New hybrid model for nonlinear systems via Takagi-Sugeno fuzzy approach

Anouar Ben Mabrouk; Abdulaziz Alanazi; Zaid Bassfar; Dalal Alanazi; Anouar Ben Mabrouk; Abdulaziz Alanazi; Zaid Bassfar; Dalal Alanazi

doi:10.3934/math.20241128

AIMS Mathematics

2024, Volume 9, Issue 9: 23197-23220. doi: 10.3934/math.20241128

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Research article

New hybrid model for nonlinear systems via Takagi-Sugeno fuzzy approach

1.
Department of Mathematics, Faculty of Science, University of Tabuk, King Faisal road, Tabuk 71491, Saudi Arabia
2.
Faculty of Computing and Information Technology, University of Tabuk, King Faisal road, Tabuk 71491, Saudi Arabia

Received: 27 May 2024 Revised: 12 July 2024 Accepted: 23 July 2024 Published: 30 July 2024
MSC : 34A07, 65T60, 93C42, 93D05

Mathematical models, especially complex nonlinear systems, are always difficult to analyze and synthesize, and researchers need effective and suitable control methods to address these issues. In the present work, we proposed a hybrid method that combines the well-known Takagi-Sugeno fuzzy model with wavelet decomposition to investigate nonlinear systems characterized by the presence of mixed nonlinearities. Here, one nonlinearity is super-linear and convex, and other is sub-linear, concave, and singular at zero, which leads to difficulties in the analysis, as is known in PDE theory. Linear and polynomial fuzzy models were combined with wavelets to ensure an improvement in both methods for investigating such problems. The results showed a high performance compared with existing methods via error estimates and Lyapunov theory of stability. The model was applied to a prototype nonlinear Schrödinger dynamical system.
- control,
- dynamical systems,
- nonlinear systems,
- fuzzy model,
- wavelets,
- Lyapunov stability
Citation: Anouar Ben Mabrouk, Abdulaziz Alanazi, Zaid Bassfar, Dalal Alanazi. New hybrid model for nonlinear systems via Takagi-Sugeno fuzzy approach[J]. AIMS Mathematics, 2024, 9(9): 23197-23220. doi: 10.3934/math.20241128

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Abstract

Mathematical models, especially complex nonlinear systems, are always difficult to analyze and synthesize, and researchers need effective and suitable control methods to address these issues. In the present work, we proposed a hybrid method that combines the well-known Takagi-Sugeno fuzzy model with wavelet decomposition to investigate nonlinear systems characterized by the presence of mixed nonlinearities. Here, one nonlinearity is super-linear and convex, and other is sub-linear, concave, and singular at zero, which leads to difficulties in the analysis, as is known in PDE theory. Linear and polynomial fuzzy models were combined with wavelets to ensure an improvement in both methods for investigating such problems. The results showed a high performance compared with existing methods via error estimates and Lyapunov theory of stability. The model was applied to a prototype nonlinear Schrödinger dynamical system.

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