Extended dissipativity-based sampled-data controller design for fuzzy distributed parameter systems

R. Suresh; S. Saravanan; Hemen Dutta; R. Vadivel; Nallappan Gunasekaran

doi:10.3934/mfc.2024006

Article Contents

2025, Volume 8, Issue 4: 540-562. Doi: 10.3934/mfc.2024006

This issue Previous Article Fixed point property for nonexpansive mappings on large classes in Köthe-Toeplitz duals of certain difference sequence spaces Next Article Existence and controllability results of semilinear Sobolev type difference equation

Extended dissipativity-based sampled-data controller design for fuzzy distributed parameter systems

1.
Department of Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur 602117, India
2.
Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600 069, Tamilnadu, India
3.
Department of Mathematics, Gauhati University, Guwahati, India
4.
Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat University, Phuket-83000, Thailand
5.
Eastern Michigan Joint College of Engineering, Beibu Gulf University, Qinzhou 535011, China

^*Corresponding authors: R. Vadivel and Nallappan Gunasekaran
^*Corresponding authors: R. Vadivel and Nallappan Gunasekaran

Received: November 2023

Revised: January 2024

Early access: February 22, 2024

Published online: February 22, 2024

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Abstract

This paper presents an extended dissipativity-based sampled-data controller design for Fuzzy distributed parameter systems (DPSs). Then the proposed DPSs solved by integral inequality techniques to construct a Lyapunov-Krasovskii functional (LKF) that demonstrates the stability and stabilization of the dissipativity performance of DPSs. The fuzzy-based sampled-data control (FSDC) scheme is obtained by solving linear matrix inequalities (LMIs), ensuring that the closed-loop system is extended dissipativity. The FSDC can be adjusted to achieve various performance goals, such as $ L_2-L_{\infty} $, $ H_{\infty} $, passivity, and $ (Q, S, R)-\gamma $-dissipative performance for DPSs. The proposed method is verified through simulations using the MATLAB LMI control toolbox.

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Mathematics Subject Classification: Primary: 93C57, 93C10; Secondary: 35R13.

Citation:

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Figure 1. Trajectories of states for example 5.1 with $ L_2 $-$ L_\infty $ performance

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Figure 2. Trajectories of control input for the Example 5.1 with $ L_2 $-$ L_\infty $ performance

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Figure 3. Trajectories of states for the Example 5.1 with $ H_\infty $ performance

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Figure 4. Trajectories of control input for the Example 5.1 with $ H_\infty $ performance

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Figure 5. Trajectories of states for example 5.1 with passivity performance

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Figure 6. Trajectories of control input for the example 5.1 with passivity performance

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Figure 7. Trajectories of states for example 5.1 with $ (Q,S,R)-\gamma $-dissipativity performance

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Figure 8. Trajectories of control input for the Example 5.1 $ (Q,S,R)-\gamma $-dissipativity performance

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Figure 9. State responses of the system (43) with $ L_2-L_\infty $ performances

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Figure 10. State responses of the system (43) with $ H_\infty $ performances

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Figure 11. State curves of the system (43) with passivity performances

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Figure 12. State responses of the system with (43) $ (Q,S,R) $-$ \gamma $ Dissipativity performances

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Table 1. Four Cases of the ED Problems:

Performance/R	$ R_1 $	$ R_2 $	$ R_3 $	$ R_4 $
$ L_2-L_\infty $	0	0	$ \gamma^2I $	$ I $
$ H_\infty $	-$ I $	0	$ \gamma^2I $	$ 0 $
Passivity	0	$ I $	$ \gamma I $	0
Dissipativity	-0.5$ I $	$ I $	$ 0.2I-\gamma I $	$ 0 $

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Table 2. Different minimums $ \gamma $ for various $ \hslash $ and fixed $ \mu = 0.3 $ in Example 5.2

$ \hslash $	0.01	0.075	0.1	0.15
$ \gamma $	0.6742	0.3245	0.2145	0.0945

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Table 3. Different minimum $ \gamma $ for various $ \hslash $ and fixed $ \mu = 0.3 $ in Example 5.2

$ \hslash $	0.01	0.075	0.1	0.15	0.25
$ \gamma $	0.9012	0.8745	0.7000	0.5228	0.4512

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Table 4. Allowable maximum $ \hslash $ for various $ \mu $ and fixed $ \gamma $ = 0.7 in Example 5.2

$ \mu $	0.1	0.15	0.2	0.25	0.3
$ \hslash $	0.0640	0.0787	0.1513	0.1720	0.2324

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Table 5. Allowable maximum $ \hslash $ for various $ \mu $ and fixed $ \gamma $ = 0.7 in Example 5.2

$ \mu $	0.1	0.2	0.3	0.4	0.5
$ \hslash $	0.2010	0.1542	0.0840	0.0536	0.0402

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