Fractional-Order Systems: Control, Modeling and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 31 August 2024 | Viewed by 7233

Special Issue Editors


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Guest Editor
School of Sciences, Yanshan University, Qinhuangdao 066004, China
Interests: fractional differential equations; numerical solution; fractional order model of viscoelastic materials

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Guest Editor
College of Science, North China University of Science and Technology, Tangshan 063000, China
Interests: artificial intelligence; network security; big data modeling; numerical calculation; green metallurgy; precision medicine
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Guest Editor
Department of Mathematics, Taiyuan Normal University, Jinzhong 030619, China
Interests: numerical algorithm of fractional calculus; fractional dynamics system

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Guest Editor
Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China
Interests: numerical methods for fractional calculus; fractional order system and control

Special Issue Information

Dear Colleagues,

The purpose of this journal is to promote the development of fractional calculus theory and its applications and to better display fractional calculus theory and cutting-edge achievements to researchers. Compared with integer order calculus, fractional calculus is more accurate for solving complex problems. With the development of science and technology and the deep exploration of fractional calculus, the applications of fractional calculus have drawn much attention and shown the increasingly important role of fractional calculus in various scientific fields. For example, considering the perspective of fractal theory and the combination with fractional order control theory, fractional order systems have been introduced into the scope of fractal elements. Therefore, this Special Issue focuses on the topics on numerical methods of fractional calculus, modeling of fractional viscoelastic material, fractional order dynamical systems, fractional order control, fractional order system identification, fractional order robots, and so on.

The topics of this Special Issue can cover nonlinear system theory, research of control methods using new analytical tools, and modeling and application of nonlinear dynamics problems with new methods. This Special Issue will show the important theoretical significance and practical values of fractional calculus.

Prof. Dr. Yiming Chen
Prof. Dr. Aimin Yang
Dr. Jiaquan Xie
Dr. Yanqiao Wei
Guest Editors

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Keywords

  • fractional differential equations
  • numerical methods
  • fractional order dynamical system
  • fractional order control
  • fractional viscoelastic material modeling
  • fractional order system identification
  • fractional order robot system

Published Papers (4 papers)

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Research

10 pages, 770 KiB  
Article
Advancing Fractional Riesz Derivatives through Dunkl Operators
by Fethi Bouzeffour
Mathematics 2023, 11(19), 4073; https://doi.org/10.3390/math11194073 - 25 Sep 2023
Viewed by 838
Abstract
The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known [...] Read more.
The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform. Full article
(This article belongs to the Special Issue Fractional-Order Systems: Control, Modeling and Applications)
13 pages, 283 KiB  
Article
Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative
by Songshu Liu
Mathematics 2022, 10(17), 3213; https://doi.org/10.3390/math10173213 - 5 Sep 2022
Viewed by 1312
Abstract
This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source [...] Read more.
This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source problem. In the theoretical results, we propose a priori and a posteriori regularization parameter choice rules and obtain the convergence estimates. Full article
(This article belongs to the Special Issue Fractional-Order Systems: Control, Modeling and Applications)
13 pages, 29770 KiB  
Article
Research on the Period-Doubling Bifurcation of Fractional-Order DCM Buck–Boost Converter Based on Predictor-Corrector Algorithm
by Lingling Xie, Jiahao Shi, Junyi Yao and Di Wan
Mathematics 2022, 10(12), 1993; https://doi.org/10.3390/math10121993 - 9 Jun 2022
Cited by 7 | Viewed by 1534
Abstract
DC–DC converters are widely used. They are a typical class of strongly nonlinear time-varying systems that show rich nonlinear phenomena under certain working conditions. Therefore, an in-depth study of their nonlinear phenomena, characteristics, and generation mechanism is of great practical significance for gaining [...] Read more.
DC–DC converters are widely used. They are a typical class of strongly nonlinear time-varying systems that show rich nonlinear phenomena under certain working conditions. Therefore, an in-depth study of their nonlinear phenomena, characteristics, and generation mechanism is of great practical significance for gaining a deep understanding of this kind of switching converter, revealing the essence of these nonlinear phenomena and then optimizing the design of this kind of converter. Based on the fact that most of the inductance and capacitance are fractional-order, the nonlinear dynamic characteristics of the fractional-order (FO) DCM buck–boost converter are researched in this paper. The main research work and achievements of this paper include: (1) using the predictor–corrector method of fractional calculus, which is not limited by fractional order and can directly calculate the accurate values of the inductance current and capacitor voltage of the fractional converter; the predictor–corrector model of the FO converter is established. (2) The bifurcation diagrams are obtained based on this model, and the period-doubling bifurcation and chaotic behavior of the FO buck–boost converter are analyzed. (3) The phase diagrams are obtained and verified to the point that period-doubling bifurcation occurs; then, some conclusions are drawn. The results show that under certain operating and parameters conditions, the FO buck–boost converter will appear as a bifurcation and chaotic nonlinear phenomenon. Under the condition of the same circuit parameters, the stability parameter domains of the integer-order buck–boost converter and the FO buck–boost converter are different. Compared with the integer-order converter, the parameter stability region of the FO buck–boost converter is bigger. The FO buck–boost converter is more accurate at describing the nonlinear dynamic characteristics. Furthermore, the predictor–corrector method can also be applied to other FO power converters and provides theoretical guidance for converter parameter optimization and controller design. Full article
(This article belongs to the Special Issue Fractional-Order Systems: Control, Modeling and Applications)
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19 pages, 5303 KiB  
Article
Aerodynamic Heating Ground Simulation of Hypersonic Vehicles Based on Model-Free Control Using Super Twisting Nonlinear Fractional Order Sliding Mode
by Xiaodong Lv, Guangming Zhang, Mingxiang Zhu, Zhihan Shi, Zhiqing Bai and Igor V. Alexandrov
Mathematics 2022, 10(10), 1664; https://doi.org/10.3390/math10101664 - 12 May 2022
Cited by 2 | Viewed by 1563
Abstract
In this article, a model-free control (MFC) using super twisting nonlinear fractional order sliding mode for aerodynamic heating ground simulation of hypersonic vehicles (AHGSHV) is proposed. Firstly, the mathematical model of AHGSHV is built up. To reduce order and simplify the dynamic model [...] Read more.
In this article, a model-free control (MFC) using super twisting nonlinear fractional order sliding mode for aerodynamic heating ground simulation of hypersonic vehicles (AHGSHV) is proposed. Firstly, the mathematical model of AHGSHV is built up. To reduce order and simplify the dynamic model of AHGSHV, an ultra-local model of MFC is taken into consideration. Then, time delay estimation can be used to estimate systematic uncertainties and external unknown disturbances. On the basis of the original fractional order sliding mode surface, the nonlinear function fal is introduced to design the nonlinear fractional order sliding mode surface, which can guarantee stability, increase convergence rate, and reduce static error and saturation error. In addition, the super twisting reaching law is used to improve the control performance of the reaching phase, resulting from the existence of sign function in the integral term, and it can effectively reduce the high-frequency chattering. Moreover, the Lyapunov function is used to prove the stability of the whole system. Finally, several numerical simulations show that the designed controller has more advantages than others. Full article
(This article belongs to the Special Issue Fractional-Order Systems: Control, Modeling and Applications)
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