scholarly journals Non-asymptotic estimation for fractional integrals of noisy accelerations for fractional order vibration systems

Automatica ◽  
2022 ◽  
Vol 135 ◽  
pp. 109996
Author(s):  
Yang Tian ◽  
Zhi-Bo Wang ◽  
Da-Yan Liu ◽  
Driss Boutat ◽  
Hao-Ran Liu
Keyword(s):  
Fractional Order ◽  
Fractional Integrals ◽  
Asymptotic Estimation

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

scholarly journals DYNAMICS OF FRACTAL SOLIDS

2005 ◽  
Vol 19 (27) ◽  
pp. 4103-4114 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fractional Order ◽  
Experimental Test ◽  
Fractional Integral ◽  
Continuous Medium ◽  
Fractional Integrals ◽  
Generalized Equations ◽  
Moments Of Inertia ◽  
Medium Model ◽  
Mass Dimension ◽  
Euler's Equations

We describe the fractal solid by a special continuous medium model. We propose to describe the fractal solid by a fractional continuous medium model, where all characteristics and fields are defined everywhere in the volume but they follow some generalized equations which are derived by using fractional integrals of fractional order. The order of fractional integral can be equal to the fractal mass dimension of the solid. Fractional integrals are considered as an approximation of integrals on fractals. We suggest the approach to compute the moments of inertia for fractal solids. The dynamics of fractal solids are described by the usual Euler's equations. The possible experimental test of continuous medium model for fractal solids is considered.

On stable integral manifolds for impulsive Kolmogorov systems of fractional order

2017 ◽  
Vol 31 (15) ◽  
pp. 1750168
Author(s):  
Gani Stamov ◽  
Ivanka Stamova
Keyword(s):  
Fractional Order ◽  
Lyapunov Functions ◽  
Type System ◽  
Fractional Integrals ◽  
Order System ◽  
Kolmogorov Type ◽  
Fractal Media ◽  
Integral Manifolds ◽  
Existence And Stability ◽  
Caputo Fractional Order Derivative

In this paper, an impulsive Kolmogorov-type system using the Caputo fractional-order derivative is developed. The fractional-order system displays many interesting dynamic behaviors and fractional integrals can be used to describe the fractal media. The existence and stability of integral manifolds for the impulsive fractional model are considered. The main results are proved by means of piecewise continuous Lyapunov functions and the new fractional comparison principle. The impulses are realized at variable impulsive moments of time and can be considered as a control. Finally, an example is given to illustrate our results.

scholarly journals Design Of Multivariable Fractional Order Pid Controller Using Covariance Matrix Adaptation Evolution Strategy

2014 ◽  
Vol 24 (2) ◽  
pp. 235-251 ◽  
Author(s):  
Sudalaiandi Sivananaithaperumal ◽  
Subramanian Baskar
Keyword(s):  
Covariance Matrix ◽  
Fractional Order ◽  
Pid Controller ◽  
Order Approximation ◽  
Absolute Error ◽  
Evolution Strategy ◽  
Fractional Integrals ◽  
Integer Order ◽  
Covariance Matrix Adaptation ◽  
Adaptation Evolution

Abstract This paper presents an automatic tuning of multivariable Fractional-Order Proportional, Integral and Derivative controller (FO-PID) parameters using Covariance Matrix Adaptation Evolution Strategy (CMAES) algorithm. Decoupled multivariable FO-PI and FO-PID controller structures are considered. Oustaloup integer order approximation is used for the fractional integrals and derivatives. For validation, two Multi-Input Multi- Output (MIMO) distillation columns described byWood and Berry and Ogunnaike and Ray are considered for the design of multivariable FO-PID controller. Optimal FO-PID controller is designed by minimizing Integral Absolute Error (IAE) as objective function. The results of previously reported PI/PID controller are considered for comparison purposes. Simulation results reveal that the performance of FOPI and FO-PID controller is better than integer order PI/PID controller in terms of IAE. Also, CMAES algorithm is suitable for the design of FO-PI / FO-PID controller.

