Discussion on Barbalat Lemma extensions for conformable fractional integrals

Souahi, Abdourazek; Naifar, Omar; Makhlouf, Abdellatif Ben; Hammami, Mohamed Ali

doi:10.1080/00207179.2017.1350754

Cited by 25 publications

(24 citation statements)

References 35 publications

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“…In this case, in order to prove the convergence of the state estimates, we have exploited Lemma 3 (which is an extension of a Barbalat-type lemma [22] to the fractional-order framework. This extension has not been yet demonstrated using the Caputo derivative [24]. )…”

Section: The Faulty Operating Casementioning

confidence: 97%

“…In order to overcome this situation, in the present paper we take advantage of the conformable fractional derivative concept. Indeed, it is proved in [24] that there exists an analogue Barbalat's lemma version to the one in [22].…”

Section: Introductionmentioning

confidence: 99%

“…Two different cases are studied: a healthy operating case and a faulty operating case. When possible unknown actuator faults are considered in the system's model, an adaptive observer is used and it is necessary to use the Barbalat-like lemma, proved in [24], in order to demonstrate the convergence to zero of the state estimation error. Such a result cannot be proved using the Caputo derivative concept.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

State estimation for nonlinear conformable fractional‐order systems: A healthy operating case and a faulty operating case

Jmal¹,

Elloumi

Naifar³

et al. 2019

Asian Journal of Control

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The issue of estimating states for classical integer‐order nonlinear systems has been widely addressed in the literature. Yet, generalization of existing results to the fractional‐order framework represents a fertile area of research. Note that, recently, a new and advantageous type of fractional derivative, the conformable derivative, was defined. So far, the general query of designing observers for conformable fractional‐order systems has not been investigated. In addition, it has been proved in the literature that some important tools for stability analysis of fractional‐order systems are valid using the conformable derivative concept, but invalid using other fractional derivative concepts. Motivated by the cited facts, this paper presents a first‐state estimation scheme for fractional‐order systems under the conformable derivative concept. A healthy operating case and a faulty operating case are treated. In this paper, a version of Barbalat's lemma, which is invalid using the well‐known Caputo derivative, is exploited to prove the convergence of the estimation errors. In order to validate the theoretical results, a numerical example is studied in the simulation section.

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Section: The Faulty Operating Casementioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

State estimation for nonlinear conformable fractional‐order systems: A healthy operating case and a faulty operating case

Jmal¹,

Elloumi

Naifar³

et al. 2019

Asian Journal of Control

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show abstract

“…Zhao et al established the multivariate theory of GCFD and illustrated the conformable Maxwell equations, and theorems for Conformable Gauss's, Green's, and Stokes's Theorem, see Zhao et al 22 Though, this new local fractional derivative fails some properties as pointed out in previous studies, [23][24][25][26] it seems to account for many deficiencies of some of the earlier proposed definitions which are of great importance in applied sciences and therefore suitable for more applications. Some authors followed this work and explored potential applications in various fields such as the control theory of dynamical systems, [27][28][29][30] mathematical biology and epidemiology, 19,[31][32][33][34] mechanics, 16,35,36 systems of linear and nonlinear conformable fractional differential equations (CFDEs), [37][38][39] quantum mechanics, 40,41 variational calculus, 42,43 arbitrary time scale problems, [44][45][46] modelling of diffusion, 47,48 stochastic process, 49,50 and optics. 51 Some analytical and numerical methods have attracted great interest and became an important tool for differential equations with CFDs, (see previous studies ).…”

Section: Introductionmentioning

confidence: 99%

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Mortazaasl

Akbarfam

2020

Math Methods in App Sciences

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In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm-Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm-Liouville theory. In the class of r-CFSLPs, we discuss two types of CFSLPs which include left-and right-sided CFDs, each of order ∈ (n, n + 1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed-point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s-CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order ∈ (0, 1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs-I and-II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved. KEYWORDS conformable fractional derivatives and integrals, eigenvalues and eigenfunctions, fixed-point theorem, fractional diffusion, fractional Sturm-Liouville operators, transform methods JEL CLASSIFICATION

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“…Naifar et al 40 investigated the global practical Mittag-Leffler stabilization of fractional-order delayed nonlinear systems by means of observers. Souahi et al 41 studied the Barbalat lemma for conformable fractional integrals, and these lemma are easily applied to control theory, especially adaptive control and adaptive observers.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis and robust synchronization of fractional‐order competitive neural networks with different time scales and impulsive perturbations

Anbalagan

Raja

Agarwal

et al. 2019

Adaptive Control & Signal

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This article mainly examine a class of robust synchronization, global stability criterion, and boundedness analysis for delayed fractional-order competitive type-neural networks with impulsive effects and different time scales. Firstly, by endowing the robust analysis skills and a new class of Lyapunov-Krasovskii functional approach, the error dynamical system is furnished to be a robust adaptive synchronization in the voice of linear matrix inequality (LMI) technique. Secondly, by ignoring the uncertain parameter terms, the existence of equilibrium points are established by means of topological degree properties, and the solution representation of the considered network model are provided. Thirdly, a novel global asymptotic stability condition is proposed in the voice of LMIs, which is less conservative. Finally, our analytical results are justified with two numerical examples with simulations.

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Discussion on Barbalat Lemma extensions for conformable fractional integrals

Cited by 25 publications

References 35 publications

State estimation for nonlinear conformable fractional‐order systems: A healthy operating case and a faulty operating case

State estimation for nonlinear conformable fractional‐order systems: A healthy operating case and a faulty operating case

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Stability analysis and robust synchronization of fractional‐order competitive neural networks with different time scales and impulsive perturbations

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