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Fractional Order Systems and Their Applications

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (23 January 2022) | Viewed by 29420

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Special Issue Editors

Dr. António Lopes

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Guest Editor

Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal
Interests: complex systems modelling; automation and robotics; fractional order systems modelling and control; data analysis and visualization
Special Issues, Collections and Topics in MDPI journals

Dr. Liping Chen

E-Mail Website
Guest Editor

School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China
Interests: fractional calculus; control theory and engineering; nonlinear dynamics; neural networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus (FC) generalizes the concepts of derivative and integral to non-integer orders. It was introduced by Leibniz (1646–1716), but remained a purely mathematical exercise for a long time, despite the original contributions of important mathematicians, physicists, and engineers. FC experienced rapid development during the last few decades both in mathematics and applied sciences, being recognized as an excellent tool to describe complex dynamics. From this perspective, several models governing physical phenomena in the area of science and engineering have been reformulated in light of FC for better reflecting their non-local, frequency- and history-dependent properties. Applications of FC include modeling of diffusion, viscoelasticity, and relaxation processes in fluid mechanics, dynamics of mechanical, electronic and biological systems, signal processing, control, and others.

The Special Issue focuses on original and new research results on fractional order theory and applications. Manuscripts addressing novel theoretical issues, as well as those on more specific applications, are welcome.

Contributions should fit the scope of the journal Fractal and Fractional and topics of interest include (but are not limited to):

Complex dynamics
Fractional calculus theory
Numerical methods
Fractals
Chaos
Fractional order control
Systems identification
Nonlinear dynamical systems
Entropy and information theory
Advanced control systems
Finance and economy dynamics
Biological systems and bioinformatics
Nonlinear waves and acoustics
Image and signal processing
Astronomy and cosmology

Prof. Dr. António M. Lopes
Dr. Liping Chen
Guest Editors

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. Fractal and Fractional is an international peer-reviewed open access monthly 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 2700 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

fractals
fractional calculus
complex systems
chaos
modeling
identification
control

Published Papers (14 papers)

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Editorial

Jump to: Research

3 pages, 209 KiB

Open AccessEditorial

Fractional Order Systems and Their Applications

by António M. Lopes and Liping Chen

Fractal Fract. 2022, 6(7), 389; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6070389 - 13 Jul 2022

Cited by 5 | Viewed by 1213

Abstract

Fractional calculus (FC) generalizes the concepts of derivative and integral to non-integer orders [...] Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

Research

Jump to: Editorial

17 pages, 481 KiB

Open AccessArticle

Asymptotic Stabilization of Delayed Linear Fractional-Order Systems Subject to State and Control Constraints

by Xindong Si, Zhen Wang, Zhibao Song and Ziye Zhang

Fractal Fract. 2022, 6(2), 67; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6020067 - 27 Jan 2022

Cited by 7 | Viewed by 1900

Abstract

Studies have shown that fractional calculus can describe and characterize a practical system satisfactorily. Therefore, the stabilization of fractional-order systems is of great significance. The asymptotic stabilization problem of delayed linear fractional-order systems (DLFS) subject to state and control constraints is studied in this article. Firstly, the existence conditions for feedback controllers of DLFS subject to both state and control constraints are given. Furthermore, a sufficient condition for invariance of polyhedron set is established by using invariant set theory. A new Lyapunov function is constructed on the basis of the constraints, and some sufficient conditions for the asymptotic stability of DLFS are obtained. Then, the feedback controller and the corresponding solution algorithms are given to ensure the asymptotic stability under state and control input constraints. The proposed solution algorithm transforms the asymptotic stabilization problem into a linear/nonlinear programming (LP/NP) problem which is easy to solve from the perspective of computation. Finally, three numerical examples are offered to illustrate the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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27 pages, 25650 KiB

Open AccessArticle

Chaos Control for a Fractional-Order Jerk System via Time Delay Feedback Controller and Mixed Controller

by Changjin Xu, Maoxin Liao, Peiluan Li, Lingyun Yao, Qiwen Qin and Youlin Shang

Fractal Fract. 2021, 5(4), 257; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040257 - 05 Dec 2021

Cited by 32 | Viewed by 2475

Abstract

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PD^σ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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20 pages, 362 KiB

Open AccessArticle

A Study of Coupled Systems of ψ-Hilfer Type Sequential Fractional Differential Equations with Integro-Multipoint Boundary Conditions

by Ayub Samadi, Cholticha Nuchpong, Sotiris K. Ntouyas and Jessada Tariboon

Fractal Fract. 2021, 5(4), 162; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040162 - 10 Oct 2021

Cited by 6 | Viewed by 1196

Abstract

In this paper, the existence and uniqueness of solutions for a coupled system of ψ-Hilfer type sequential fractional differential equations supplemented with nonlocal integro-multi-point boundary conditions is investigated. The presented results are obtained via the classical Banach and Krasnosel’skiĭ’s fixed point theorems [...] Read more.

