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Mathematics
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Fractional Order Systems
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Fractional Order Systems

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 April 2019) | Viewed by 27776

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Ivo Petráš

E-Mail Website
Guest Editor
Faculty of Mining, Ecology, Process Control and Geotechnologies (FBERG), Technical University of Kosice (TUKE), Kosice, Slovakia
Interests: fractional calculus and its applications; dynamical systems; chaos theory; control theory; mathematical modelling; simulations; process control; automation; signal processing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is well known that fractional calculus is recognized since the regular calculus with the first written reference dated in September 1695 in letter from Leibniz to L’Hospital. Nowadays, the fractional calculus has a wide area of applications, for example, physics, chemistry, bioengineering, chaos theory, control systems engineering, and many others. In all those applications we deal, in general, with fractional order systems. Moreover, the fractional calculus plays an important role even in the complex systems and therefore allows us to use a better description of some real-world phenomena. Based on this fact, the fractional order systems are ubiquitous as well as whole real world around us is fractional, not integer one. Due to this reason it is so urgent consider almost all systems as the fractional order systems. 

This Special Issue is focused on the theory and multidisciplinary applications of fractional order systems in science and engineering, and will accept only high-quality survey, and/or original research papers.

Prof. Dr. Ivo Petráš
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fractional calculus and its applications
  • Fractional differential equations
  • Fractional order dynamical systems
  • Fractional order control
  • Fractional nonlinear and chaotic systems

Published Papers (7 papers)

