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Robust dynamic sliding mode control design for interval type-2 fuzzy systems

  • *Corresponding authors: Oh-Min Kwon; Seong-Gon Choi

    *Corresponding authors: Oh-Min Kwon; Seong-Gon Choi 
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  • This paper discusses the problem of stabilization of interval type-2 fuzzy systems with uncertainties, time delay and external disturbance using a dynamic sliding mode controller. The sliding surface function, which is based on both the system's state and control input vectors, is used during the control design process. The sliding mode dynamics are presented by defining a new vector that augments the system state and control vectors. First, the reachability of the addressed sliding mode surface is demonstrated. Second, the required sufficient conditions for the system's stability and the proposed control design are derived by using extended dissipative theory and an asymmetric Lyapunov-Krasovskii functional approach. Unlike some existing sliding mode control designs, the one proposed in this paper does not require the control coefficient matrices of all linear subsystems to be the same, reducing the method's conservatism. Finally, numerical examples are provided to demonstrate the viability and superiority of the proposed design method.

    Mathematics Subject Classification: Primary: 93C10, 93C42; Secondary: 93D09.


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  • Figure 1.  IT2 fuzzy membership functions

    Figure 2.  State responses of the closed-loop system and control input in $ L_{2}-L_{\infty} $ case

    Figure 3.  State responses of the closed-loop system and control input in $ H_{\infty} $ case

    Figure 4.  State responses of the closed-loop system and control input in passivity case

    Figure 5.  State responses of the closed-loop system and control input in dissipativity case

    Figure 6.  State responses of the open-loop system

    Table 1.  Four cases of extended dissipative performance

    Performance $ \vartheta_{1} $ $ \vartheta_{2} $ $ \vartheta_{3} $ $ \vartheta_{4} $
    $ L_2-L_{\infty} $ 0 0 $ \lambda^{2}I $ $ I $
    $ H_{\infty} $ $ -I $ 0 $ \lambda^{2}I $ 0
    Passivity 0 $ I $ $ \lambda I $ 0
    Dissipativity $ Q=-I $ $ S=I $ $ R=(2-\lambda)I $ 0
     | Show Table
    DownLoad: CSV

    Table 2.  Minimum disturbance attenuation index for different $ d $

    Performance $ d=0.4 $ $ d=0.5 $ $ d=0.6 $ $ d=0.7 $ $ d=0.8 $ $ d=0.9 $ $ d=1.0 $
    $ L_{2}-L_{\infty} $ 0.8142 0.8200 0.8204 0.8371 0.8643 0.9008 0.9339
    $ H_{\infty} $ 0.8021 1.1488 1.1654 1.1895 1.2163 1.2440 1.2828
    Passivity 1.7813 1.7984 1.8144 1.8454 1.8931 1.9716 2.0705
    Dissipativity 0.9158 0.9166 0.9231 0.9271 0.9281 0.9282 0.9356
     | Show Table
    DownLoad: CSV

    Table 3.  Sliding surface gain matrices $ \mathcal{H}_{x} $ and $ \mathcal{H}_{u} $ for $ d = 0.4 $

    Performance $ \mathcal{H}_{x} $ $ \mathcal{H}_{u} $
    $ L_2-L_{\infty} $ [0.1660 -2.1859 -3.1432] 9.5060
    $ H_{\infty} $ [-1.3400 -5.3997 -7.5747] 19.4401
    Passivity [-0.2907 -1.2659 -1.8723] 5.0976
    Dissipativity [-0.2416 -1.5850 -2.4005] 6.8844
     | Show Table
    DownLoad: CSV

    Table 4.  Minimum $ H_{\infty} $ disturbance attenuation index for various $ d $

    $ d $ $ 0.5 $ $ 1.0 $ $ 1.5 $ $ 2.0 $ $ 2.5 $
    $ \lambda_{\min} $ 0.0044 0.0102 0.0171 0.0286 0.0322
     | Show Table
    DownLoad: CSV
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