scholarly journals Integer-and Fractional-Order Integral and Derivative Two-Port Summations: Practical Design Considerations

2019 ◽  
Vol 10 (1) ◽  
pp. 54 ◽  
Author(s):  
Roman Sotner ◽  
Ondrej Domansky ◽  
Jan Jerabek ◽  
Norbert Herencsar ◽  
Jiri Petrzela ◽  
...  
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Constant Phase Element ◽  
State Of The Art ◽  
Active Components ◽  
Integer Order ◽  
Order Part ◽  
Fractional Order Integral ◽  
Practical Recommendations ◽  
Good Agreement

This paper targets on the design and analysis of specific types of transfer functions obtained by the summing operation of integer-order and fractional-order two-port responses. Various operations provided by fractional-order, two-terminal devices have been studied recently. However, this topic needs to be further studied, and the topologies need to be analyzed in order to extend the state of the art. The studied topology utilizes the passive solution of a constant-phase element (with order equal to 0.5) implemented by parallel resistor–capacitor circuit (RC) sections operating as a fractional-order two-port. The integer-order part is implemented by operational amplifier-based lossless integrators and differentiators in branches with electronically adjustable gain, useful for time constant tuning. Four possible cases of the fractional-order and integer-order two-port interconnections are analyzed analytically, by PSpice simulations and also experimentally in the frequency range between 10 Hz and 1 MHz. Standard discrete active components are used in this design for laboratory verification. Practical recommendations for construction and also particular solutions overcoming possible issues with instability and DC offsets are also given. Experimental and simulated results are in good agreement with theory.

scholarly journals From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

Analogue Fractional-Order Generalized Memristive Devices

2009 ◽  
Author(s):  
Calvin Coopmans ◽  
Ivo Petra´sˇ ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Functional Relation ◽  
Theoretical Description ◽  
Practical Realization ◽  
Integer Order ◽  
Order Case ◽  
Memristive Systems ◽  
First Time ◽  
Fractional Order Integral ◽  
Missing Element

Memristor is a new electrical element which has been predicted and described in 1971 by Leon O. Chua and for the first time realized by HP laboratory in 2008. Chua proved that memristor behavior could not be duplicated by any circuit built using only the other three elements (resistor, capacitor, inductor), which is why the memristor is truly fundamental. Memristor is a contraction of memory resistor, because that is exactly its function: to remember its history. The memristor is a two-terminal device whose resistance depends on the magnitude and polarity of the voltage applied to it and the length of time that voltage has been applied. The missing element—the memristor, with memristance M—provides a functional relation between charge and flux, dφ = Mdq. In this paper, for the first time, the concept of (integer-order) memristive systems is generalized to non-integer order case using fractional calculus. We also show that the memory effect of such devices can be also used for an analogue implementation of the fractional-order operator, namely fractional-order integral and fractional-order derivatives. This kind of operators are useful for realization of the fractional-order controllers. We present theoretical description of such implementation and we proposed the practical realization and did some experiments as well.

A Class of Different Fractional-Order Chaotic (Hyperchaotic) Complex Duffing-Van Der Pol Models and Their Circuits Implementations

2021 ◽  
Author(s):  
Gamal Mahmoud ◽  
Tarek Abed-Elhameed ◽  
Motaz Elbadry
Keyword(s):  
Fixed Points ◽  
Lyapunov Exponents ◽  
Fractional Order ◽  
Transfer Functions ◽  
Viscoelastic Beam ◽  
Electronic Circuits ◽  
Van Der Pol ◽  
Shape Model ◽  
Predictor Corrector ◽  
Good Agreement

Abstract In this paper, we introduce three versions of fractional-order chaotic (or hyperchaotic) complex Duffing-van der Pol models. The dynamics of these models including their fixed points and their stability is investigated. Using the predictor-corrector method and Lyapunov exponents we calculate numerically the intervals of their parameters at which chaotic, hyperchaotic solutions and solutions that approach fixed points exist. These models appear in several applications in physics and engineering, e.g., viscoelastic beam and electronic circuits. The electronic circuits of these models with different fractional-order are proposed. We determine the approximate transfer functions for novel values of fractional-order and find the equivalent tree shape model (TSM). This TSM is used to build circuits simulations of our models}. A good agreement is found between both numerical and simulations results. Other circuits diagrams can be similarly designed for other fractional-order models.

scholarly journals Root Locus Practical Sketching Rules for Fractional-Order Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
António M. Lopes ◽  
J. A. Tenreiro Machado
Keyword(s):  
System Dynamics ◽  
Fractional Order ◽  
Transfer Functions ◽  
Important Case ◽  
Fractional Order Systems ◽  
Root Locus ◽  
Order Problem ◽  
Integer Order ◽  
The Subject ◽  
Zeros And Poles

For integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.

scholarly journals Realization of a fractional-order RLC circuit via constant phase element

2021 ◽  
Author(s):  
Riccardo Caponetto ◽  
Salvatore Graziani ◽  
Emanuele Murgano
Keyword(s):  
Experimental Data ◽  
Carbon Black ◽  
Fractional Order ◽  
Constant Phase Element ◽  
Polymeric Matrix ◽  
Fractional Order Systems ◽  
New Device ◽  
Rlc Circuit ◽  
Simulation Results

AbstractIn the paper, a fractional-order RLC circuit is presented. The circuit is realized by using a fractional-order capacitor. This is realized by using carbon black dispersed in a polymeric matrix. Simulation results are compared with the experimental data, confirming the suitability of applying this new device in the circuital implementation of fractional-order systems.

scholarly journals Invariant Image Representation Using Novel Fractional-Order Polar Harmonic Fourier Moments

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1544
Author(s):  
Chunpeng Wang ◽  
Hongling Gao ◽  
Meihong Yang ◽  
Jian Li ◽  
Bin Ma ◽  
...  
Keyword(s):  
Image Reconstruction ◽  
Fractional Order ◽  
Continuous Functions ◽  
Kernel Functions ◽  
Superior Performance ◽  
Image Description ◽  
Orthogonal Moments ◽  
Integer Order ◽  
Geometric Invariance ◽  
Order Continuous

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.

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.

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