Innovative observer design for nonlinear systems using Caputo fractional derivative with respect to another function

Kareem Alanazi; Omar Naifar; Raouf Fakhfakh; Abdellatif Ben Makhlouf; Kareem Alanazi; Omar Naifar; Raouf Fakhfakh; Abdellatif Ben Makhlouf

doi:10.3934/math.20241686

AIMS Mathematics

2024, Volume 9, Issue 12: 35533-35550. doi: 10.3934/math.20241686

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Innovative observer design for nonlinear systems using Caputo fractional derivative with respect to another function

1.
Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka, Saudi Arabia; Email: k.alanazi@ju.edu.sa, rfakhfakh@ju.edu.sa
2.
Sfax University, Engineering National School, Electrical Engineering Department, Control and Energy Management Laboratory (CEM lab), BP 1173, Sfax 3038, Tunisia
3.
Sfax University, Faculty of Sciences of Sfax, Department of Mathematics, BP 1171, Sfax 3000, Tunisia; Email: abdellatif.benmakhlouf@fss.usf.tn

Received: 16 October 2024 Revised: 07 December 2024 Accepted: 16 December 2024 Published: 20 December 2024
MSC : 34A08, 93B07, 93B52, 93C10

This work introduces a novel control framework using the Caputo fractional derivative (CFD) with respect to another function—a derivative that has not been thoroughly treated in control theory. By extending the widely recognized Caputo-Hadamard (CH) fractional-order derivative, we address its utility in nonlinear systems. The core of our contribution is the practical stability for systems governed by this derivative, which ensures convergence toward a bounded region around the origin. Additionally, we extend the Lipschitz condition (LC) to the one-sided Lipschitz (OSL) condition for observer design and observer based-control design in fractional-order systems, ensuring its practical stability. Finally, three numerical examples validate the effectiveness of our proposed framework, providing practical insights for control theory advancements.
- observer-based control,
- practical stability,
- CFD with respect to another function,
- control,
- observer
Citation: Kareem Alanazi, Omar Naifar, Raouf Fakhfakh, Abdellatif Ben Makhlouf. Innovative observer design for nonlinear systems using Caputo fractional derivative with respect to another function[J]. AIMS Mathematics, 2024, 9(12): 35533-35550. doi: 10.3934/math.20241686

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Abstract

This work introduces a novel control framework using the Caputo fractional derivative (CFD) with respect to another function—a derivative that has not been thoroughly treated in control theory. By extending the widely recognized Caputo-Hadamard (CH) fractional-order derivative, we address its utility in nonlinear systems. The core of our contribution is the practical stability for systems governed by this derivative, which ensures convergence toward a bounded region around the origin. Additionally, we extend the Lipschitz condition (LC) to the one-sided Lipschitz (OSL) condition for observer design and observer based-control design in fractional-order systems, ensuring its practical stability. Finally, three numerical examples validate the effectiveness of our proposed framework, providing practical insights for control theory advancements.

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