Introduction of new kernels and new models to solve the drawbacks of fractional integration/differentiation operators and classical fractional-order models

2022 ◽  
pp. 551-586
Author(s):  
Jocelyn Sabatier
Keyword(s):  
Fractional Order ◽  
Fractional Integration ◽  
New Models

CONTINUOUS-TIME FRACTIONAL ORDER LINEAR SYSTEMS IDENTIFICATION USING CHEBYSHEV WAVELET

2020 ◽  
Vol 6 (8(77)) ◽  
pp. 23-28
Author(s):  
Shuen Wang ◽  
Ying Wang ◽  
Yinggan Tang
Keyword(s):  
Linear Systems ◽  
Fractional Order ◽  
Continuous Time ◽  
Fractional Integration ◽  
Operational Matrices ◽  
Computation Complexity ◽  
Order Linear ◽  
Fractional Integration Operator ◽  
Systems Identification ◽  
Chebyshev Wavelet

In this paper, the identification of continuous-time fractional order linear systems (FOLS) is investigated. In order to identify the differentiation or- ders as well as parameters and reduce the computation complexity, a novel identification method based on Chebyshev wavelet is proposed. Firstly, the Chebyshev wavelet operational matrices for fractional integration operator is derived. Then, the FOLS is converted to an algebraic equation by using the the Chebyshev wavelet operational matrices. Finally, the parameters and differentiation orders are estimated by minimizing the error between the output of real system and that of identified systems. Experimental results show the effectiveness of the proposed method.

Initialization of Identification of Fractional Model by Output-Error Technique

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Abir Khadhraoui ◽  
Khaled Jelassi ◽  
Jean-Claude Trigeassou ◽  
Pierre Melchior
Keyword(s):  
Numerical Simulation ◽  
Least Squares ◽  
Fractional Order ◽  
Fractional Integration ◽  
New Approach ◽  
Fractional Model ◽  
Output Error

A bad initialization of output-error (OE) technique can lead to an inappropriate identification results. In this paper, we introduce a solution to this problem; the basic idea is to estimate the parameters and the fractional order of the noninteger system by a new approach of least-squares (LS) method based on repeated fractional integration to initialize OE technique. It will be shown that LS method offers a good initialization to OE algorithm and leads to acceptable identification results. The performance of the proposed method is shown through numerical simulation examples.

scholarly journals Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?

2020 ◽  
Vol 4 (3) ◽  
pp. 40
Author(s):  
Jocelyn Sabatier
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Current Trend ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Definition Of ◽  
A Current

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.

scholarly journals Absorptive Repetitive Filters with Operators of Generalized Fractional Calculus

2013 ◽  
Vol 13 (4) ◽  
pp. 42-53 ◽  
Author(s):  
Nina Nikolova ◽  
Emil Nikolov
Keyword(s):  
Comparative Analysis ◽  
Fractional Calculus ◽  
Control Systems ◽  
Fractional Order ◽  
Fractional Integration ◽  
Frequency Characteristics ◽  
Rational Approximations ◽  
Integer Order ◽  
New Class ◽  
Generalized Fractional Calculus

Abstract : An essentially new class of repetitive fractional disturbance absorptive filters in disturbances absorbing control systems is proposed in the paper. Systematization of the standard repetitive fractional disturbance absorptive filters of this class is suggested. They use rational approximations of the operators for fractional integration in the theory of fractional calculus. The paper discusses the possibilities for repetitive absorbing of the disturbances with integer order filters and with fractional order filters. The results from the comparative analysis of their frequency characteristics are given below.

What Euler could further write, or the unnoticed “big bang” of the fractional calculus

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Igor Podlubny
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
English Translation ◽  
Inverse Relationship ◽  
Short Communication ◽  
Fractional Integration ◽  
Big Bang ◽  
Fractional Differentiation ◽  
Famous Paper ◽  
Latin Version

AbstractIn this short communication, an attempt is made to continue beyond paragraph 29 of Euler’s famous paper in Vol. 5 of Comment. Acad. Sci. Petropol. (1738), using his style of storytelling to extrapolate the audacity of his approach from fractional differentiation to fractional integration. To add the authenticity and the amusement to the imitation, the emulated paragraphs 30–32 are first presented in Latin version followed by the English translation.This reconstruction aims to demonstrate that Euler could consider not only fractional differentiation, but also fractional-order integration and its inverse relationship with differentiation of the same fractional order.

scholarly journals Parameter Estimation of the Lotka–Volterra Model with Fractional Order Based on the Modulation Function and Its Application

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ying Hao ◽  
Mingshun Guo
Keyword(s):  
Parameter Estimation ◽  
Fractional Order ◽  
Fractional Integration ◽  
Linear Equations ◽  
Volterra Model ◽  
Modulation Function ◽  
Implementation Program ◽  
Shifted Chebyshev Polynomial ◽  
Improved Model ◽  
Universal Applicability

