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

Ramasamy Saravanakumar; Yang Cao; Ali Kazemy; Quanxin Zhu

doi:10.3934/dcdss.2022082

Article Contents

2022, Volume 15, Issue 11: 3313-3330. Doi: 10.3934/dcdss.2022082

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

1.
Graduate School of Advanced Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739-8527, Japan
2.
School of Cyber Science and Engineering, Southeast University, Nanjing 211189, China
3.
Department of Electrical Engineering, Tafresh University, Tafresh 39518-79611, Iran
4.
MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
5.
Key Laboratory of Control and Optimization of Complex Systems, College of Hunan Province, Hunan Normal University, Changsha 410081, China

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

Received: December 31, 2021

Revised: January 31, 2022

Early access: March 2022

Published: September 2022

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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Abstract

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.

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

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Figure 1. Synchronization error (Example 1)

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Figure 2. Generated control signals (Example 1)

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Figure 3. Synchronization error (Example 2)

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Figure 4. Generated control signals (Example 2)

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

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

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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] $

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