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

Ramasamy Kavikumar; Boomipalagan Kaviarasan; Yong-Gwon Lee; Oh-Min Kwon; Rathinasamy Sakthivel; Seong-Gon Choi

doi:10.3934/dcdss.2022014

Article Contents

2022, Volume 15, Issue 7: 1839-1858. Doi: 10.3934/dcdss.2022014

This issue Previous Article Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3) Next Article Composite control with observers for a class of stochastic systems with multiple disturbances

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

1.
School of Electrical Engineering, Chungbuk National University, Cheongju, 28644, South Korea
2.
Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India
3.
School of Information and Communication Engineering, Chungbuk National University, Cheongju 28644, South Korea

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

Received: October 31, 2021

Revised: November 30, 2021

Published: July 2022

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Abstract

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.

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

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Figure 1. IT2 fuzzy membership functions

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

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

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

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

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Figure 6. State responses of the open-loop system

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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

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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

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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

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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

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