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.

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.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

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 Fractional Order Models of Industrial Pneumatic Controllers

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Abolhassan Razminia ◽  
Dumitru Baleanu
Keyword(s):  
Process Control ◽  
Fractional Calculus ◽  
Numerical Simulations ◽  
Control Systems ◽  
Fractional Order ◽  
Dynamical Models ◽  
Industrial Automation ◽  
New Approach ◽  
Process Control Systems ◽  
Theoretical Results

This paper addresses a new approach for modeling of versatile controllers in industrial automation and process control systems such as pneumatic controllers. Some fractional order dynamical models are developed for pressure and pneumatic systems with bellows-nozzle-flapper configuration. In the light of fractional calculus, a fractional order derivative-derivative (FrDD) controller and integral-derivative (FrID) are remodeled. Numerical simulations illustrate the application of the obtained theoretical results in simple examples.

The Filtering Mechanism Modeling and Simulation of Passive Power Filter Based on Differentiator Circuit

2012 ◽  
Vol 430-432 ◽  
pp. 1593-1596
Author(s):  
Wan Neng Yu ◽  
Su Wen Li ◽  
Min Ying Zheng
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Integer Order ◽  
Filter Performance ◽  
Power Filter ◽  
Fractional Order Differential Equations ◽  
Passive Power ◽  
The Mathematical Model

Traditional continuous-time filters are of integer order which the power loss of passive power filter is general very much. However, using fractional calculus, filters may also be represented by the more general fractional-order differential equations. In this work, firstly, the passive elements were described with fractional-order differential equations depending on the introduction of fractional calculus application research. Secondly, the mathematical model of fractional-order filters was derived and discussed which includes high impedance at a certain frequency and low impedance at others, and the integer-order filters are only a tight subset of fractional-order filters that are testified. At last, the filter design idea to the fractional-order domain is developed and the better filter performance of the fractional-order passive power filter is validated by the mathematical model analysis and simulation results.

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 Fractional-order LCL filters: principle, frequency characteristics, and their analysis

2021 ◽  
Author(s):  
Junhua Xu ◽  
Xiaocong Li ◽  
Xueli Luo ◽  
Liliang Hou ◽  
Jianbo Qin ◽  
...  
Keyword(s):  
Fractional Order ◽  
Theoretical Basis ◽  
Digital Simulation ◽  
Frequency Characteristics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operating Characteristics ◽  
Theoretical Derivation ◽  
Integer Order ◽  
Necessary And Sufficient

Abstract In this paper, a fractional-order LCL (FOLCL) filter is constructed by introducing fractional-order inductors (FOIs) and fractional-order capacitors (FOCs) to replace the inductors and capacitors in a traditional integer-order LCL (IOLCL) filter, respectively. The principle and frequency characteristics of an FOLCL filter are systematically studied, and five important properties are derived and demonstrated in-depth. One of the most important achievements is that we discover the necessary and sufficient condition for the existence of resonance for an FOLCL filter, that is, the sum of the order of the FOIs and the FOC is equal to 2, which provides a theoretical basis for avoiding the resonance of an FOLCL filter effectively in design. The correctness of the theoretical derivation and analysis are verified by digital simulation. Compared with an IOLCL filter, an FOLCL filter presents more flexible and diverse operating characteristics and has a broader application prospect.

The Frequency Research of Fractional Order PIλDμ Controller

2013 ◽  
Vol 278-280 ◽  
pp. 1521-1524
Author(s):  
Hai Qun Wang ◽  
Ling Meng
Keyword(s):  
Complex Systems ◽  
Frequency Domain ◽  
Control Systems ◽  
Fractional Order ◽  
Industrial Production ◽  
Pid Controller ◽  
Time Analysis ◽  
Integer Order ◽  
Integral Order

Since most of the control systems in real industrial production are fractional，there is necessery to propose fractional order PIλDμ controller, which extend the traditional integer-order PID controller to fractional order, it has increased two free degree: Integral-order λ and differential-order μ, to more accurately control those complex systems. At the same time analysis the performance of fractional order PIλDμ controller in frequency domain. Especially,the two degrees’ effect to controller.

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