The Lotka–Volterra model is widely applied in various fields, and parameter estimation is important in its application. In this study, the Lotka–Volterra model with universal applicability is established by introducing the fractional order. Modulation function is multiplied by both sides of the Lotka–Volterra model, and the model is converted into linear equations with parameters to be estimated by the fractional integration method. The parameters are obtained by solving the equations. The state of the system is estimated by shifted Chebyshev polynomial. Last, the implementation program of the model is compiled. The concrete implementation method of the improved model is proposed by an example in this study.

scholarly journals Multistability Emergence through Fractional-Order-Derivatives in a PWL Multi-Scroll System

Electronics ◽  
2020 ◽  
Vol 9 (6) ◽  
pp. 880 ◽  
Author(s):  
José Luis Echenausía-Monroy ◽  
Guillermo Huerta-Cuellar ◽  
Rider Jaimes-Reátegui ◽  
Juan Hugo García-López ◽  
Vicente Aboites ◽  
...  
Keyword(s):  
Fractional Order ◽  
Fractional Integration ◽  
Basin Of Attraction ◽  
Parameter Variation ◽  
Order System ◽  
Bifurcation Parameter ◽  
Stable States ◽  
Integer Order ◽  
The Stability ◽  
Integration Order

In this paper, the emergence of multistable behavior through the use of fractional-order-derivatives in a Piece-Wise Linear (PWL) multi-scroll generator is presented. Using the integration-order as a bifurcation parameter, the stability in the system is modified in such a form that produces a basin of attraction segmentation, creating many stable states as scrolls are generated in the integer-order system. The results here presented reproduce the same phenomenon reported in systems with integer-order derivatives, where the multistable regimen is obtained through a parameter variation. The multistable behavior reported is also validated through electronic simulation. The presented results are not only applicable in engineering fields, but they also enrich the analysis and the understanding of the implications of using fractional integration orders, boosting the development of further and better studies.

Differential and integral relations in the class of multi-index Mittag-Leffler functions

2018 ◽  
Vol 21 (1) ◽  
pp. 254-265 ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Integration ◽  
Multi Index ◽  
Generalized Fractional Calculus ◽  
Special Cases ◽  
Integral Relations ◽  
Derivatives Of ◽  
Prabhakar Function

Abstract As recently observed by Bazhlekova and Dimovski [1], the n-th derivative of the 2-parametric Mittag-Leffler function gives a 3-parametric Mittag-Leffler function, known as the Prabhakar function. Following this analogy, the n-th derivative of the (2m-index) multi-index Mittag-Leffler functions [6] is obtained, and it turns out that it is expressed in terms of the (3m-index) Mittag-Leffler functions [10, 11]. Further, some special cases of the fractional order Riemann-Liouville and Erdélyi-Kober integrals of the Mittag-Leffler functions are calculated and interesting relations are proved. Analogous relations happen to connect the 3m-Mittag-Leffler functions with the integrals and derivatives of 2m-Mittag-Leffler functions. Finally, multiple Erdélyi-Kober fractional integration operators, as operators of the generalized fractional calculus [5], are shown to relate the 2m- and 3m-parametric Mittag-Leffler functions.

Complexity Analysis of Time Series Based on Generalized Fractional Order Refined Composite Multiscale Dispersion Entropy

2020 ◽  
Vol 30 (14) ◽  
pp. 2050211
Author(s):  
Yu Wang ◽  
Pengjian Shang
Keyword(s):  
Time Series ◽  
Fractional Order ◽  
Complexity Analysis ◽  
Financial Time Series ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Entropy Model ◽  
Intrinsic Nature ◽  
New Models ◽  
Stability And Accuracy

Based on the dispersion entropy model, combined with multiscale analysis method and fractional order information entropy theory, this paper proposes new models — the generalized fractional order multiscale dispersion entropy (GMDE) and the generalized fractional order refined composite multiscale dispersion entropy (GRCMDE). The new models take the amplitude value information of the sequence itself into consideration, which can make better use of some key information in the sequence and have a higher stability and accuracy. In addition, extending the algorithm to generalized fractional order can make the model better capture the small evolution of the signal data, which is more advantageous for studying the dynamic characteristics of complex systems. This paper verifies the effectiveness of the new models by combining theoretical analysis with empirical research, and further studies the complexity of the financial system and the nature of its multiple time scales. The results show that the proposed GMDE, GRCMDE can better detect the intrinsic nature of financial time series and can distinguish the financial market complexity of different countries.

scholarly journals Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hasib Khan ◽  
Hossein Jafari ◽  
Rahmat Ali Khan ◽  
Haleh Tajadodi ◽  
Sarah Jane Johnston
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Bernstein Polynomials ◽  
Operational Matrices ◽  
Algebraic Equations

In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

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