Generalized conformable operators: Application to the design of nonlinear observers

Fidel Meléndez-Vázquez; Guillermo Fernández-Anaya; Aldo Jonathan Muñóz-Vázquez; Eduardo Gamaliel Hernández-Martínez; Fidel Meléndez-Vázquez; Guillermo Fernández-Anaya; Aldo Jonathan Muñóz-Vázquez; Eduardo Gamaliel Hernández-Martínez

doi:10.3934/math.2021749

AIMS Mathematics

2021, Volume 6, Issue 11: 12952-12975. doi: 10.3934/math.2021749

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Research article

Generalized conformable operators: Application to the design of nonlinear observers

1.
Department of Physics and Mathematics, Universidad Iberoamericana, Ciudad de México, Prol. Paseo de la Reforma 880, Lomas de Santa Fe, Álvaro Obregón, Mexico City 01219, México
2.
Department of Multidisciplinary Engineering, Texas A & M University, Higher Education Center, 6200 Tres Lagos Blvd., McAllen, TX 78504, USA
3.
Institute of Applied Research and Technology, Universidad Iberoamericana, Ciudad de México, Prol. Paseo de la Reforma 880, Lomas de Santa Fe, Álvaro Obregón, Mexico City 01219, México

Received: 26 July 2021 Accepted: 07 September 2021 Published: 13 September 2021
MSC : 93D23, 93B53

In this work, a pair of observers are proposed for a class of nonlinear systems whose dynamics involve a generalized differential operator that encompasses the conformable derivatives. A generalized conformable exponential stability function, based on this derivative, is introduced in order to prove some Lyapunov-like theorems. These theorems help to verify the stability of the observers proposed, which is exponential in a generalized sense. The performance of the observation scheme is evaluated by means of numerical simulations. Moreover, a comparison of the results obtained with integer, fractional, and generalized conformable derivatives is made.
- generalized conformable derivative,
- exponential stability,
- Lyapunov stability,
- nonlinear quadratic regulator,
- high-gain observer
Citation: Fidel Meléndez-Vázquez, Guillermo Fernández-Anaya, Aldo Jonathan Muñóz-Vázquez, Eduardo Gamaliel Hernández-Martínez. Generalized conformable operators: Application to the design of nonlinear observers[J]. AIMS Mathematics, 2021, 6(11): 12952-12975. doi: 10.3934/math.2021749

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Abstract

In this work, a pair of observers are proposed for a class of nonlinear systems whose dynamics involve a generalized differential operator that encompasses the conformable derivatives. A generalized conformable exponential stability function, based on this derivative, is introduced in order to prove some Lyapunov-like theorems. These theorems help to verify the stability of the observers proposed, which is exponential in a generalized sense. The performance of the observation scheme is evaluated by means of numerical simulations. Moreover, a comparison of the results obtained with integer, fractional, and generalized conformable derivatives is made.

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