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Sampled-data based extended dissipative synchronization of stochastic complex dynamical networks

  • ** Co-corresponding author: Yang Cao

    ** Co-corresponding author: Yang Cao; 

    ** Co-corresponding author: Yang Cao* Corresponding author: Quanxin Zhu

    * Corresponding author: Quanxin Zhu

This work was supported by Grant-in-Aid for Research Activity Start-up (20K23328), funded by Japan Society for the Promotion of Science (JSPS), the National Natural Science Foundation of China (62173139, 62103103), the Science and Technology Innovation Program of Hunan Province (2021RC4030), Hunan Provincial Science and Technology Project Foundation (2019RS1033) and the Natural Science Foundation of Jiangsu Province of China (BK20210223)

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  • In this paper, extended dissipative (ED) synchronization is considered for stochastic complex dynamical networks (SCDNs) with variable coupling delay via sampled-data control (SDC). First, a suitable Lyapunov–Krasovskii functional (LKF) is constructed, then a new synchronization criterion is obtained through stochastic integral inequality (SII) and linear matrix inequality (LMI) techniques. Moreover, the ED synchronization criteria are established, which consolidates passivity, dissipativity, $ H_\infty $, and $ L_2-L_\infty $ performances in a unified structure. SDC gain matrices are also designed for each performance in ED criteria. Finally, the feasibility and usefulness of the derived theoretical results are shown through numerical simulations.

    Mathematics Subject Classification: Primary: 37H30, 93C57, 93E15.


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  • Figure 1.  Synchronization error (Example 1)

    Figure 2.  Generated control signals (Example 1)

    Figure 3.  Synchronization error (Example 2)

    Figure 4.  Generated control signals (Example 2)

    Figure 5.  Synchronization error of the first node for different cases (Example 2)

    Figure 6.  Generated control signals of the first node for different cases (Example 2)

    Table 1.  Matrix parameters for each cases in ED synchronization

    Criteria $ \Sigma_1 $ $ \Sigma_2 $ $ \Sigma_3 $ $ \Sigma_4 $
    Passivity 0 1 $ \gamma = 0.5 $ 0
    Dissipativity -1 1 2 0
    $ H_\infty $ performance -1 0 $ \gamma^2 $ 0
    $ L_2 $-$ L_\infty $ performance 0 0 $ \gamma^2 $ 1
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    Table 2.  SDC gain matrices for each performance in ED synchronization

    Criteria SDC gain matrices
    Passivity $ \begin{array}{l} K_1 = K_5 = \left[ \begin{array} {ccc} -0.8104 &-0.1674\\0.0499&-1.6699 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} 0.6884&-0.1835\\0.0605&-1.6412 \end{array} \right] , \\ \;\;\;\;\; K_3 = \left[ \begin{array} {ccc} 0.6969&-0.1761\\0.0581&-1.6111 \end{array} \right] \end{array}$
    Dissipativity $\begin{array}{l} K_1 = K_5 = \left[ \begin{array} {ccc} -0.7750&-0.1747\\-0.0155&-1.6717 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} -0.7588&-0.1733\\-0.0169&-1.6445 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.7567&-0.1693\\-0.0167&-1.6214 \end{array} \right]\end{array} $
    $ H_\infty $ performance $ \begin{array}{l}K_1 = K_5 = \left[ \begin{array} {ccc} -0.7310&-0.1789\\0.0532&-1.6689 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} 0.5810&-0.2004\\0.0642&-1.6405 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.5843&-0.1930\\0.0651&-1.6107 \end{array} \right] \end{array}$
    $ L_2 $-$ L_\infty $ performance $ \begin{array}{l}K_1 = K_5 = \left[ \begin{array} {ccc} -0.7715&-0.1847\\0.0286&-1.6624 \end{array} \right], \;\;\; K_2 = K_4 = \left[ \begin{array} {ccc} -0.6395&-0.2060\\0.0318&-1.6327 \end{array} \right] , \\ \;\;\; K_3 = \left[ \begin{array} {ccc} -0.6434&-0.1989\\0.0315&-1.6036 \end{array} \right] \end{array}$
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    Table 3.  SDC gain matrices for each performance in ED synchronization

    Criteria SDC gain matrices
    Passivity $ K_1 = \left[ \begin{array} {ccc} -1.4426& -0.0287 \\ -0.0390& -1.5145 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -1.2444& 0.0080 \\ 0.0350& -1.3375 \end{array} \right] $
    Dissipativity $ K_1 = \left[ \begin{array} {ccc} -1.2143& -0.1133 \\ -0.1427& -1.4406 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -0.8945& -0.1379\\ -0.1690& -1.2041 \end{array} \right] $
    $ H_\infty $ performance $ K_1 = \left[ \begin{array} {ccc} -1.5024& 0.0015 \\ 0.0212& -1.5711 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -1.2859& 0.0376\\ 0.1014& -1.3978 \end{array} \right] $
    $ L_2 $-$ L_\infty $ performance $ K_1 = \left[ \begin{array} {ccc} -1.3752& -0.0691 \\ -0.0757& -1.4975 \end{array} \right], \ K_2 = \left[ \begin{array} {ccc} -0.9550& -0.1274\\ -0.1197& -1.2459 \end{array} \right] $
     | Show Table
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