Observer design for practical exponential stabilization of polynomial fuzzy time-delay models

Slim Dhahri; Essia Ben Alaia; Afrah Alanazi; Hamdi Gassara; Sahar Almenwer; Slim Dhahri; Essia Ben Alaia; Afrah Alanazi; Hamdi Gassara; Sahar Almenwer

doi:10.3934/math.2026720

AIMS Mathematics

2026, Volume 11, Issue 6: 17635-17652. doi: 10.3934/math.2026720

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Research article

Observer design for practical exponential stabilization of polynomial fuzzy time-delay models

1.
Department of Computer Engineering and Networks, College of Computer and Information Sciences, Jouf University, Saudi Arabia
2.
Department of Information System, College of Computer and Information Sciences, Jouf University, Saudi Arabia
3.
Laboratory of Sciences and Technology of Automatic Control and Computer Engineering, National School of Engineering of Sfax, University of Sfax, Post Box 1173, 3038 Sfax, Tunisia
4.
Department of Computer Science, College of Computer Science and Information, Jouf University, Saudi Arabia

Received: 23 January 2026 Revised: 14 May 2026 Accepted: 28 May 2026 Published: 17 June 2026
MSC : 93C10, 93C43, 93D23

In this paper, we study the practical exponential stability (PES) of nonlinear time-delay (TD) systems with bounded exogenous input (BEI). Unlike existing results, this work is the first to provide a sum-of-squares (SOS)-based design to ensure the PES of the nonlinear TD system. First, polynomial fuzzy (PF) modeling is used to characterize the nonlinear behavior of the system. Specifically, we consider a class of models in which the premise variables are measurable, and the polynomial matrices depend on any measurable variable. Second, we propose a PF observer-based (PFOB) control strategy that guarantees the system's PES under the condition that the system is observable, even when some state variables are not directly measurable. Next, the PFOB controller is designed via the SOS approach to ensure the PES of the augmented system formed by the state and the state estimation error. This study is the first to incorporate recently proposed relaxed conditions for parameterized linear matrix inequalities formulated in a double-sum representation into the SOS approach. Finally, a numerical case study validates the PFOB control strategy.
- global uniform practical exponential stability,
- sum-of-squares approach,
- polynomial fuzzy models,
- polynomial fuzzy observer,
- time delay systems
Citation: Slim Dhahri, Essia Ben Alaia, Afrah Alanazi, Hamdi Gassara, Sahar Almenwer. Observer design for practical exponential stabilization of polynomial fuzzy time-delay models[J]. AIMS Mathematics, 2026, 11(6): 17635-17652. doi: 10.3934/math.2026720

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Abstract

In this paper, we study the practical exponential stability (PES) of nonlinear time-delay (TD) systems with bounded exogenous input (BEI). Unlike existing results, this work is the first to provide a sum-of-squares (SOS)-based design to ensure the PES of the nonlinear TD system. First, polynomial fuzzy (PF) modeling is used to characterize the nonlinear behavior of the system. Specifically, we consider a class of models in which the premise variables are measurable, and the polynomial matrices depend on any measurable variable. Second, we propose a PF observer-based (PFOB) control strategy that guarantees the system's PES under the condition that the system is observable, even when some state variables are not directly measurable. Next, the PFOB controller is designed via the SOS approach to ensure the PES of the augmented system formed by the state and the state estimation error. This study is the first to incorporate recently proposed relaxed conditions for parameterized linear matrix inequalities formulated in a double-sum representation into the SOS approach. Finally, a numerical case study validates the PFOB control strategy.

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