filtering for multirate nonlinear sampled-data systems using T-S fuzzy models. Digit. Signal Process. 23(1), 418–426 (2013)

    Article  MathSciNet  Google Scholar 

  13. H.K. Lam, H. Li, C. Deters, H. Wurdemann, E. Secco, K. Althoefer, Control design for interval type-2 fuzzy systems under imperfect premise matching. IEEE Trans. Ind. Electron. 61(2), 956–968 (2014)

    Article  Google Scholar 

  14. H.K. Lam, S.H. Tsai, Stability analysis of polynomial-fuzzy-model-based control systems with mismatched premise membership functions. IEEE Trans. Fuzzy Syst. 22(1), 223–229 (2014)

    Article  Google Scholar 

  15. H. Li, H. Gao, P. Shi, X. Zhao, Fault-tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach. Automatica 50(7), 1825–1834 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Li, X. Jing, H. Karimi, Output-feedback based \(H_\infty \) control for active suspension systems with control delay. IEEE Trans. Ind. Electron. 61(1), 436–446 (2014)

    Article  Google Scholar 

  17. H. Li, X. Jing, H.K. Lam, P. Shi, Fuzzy sampled-data control for uncertain vehicle suspension systems. IEEE Trans. Cybern. 44(7), 1111–1126 (2014)

    Article  Google Scholar 

  18. Z. Li, H. Shu, X. Kan, \(L_2-L_\infty \) reliable control for discrete time-delay systems with fractional uncertainties and saturated package losses. IET Control Theory Appl. 8(11), 891–900 (2014)

    Article  MathSciNet  Google Scholar 

  19. C. Qin, Z. Xiang, H.R. Karimi, Robust stability and \(H_\infty \) stabilization of switched systems with time-varying delays using delta operator approach. Math. Probl. Eng. (2013). doi:10.1155/2013/230576

  20. R. Sakthivel, P. Vadivel, K. Mathiyalagan, A. Arunkumar, Fault-distribution dependent reliable \(H_\infty \) control for TS fuzzy systems. J. Dyn. Sys. Meas. Control 136(2), 021021 (2014)

    Article  Google Scholar 

  21. R. Sakthivel, P. Vadivel, K. Mathiyalagan, J.H. Park, Robust control for fuzzy nonlinear uncertain systems with discrete and distributed time delays. Z. Naturforsch. A 69a, 569–580 (2014)

    Google Scholar 

  22. H. Shen, Z. Wang, J. Wang, Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback. J. Franklin Inst. 351(7), 3797–3817 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. X. Xie, S. Yin, H. Gao, O. Kaynak, Asymptotic stability and stabilisation of uncertain delta operator systems with time-varying delays. IET Control Theory Appl. 7(8), 1071–1078 (2013)

    Article  MathSciNet  Google Scholar 

  25. H. Yang, X. Li, Z. Liu, L. Zhao, Robust fuzzy-scheduling control for nonlinear systems subject to actuator saturation via delta operator approach. Inf. Sci. 272(10), 158–172 (2014)

    Article  MathSciNet  Google Scholar 

  26. H. Yang, P. Shi, X. Li, Z. Li, Fault-tolerant control for a class of T-S fuzzy systems via delta operator approach. Signal Process. 98, 166–173 (2014)

    Article  Google Scholar 

  27. H. Yang, P. Shi, J. Zhang, J. Qiu, Robust \(H_\infty \) control for a class of discrete time fuzzy systems via delta operator approach. Inf. Sci. 184(1), 230–245 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. X. Yang, L. Wu, H.K. Lam, X. Su, Stability and stabilization of discrete-time T-S fuzzy systems with stochastic perturbation and time-varying delay. IEEE Trans. Fuzzy Syst. 22(1), 124–138 (2014)

    Article  Google Scholar 

  29. Y. Yin, P. Shi, F. Liu, K.L. Teo, Robust \(L_2-L_\infty \) filtering for a class of dynamical systems with nonhomogeneous markov jump process. Int. J. Syst. Sci. 46(4), 599–608 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. C. Zhang, G. Feng, J. Qiu, W.A. Zhang, T-S fuzzy-model-based piecewise \(H_\infty \) output feedback controller design for networked nonlinear systems with medium access constraint. Fuzzy Sets Syst. 248, 86–105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. X. Zhao, L. Zhang, P. Shi, H. Karimi, Novel stability criteria for T-S fuzzy systems. IEEE Trans. Fuzzy Syst. 22(2), 313–323 (2014)

    Article  Google Scholar 

  32. Q. Zhou, X. Shao, H.R. Karimi, J. Zhu, Stability of genetic regulatory networks with time-varying delay: delta operator method. Neurocomputing 149(Part B), 490–495 (2015)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440 746, Republic of Korea

    R. Sakthivel

  2. Department of Mathematics, Sri Ramakrishna Institute of Technology, Coimbatore, 641 010, Tamilnadu, India

    R. Sakthivel

  3. Department of Applied Mathematics, Kongju National University, Gongju-si, Chungcheongnam-do, 314-701, Republic of Korea

    Yong-Ki Ma

  4. Department of Mathematics, Anna University-Reginal Centre, Coimbatore, 641 046, Tamilnadu, India

    M. Rathika, Srimanta Santra & K. Mathiyalagan

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  1. R. Sakthivel
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  2. M. Rathika
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  3. Srimanta Santra
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  4. Yong-Ki Ma
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  5. K. Mathiyalagan
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Correspondence to Yong-Ki Ma.

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Sakthivel, R., Rathika, M., Santra, S. et al. Stabilization of Discrete-time Fuzzy Systems Via Delta Operators and its Application to Truck–Trailer Model. Circuits Syst Signal Process 35, 2373–2389 (2016). https://doi.org/10.1007/s00034-015-0154-x

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