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

21 pages, 1175 KiB

Open AccessArticle

Guaranteed Cost Leaderless Consensus Protocol Design for Fractional-Order Uncertain Multi-Agent Systems with State and Input Delays

by Yingming Tian, Qin Xia, Yi Chai, Liping Chen, António M. Lopes and YangQuan Chen

Fractal Fract. 2021, 5(4), 141; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040141 - 28 Sep 2021

Cited by 6 | Viewed by 1460

Abstract

This paper addresses the guaranteed cost leaderless consensus of delayed fractional-order (FO) multi-agent systems (FOMASs) with nonlinearities and uncertainties. A guaranteed cost function for FOMAS is proposed to simultaneously consider consensus performance and energy consumption. By employing the linear matrix inequality approach and the FO Razumikhin theorem, a delay-dependent and order-dependent consensus protocol is formulated for FOMASs with input delay. The proposed protocol not only guarantees the robust stability of the closed-loop system error but also ensures that the performance degradation caused by the system uncertainty is lesser than that obtained with other approaches. Two numerical examples are provided in order to verify the effectiveness and accuracy of the proposed protocol. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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17 pages, 4353 KiB

Open AccessArticle

Dynamics of Fractional-Order Digital Manufacturing Supply Chain System and Its Control and Synchronization

by Yingjin He, Song Zheng and Liguo Yuan

Fractal Fract. 2021, 5(3), 128; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030128 - 17 Sep 2021

Cited by 8 | Viewed by 1726

Abstract

Digital manufacturing is widely used in the production of automobiles and aircrafts, and plays a profound role in the whole supply chain. Due to the long memory property of demand, production, and stocks, a fractional-order digital manufacturing supply chain system can describe their dynamics more precisely. In addition, their control and synchronization may have potential applications in the management of real-word supply chain systems to control uncertainties that occur within it. In this paper, a fractional-order digital manufacturing supply chain system is proposed and solved by the Adomian decomposition method (ADM). Dynamical characteristics of this system are studied by using a phase portrait, bifurcation diagram, and a maximum Lyapunov exponent diagram. The complexity of the system is also investigated by means of SE complexity and

C_{0}

complexity. It is shown that the complexity results are consistent with the bifurcation diagrams, indicating that the complexity can reflect the dynamical properties of the system. Meanwhile, the importance of the fractional-order derivative in the modeling of the system is shown. Moreover, to further investigate the dynamics of the fractional-order supply chain system, we design the feedback controllers to control the chaotic supply chain system and synchronize two supply chain systems, respectively. Numerical simulations illustrate the effectiveness and applicability of the proposed methods. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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20 pages, 1283 KiB

Open AccessArticle

Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations

by Mohamed A. Abdelkawy, Ahmed Z. M. Amin, Mohammed M. Babatin, Abeer S. Alnahdi, Mahmoud A. Zaky and Ramy M. Hafez

Fractal Fract. 2021, 5(3), 115; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030115 - 09 Sep 2021

Cited by 5 | Viewed by 1622

Abstract

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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15 pages, 830 KiB

Open AccessArticle

Controllability for Fuzzy Fractional Evolution Equations in Credibility Space

by Azmat Ullah Khan Niazi, Naveed Iqbal, Rasool Shah, Fongchan Wannalookkhee and Kamsing Nonlaopon

Fractal Fract. 2021, 5(3), 112; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030112 - 08 Sep 2021

Cited by 26 | Viewed by 1614

Abstract

This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order [...] Read more.

β \in (1, 2)

_{0}^{C} D_{t}^{β} u (t, ζ) = A u (t, ζ) + f (t, u (t, ζ)) d C_{t} + B x (t) C x (t) d t

with initial conditions

u (0) = u_{0}, u^{'} (0) = u_{1},

where

u (t, ζ)

takes values from

U (\subset E_{N}), V (\subset E_{N})

is the other bounded space, and

E_{N}

represents the set of all upper semi-continuously convex fuzzy numbers on

R

. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

27 pages, 6863 KiB

Open AccessArticle

Control and Robust Stabilization at Unstable Equilibrium by Fractional Controller for Magnetic Levitation Systems

by Banu Ataşlar-Ayyıldız, Oğuzhan Karahan and Serhat Yılmaz

Fractal Fract. 2021, 5(3), 101; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030101 - 20 Aug 2021