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Research

19 pages, 1251 KiB  
Article
Optimal Randomness in Swarm-Based Search
by Jiamin Wei, YangQuan Chen, Yongguang Yu and Yuquan Chen
Mathematics 2019, 7(9), 828; https://doi.org/10.3390/math7090828 - 06 Sep 2019
Cited by 29 | Viewed by 3064
Abstract
Lévy flights is a random walk where the step-lengths have a probability distribution that is heavy-tailed. It has been shown that Lévy flights can maximize the efficiency of resource searching in uncertain environments and also the movements of many foragers and wandering animals [...] Read more.
Lévy flights is a random walk where the step-lengths have a probability distribution that is heavy-tailed. It has been shown that Lévy flights can maximize the efficiency of resource searching in uncertain environments and also the movements of many foragers and wandering animals have been shown to follow a Lévy distribution. The reason mainly comes from the fact that the Lévy distribution has an infinite second moment and hence is more likely to generate an offspring that is farther away from its parent. However, the investigation into the efficiency of other different heavy-tailed probability distributions in swarm-based searches is still insufficient up to now. For swarm-based search algorithms, randomness plays a significant role in both exploration and exploitation or diversification and intensification. Therefore, it is necessary to discuss the optimal randomness in swarm-based search algorithms. In this study, cuckoo search (CS) is taken as a representative method of swarm-based optimization algorithms, and the results can be generalized to other swarm-based search algorithms. In this paper, four different types of commonly used heavy-tailed distributions, including Mittag-Leffler distribution, Pareto distribution, Cauchy distribution, and Weibull distribution, are considered to enhance the searching ability of CS. Then four novel CS algorithms are proposed and experiments are carried out on 20 benchmark functions to compare their searching performances. Finally, the proposed methods are used to system identification to demonstrate the effectiveness. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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13 pages, 649 KiB  
Article
Audio Signal Processing Using Fractional Linear Prediction
by Tomas Skovranek and Vladimir Despotovic
Mathematics 2019, 7(7), 580; https://doi.org/10.3390/math7070580 - 29 Jun 2019
Cited by 3 | Viewed by 3272
Abstract
Fractional linear prediction (FLP), as a generalization of conventional linear prediction (LP), was recently successfully applied in different fields of research and engineering, such as biomedical signal processing, speech modeling and image processing. The FLP model has a similar design as the conventional [...] Read more.
Fractional linear prediction (FLP), as a generalization of conventional linear prediction (LP), was recently successfully applied in different fields of research and engineering, such as biomedical signal processing, speech modeling and image processing. The FLP model has a similar design as the conventional LP model, i.e., it uses a linear combination of “fractional terms” with different orders of fractional derivative. Assuming only one “fractional term” and using limited number of previous samples for prediction, FLP model with “restricted memory” is presented in this paper and the closed-form expressions for calculation of FLP coefficients are derived. This FLP model is fully comparable with the widely used low-order LP, as it uses the same number of previous samples, but less predictor coefficients, making it more efficient. Two different datasets, MIDI Aligned Piano Sounds (MAPS) and Orchset, were used for the experiments. Triads representing the chords composed of three randomly chosen notes and usual Western musical chords (both of them from MAPS dataset) served as the test signals, while the piano recordings from MAPS dataset and orchestra recordings from the Orchset dataset served as the musical signal. The results show enhancement of FLP over LP in terms of model complexity, whereas the performance is comparable. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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16 pages, 1504 KiB  
Article
Back to Basics: Meaning of the Parameters of Fractional Order PID Controllers
by Inés Tejado, Blas M. Vinagre, José Emilio Traver, Javier Prieto-Arranz and Cristina Nuevo-Gallardo
Mathematics 2019, 7(6), 530; https://doi.org/10.3390/math7060530 - 11 Jun 2019
Cited by 49 | Viewed by 5544
Abstract
The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the [...] Read more.
The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the actual (proportional), the accumulated (integral), and the predicted (derivative) values; the three gains depend on the magnitude of the error, the time required to eliminate the accumulated error, and the prediction horizon of the error. This paper explores the new meaning of integral and derivative actions, and gains, derived by the consideration of non-integer integration and differentiation orders, i.e., for fractional order PID controllers. The integral term responds with selective memory to the error because of its non-integer order λ , and corresponds to the area of the projection of the error curve onto a plane (it is not the classical area under the error curve). Moreover, for a fractional proportional-integral (PI) controller scheme with automatic reset, both the velocity and the shape of reset can be modified with λ . For its part, the derivative action refers to the predicted future values of the error, but based on different prediction horizons (actually, linear and non-linear extrapolations) depending on the value of the differentiation order, μ . Likewise, in case of a proportional-derivative (PD) structure with a noise filter, the value of μ allows different filtering effects on the error signal to be attained. Similarities and differences between classical and fractional PIDs, as well as illustrative control examples, are given for a best understanding of new possibilities of control with the latter. Examples are given for illustration purposes. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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9 pages, 494 KiB  
Article
Fractional Calculus as a Simple Tool for Modeling and Analysis of Long Memory Process in Industry
by Ivo Petráš and Ján Terpák
Mathematics 2019, 7(6), 511; https://doi.org/10.3390/math7060511 - 04 Jun 2019
Cited by 28 | Viewed by 4280
Abstract
This paper deals with the application of the fractional calculus as a tool for mathematical modeling and analysis of real processes, so called fractional-order processes. It is well-known that most real industrial processes are fractional-order ones. The main purpose of the article is [...] Read more.
This paper deals with the application of the fractional calculus as a tool for mathematical modeling and analysis of real processes, so called fractional-order processes. It is well-known that most real industrial processes are fractional-order ones. The main purpose of the article is to demonstrate a simple and effective method for the treatment of the output of fractional processes in the form of time series. The proposed method is based on fractional-order differentiation/integration using the Grünwald–Letnikov definition of the fractional-order operators. With this simple approach, we observe important properties in the time series and make decisions in real process control. Finally, an illustrative example for a real data set from a steelmaking process is presented. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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11 pages, 11340 KiB  
Article
Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact
by Bohdan Datsko, Igor Podlubny and Yuriy Povstenko
Mathematics 2019, 7(5), 433; https://doi.org/10.3390/math7050433 - 16 May 2019
Cited by 15 | Viewed by 4465
Abstract
The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the [...] Read more.
The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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17 pages, 617 KiB  
Article
Adaptive Pinning Synchronization of Fractional Complex Networks with Impulses and Reaction–Diffusion Terms
by Xudong Hai, Guojian Ren, Yongguang Yu and Conghui Xu
Mathematics 2019, 7(5), 405; https://doi.org/10.3390/math7050405 - 07 May 2019
Cited by 7 | Viewed by 2180
Abstract
In this paper, a class of fractional complex networks with impulses and reaction–diffusion terms is introduced and studied. Meanwhile, a class of more general network structures is considered, which consists of an instant communication topology and a delayed communication topology. Based on the [...] Read more.
In this paper, a class of fractional complex networks with impulses and reaction–diffusion terms is introduced and studied. Meanwhile, a class of more general network structures is considered, which consists of an instant communication topology and a delayed communication topology. Based on the Lyapunov method and linear matrix inequality techniques, some sufficient criteria are obtained, ensuring adaptive pinning synchronization of the network under a designed adaptive control strategy. In addition, a pinning scheme is proposed, which shows that the nodes with delayed communication are good candidates for applying controllers. Finally, a numerical example is given to verify the validity of the main results. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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16 pages, 2953 KiB  
Article
Fractional Order Complexity Model of the Diffusion Signal Decay in MRI
by Richard L. Magin, Hamid Karani, Shuhong Wang and Yingjie Liang
Mathematics 2019, 7(4), 348; https://doi.org/10.3390/math7040348 - 12 Apr 2019
Cited by 23 | Viewed by 4087
Abstract
Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can [...] Read more.
Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can be encoded in the attenuation of the observed MRI signal through the fractional order of the time- and space-derivatives. Specifically, in solving the Bloch-Torrey equation, fractional order imaging biomarkers are identified that connect the continuous time random walk model of Brownian motion to the structure and composition of cells, cell membranes, proteins, and lipids. In this way, the decay of the induced magnetization is influenced by the micro- and meso-structure of tissues, such as the white and gray matter of the brain or the cortex and medulla of the kidney. Fractional calculus provides new functions (Mittag-Leffler and Kilbas-Saigo) that characterize tissue in a concise way. In this paper, we describe the exponential, stretched exponential, and fractional order models that have been proposed and applied in MRI, examine the connection between the model parameters and the underlying tissue structure, and explore the potential for using diffusion-weighted MRI to extract biomarkers associated with normal growth, aging, and the onset of disease. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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