scholarly journals Fractional Sums and Differences with Binomial Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Fahd Jarad ◽  
Ravi P. Agarwal
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Binomial Coefficients ◽  
Discrete Version ◽  
Binomial Theorem ◽  
Left And Right ◽  
Fractional Integrals And Derivatives ◽  
Dual Identities

In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

scholarly journals Fractional integrals of generalised functions

1972 ◽  
Vol 14 (1) ◽  
pp. 30-37 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Fractional Order ◽  
Fractional Integrals ◽  
Fractional Powers ◽  
Pointwise Multiplication ◽  
Convolution Of Distributions ◽  
Independent Variable ◽  
Image Position

The concept of integrals of fractional order of a function f, defined by if Reα > 0, can be extended to generalised functions in the framework of the theory of convolution of distributions. The resulting theory [2, Chap. I §5.5] is very satisfactory for many purposes but there are circumstances in which it is not suitable. Such circumstances arise in particular if it is necessary to multiply, before or after integratrion, by non-integral powers of the variable. Pointwise multiplication by fractional powers of the independent variable does not make sense in the theory of distributions.

scholarly journals The multiple composition of the left and right fractional Riemann-Liouville integrals - analytical and numerical calculations

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6087-6099 ◽  
Author(s):  
Mariusz Ciesielski ◽  
Tomasz Blaszczyk
Keyword(s):  
Numerical Calculation ◽  
Fractional Order ◽  
Fractional Integral ◽  
Integral Operators ◽  
Numerical Calculations ◽  
Fractional Integrals ◽  
Numerical Accuracy ◽  
Fractional Integral Operators ◽  
Left And Right ◽  
The Right

New fractional integral operators of order ? ? R+ are introduced. These operators are defined as the composition of the left and right (or the right and left) Riemann-Liouville fractional order integrals. Some of their properties are studied. Analytical results of fractional integrals of several functions are presented. For a numerical calculation of fractional order integrals, two numerical procedures are given. In the final part of this paper, examples of numerical evaluations of these operators of three different functions are shown in plots and the comparison of the numerical accuracy was analyzed in tables.

Modulating Functions Based Fast and Robust Estimation for a Class of Fractional Order Vibration Systems

2021 ◽  
Author(s):  
Zhi-Bo Wang ◽  
Da-Yan Liu ◽  
Driss Boutat ◽  
Yang Tian ◽  
Hao-Ran Liu
Keyword(s):  
Fractional Order ◽  
Original Problem ◽  
Initial Conditions ◽  
Main Idea ◽  
Fractional Integrals ◽  
Initial Values ◽  
Integral Formulas ◽  
Fractional Integrals And Derivatives ◽  
Modulating Functions ◽  
Derivatives Of

Abstract This paper aims to fast and robustly estimate the fractional integrals and derivatives of positions from noisy accelerations for a class of fractional order vibration systems defined by the Caputo fractional derivative. The main idea is to convert the original issue into the estimation of the fractional integrals of accelerations and the ones of the unknown initial conditions, on the basis of the additive index law. Being proper integrals, the fractional integrals of accelerations can be estimated via a numerical method. Consequently, solving the original problem boils down to estimating the unknown initial values. To this end, the modulating functions method is adopted. By constructing appropriate modulating functions, the unknown initial values are exactly given in terms of algebraic integral formulas in different situations. Finally, two illustrations are presented to verify the correctness and robustness of the proposed estimators.

scholarly journals Green’s theorem for generalized fractional derivatives

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Green's Theorem ◽  
Green’S Theorem ◽  
Generalized Fractional Derivatives

AbstractWe study three types of generalized partial fractional order operators. An extension of Green’s theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo.

scholarly journals THE FRACTIONAL CHAPMAN–KOLMOGOROV EQUATION

2007 ◽  
Vol 21 (04) ◽  
pp. 163-174 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fractional Order ◽  
Planck Equation ◽  
Kolmogorov Equation ◽  
Fractional Integrals ◽  
Fractal Media ◽  
Fokker Planck Equation ◽  
Fokker Planck

The Chapman–Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using fractional integrals, the fractional generalization of the Chapman–Kolmogorov equation is obtained. From the fractional Chapman–Kolmogorov equation, the Fokker–Planck equation is derived.

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