Cited by 11 | Viewed by 2476

Abstract

The problem of control and stabilizing inherently non-linear and unstable magnetic levitation (Maglev) systems with uncertain equilibrium states has been studied. Accordingly, some significant works related to different control approaches have been highlighted to provide robust control and enhance the performance of the Maglev system. This work examines a method to control and stabilize the levitation system in the presence of disturbance and parameter variations to minimize the magnet gap deviation from the equilibrium position. To fulfill the stabilization and disturbance rejection for this non-linear dynamic system, the fractional order PID, fractional order sliding mode, and fractional order Fuzzy control approaches are conducted. In order to design the suitable control outlines based on fractional order controllers, a tuning hybrid method of GWO–PSO algorithms is applied by using the different performance criteria as Integrated Absolute Error (IAE), Integrated Time Weighted Absolute Error (ITAE), Integrated Squared Error (ISE), and Integrated Time Weighted Squared Error (ITSE). In general, these objectives are used by targeting the best tuning of specified control parameters. Finally, the simulation results are presented to determine which fractional controllers demonstrate better control performance, achieve fast and robust stability of the closed-loop system, and provide excellent disturbance suppression effect under nonlinear and uncertainty existing in the processing system. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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17 pages, 2851 KiB

Open AccessArticle

State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter

by Liping Chen, Yu Chen, António M. Lopes, Huifang Kong and Ranchao Wu

Fractal Fract. 2021, 5(3), 91; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030091 - 08 Aug 2021

Cited by 17 | Viewed by 2129

Abstract

The covariance matrix of measurement noise is fixed in the Kalman filter algorithm. However, in the process of battery operation, the measurement noise is affected by different charging and discharging conditions and the external environment. Consequently, obtaining the noise statistical characteristics is difficult, which affects the accuracy of the Kalman filter algorithm. In order to improve the estimation accuracy of the state of charge (SOC) of lithium-ion batteries under actual working conditions, a fuzzy fractional-order unscented Kalman filter (FFUKF) is proposed. The algorithm combines fuzzy inference with fractional-order unscented Kalman filter (FUKF) to infer the measurement noise in real time and take advantage of fractional calculus in describing the dynamic behavior of the lithium batteries. The accuracy of the SOC estimation under different working conditions at three different temperatures is verified. The results show that the accuracy of the proposed algorithm is superior to those of the FUKF and extended Kalman filter (EKF) algorithms. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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24 pages, 1207 KiB

Open AccessArticle

A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect

by Emli Rahmi, Isnani Darti, Agus Suryanto and Trisilowati

Fractal Fract. 2021, 5(3), 84; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030084 - 01 Aug 2021

Cited by 15 | Viewed by 2260

Abstract

In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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13 pages, 322 KiB

Open AccessArticle

Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

by Hari Mohan Srivastava, Artion Kashuri, Pshtiwan Othman Mohammed, Dumitru Baleanu and Y. S. Hamed

Fractal Fract. 2021, 5(3), 80; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030080 - 29 Jul 2021

Cited by 7 | Viewed by 1864

Abstract

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

20 pages, 402 KiB

Open AccessArticle

Existence, Uniqueness, and E_q–Ulam-Type Stability of Fuzzy Fractional Differential Equation

by Azmat Ullah Khan Niazi, Jiawei He, Ramsha Shafqat and Bilal Ahmed

Fractal Fract. 2021, 5(3), 66; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030066 - 11 Jul 2021

Cited by 19 | Viewed by 1801

Abstract

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order

q \in (1, 2]

, [...] Read more.

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order

q \in (1, 2]

_{0}^{c} D_{0^{+}}^{q} u (t) = λ u (t) \oplus f (t, u (t)) \oplus B (t) C (t), t \in [0, T]

with initial conditions

u (0) = u_{0}, u^{'} (0) = u_{1} .

Moreover, by using direct analytic methods, the

E_{q}

–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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10 pages, 1473 KiB

Open AccessArticle

Fractals Parrondo’s Paradox in Alternated Superior Complex System

by Yi Zhang and Da Wang

Fractal Fract. 2021, 5(2), 39; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5020039 - 28 Apr 2021

Cited by 2 | Viewed by 2091

Abstract

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system [...] Read more.

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system

z_{n + 1} = β (z_{n}^{2} + c_{i}) + (1 - β) z_{n}, i = 1, 2

. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria. Full article

(This article belongs to the Special Issue Fractional Order Systems and Their Applications)

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